This book originated in the idea that open problems act as crystallization points in mathematical research. Mathematical
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Table of contents :
Introduction
Contents
Chapter 1. Is a Finite Locally Free Group Scheme Killed by Its Order?
1 Introduction
2 Finite group schen1es over fields
3 Commutative group schemes
4 Finite group schemes over Artin rings
References
Chapter 2. Lifting of Curves with Automorphisms
1 Introduction
1.1 Riemai1n's existence theorem
1.2 Lifting of covers of curves
1.3 A remark on this exposition
1.4 Notation and conventions
2 Global results
2.1 Tame covers
2.2 Wild covers
2.3 Oort groups and the Oo1't conjecture
3 The localglobal principle
3.1 Preliminaries arid etale lifting
3.2 A patching result
3.3 Proof of Theorem 3.1
4 The local lifting problein and its obstructions
4.1 The (local) KGB obstruction
4.2 The ( differential) Hurwitz tree obstruction
5 Surnniary of present local lifting problem results
5.1 Local Oort groups
5.2 Weak local Oort groups
6 Lifting techniques and examples
6.1 Birational lifts and the different criterion
6.2 Explicit lifts
6.3 SekiguchiSuwa theory
6.4 Hurwitz trees
6.5 Successive approximation
6.6 The ''Mumford method''
7 Approach using deforination theory
7.1 Setup
7.2 The localglobal principle via deformation theory
7.3 Examples of local miniversal deformation rings
8 Open probleins
8.1 Existence of local lifts
8.2 Rings of definition
8.3 Moduli/deformations/geometry of local lifts
8.4 Nonalgebraically closed residue fields
A Some algebraic preliminaries
A.1 Homological Algebra
A.2 Complete local rings
A.3 Ramification theory
References
Chapter 3. The AndreOort Conjecture
Introduction
1 The ManinMuinford conjecture
2 The AndreOort conjecture
3 Special subvarieties are linear in SerreTate canonical coordinates
4 The AndreOort conjecture: recent results
5 Generalizations: Unlikely intersections
6 Open problems and questions
References
Chapter 4. Special Subvarieties in the Torelli Locus
1 Introduction
2 An expectation
3 Known counterexatnples
4 Weyl CM fields
5 Positive characteristic
6 Jacobians in Inixed characteristic
7 Appendix: CM abelian varieties
8 Appendix: special subvarieties
9 Appendix: notations
10 Open probleins and questions
References
Chapter 5. Moduli of Abelian Varieties
0 Introduction
1 pdivisible groups
2 Stratifications and foliations
3 Newton polygon strata
4 Foliations
5 The EO stratification
6 Irreducibility
7 CM lifting
8 Generalized SerreTate coordinates
9 Some historical remarks
10 Sonie Questions
References
Chapter 6. Current Results on Newton Polygons of Curves
1 Introduction
2 Supersingular curves
3 Newton polygons and EkedahlOort types of curves
4 Notation and background
5 Stratifications of moduli spaces
6 An inductive result about Newton polygon and EkedahlOort strata
7 Open problems
References
Chapter 7. Sustained pdivisible Groups: A Foliation Retraced
1 What is a sustained pdivisible group
2 Stabilized Horn schemes for truncations of pdivisible groups
3 Descent of sustained pdivisible groups, torsors for stabilized Isom schemes and the slope filtration
4 Pointwise criterion for sustained pdivisible groups
5 Deforniation of sustained pdivisible groups and local structure of central leaves
References
Chapter 8. The Hecke Orbit Conjecture: A Survey and Outlook
1 Introduction
2 Hecke syrnnietry on Inodular varieties of PEL type
3 Sustained pdivisible groups and central leaves
4 Local structure of leaves
5 Action of the local stabilizer subgroup
6 Local rigidity for subvarieties of leaves
7 Monodroniy of Hecke invariant subvarieties
8 The Hecke orbit conjecture for A_g
9 Open questions
References
Appendix 1. Some Questions in Algebraic Geometry
References
Appendix 2. Automorphisms of Curves2005 Collection
References
Appendix 3. Questions in Arithmetic Algebraic Geometry
References
An afterthought: When do we use the word conjecture?
Advanced Lectures in Mathematics
I Editor: Frans Oort
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Advanced Lectures in Mathematics
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Editor: Frans Oort
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International Press
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G
cet
~ 0,
c
where 0 denotes the connected component of G and cet its largest etale quotient. By 01·dinary group theory, cet is a1111ihilated by its order. Since finite group schemes in characteristic zero are etale) we may assume that the characteristic of the 1·esidt1e field of R is p > 0, and that G is a local group scheme, as required. If the maximal ideal m of the Artin ring R in the seco11d question is zero, R is a field and the answer is affirrr1ative by the SGA 3 r·esult 111entioned above. In [4] it is shown that if mP ==pm== 0, then every finite free g1·oup scheme G over R is also killed by its 01~der. This happens in pa1·ticular when the maxirnal ideal m satisfies m2 = 0. In tl1is note we give proofs of the two 1nain results mentio11ecl above. In Secti 0 and t,l1at G is local. By [6, 14.4], tl1e 01·der of G is eqt1al to prrz for some 111. ~ 0 and tl1e Hopf algebra A of G is a local Artin kalgebra of dimension prn. Tl1erefore the augn1entatio11 ideal I of A satisfies JP = 0. P1·oposiiion 2.1 then implies tha.t [p''n] (I) = 0. This means that tl1e n1orpl1ism [1Jr11 ] : A A factors t.h1·011gh A/ I = A~ so tl1at G is killed by its orde1~ pm~ as rec1uired. 1
"
3
Commutative group schemes
Let G be a. finite locctlly free comrri1ttative g1·oup schen1e of 01·de1· ni over a base X. 111 1969 Deligne showed tl1e fc)llowir1g (3, p.4].
Theorem 3.1. Tlie gr·oup scl,errte G is a·1i11,ih.1i.lu.ted b~lJ m.. A11y elen1e11t x of an orclinary finite g1·oup of orcler ·n1,, has tl1e p1·opert)' tl1at x 711· is equal to tl1e net1t1·al ele1nc~r1t. For corrirn ula,ti11e grolJf)S tl1is ca11 be proved by tl1e followi11g wcllk110,v11 ,11·gt1111e11t: let P == Tix:,; b the procl11ct of c:1.ll eleme11ts of tl1e grotlp and let be an arbitrarjr elen1e11t. Tl1e11 p == X == Tixxy == yrrt ITxX == y'n P a.r1 A' being given by the dual of the comultiplication map c : A > A ®RA. Moreover A' is the Hopf algeb1·a of the Cartier dual of G, with comultiplication map c' : A' ® RA' A' equal to the dual of the multiplication map m : A © R A ~ A. This easily implies that for a11y Ralgebra S we have G(S) = {f E (A' ®RS)* : c'(f) = f ® f}.
One easily checks that the group operation of G(S) coincides with the algeb1·a multiplication in the multiplicative group (A' ®R S)* of the algebra A' © R S. See [6, 2.4]. For an Ralgebra S, the structure morphism R S gives rise to a group homomorphism G(R) G(S). When S is finite and free over R, Deligne constructs a Trace map G(S) G(R) in the other direction. To do this he uses the Norm map N : S   R, wl1ich for s E S is defined as the determinant of any representative matrix of the Rlinear multiplicationbysmap S > S. The norm is multiplicative. It induces for all Ralgebras B, norm maps Ta = idB ® N from B ®R S to B. These are functorial in the sense that for every morphism f : B   C of Ralgebras the diagran1
f
commutes. Lemma 3.2. Let S be a finite free Ralgebra and let G = Spec(A) as above. Then the norm map NA' : A' 0 R S ~ A' maps G(S) to G(R) and is a group homomorphism. G(S) C A' ® RS
l
G(R)
Proof. Suppose that a E G(S) c1 (a) =a ® a. Then we have
c
NA,
A'
C A' ® R S.
So, it is invertible and satisfies
This follows easily from. the comn1utativity of tl1e diagra1ns (*) applied to the morphisms A' A' ®RA' given by the maps a 1+ a ® 1', c1, H 1' ® a and c'(a) respectively. Here 1' denotes the unit element of the algebra A'. It is tl1e counit map eA: A   R.
Cl1apte1· 1
Is a Finite Locally Free Group Scheme Killed by Its Order?
5
Tl1e forrnula shows that NA,(a) E G(R). Si11ce tl1e group laws in G(R) and G(S) agree with algebra multiplication in A' and A' ®RS respectively, and since the norm is multiplicative, we see that N : G(S) G(R) is a group homomorphism. This proves the le1nma. Proof of T .h eorem 3.1. Let m denote the order of G. In other words, mis the Rrank of A. '\Ve must sho"r that for every Ralgebra S and any u E G(S) the rrith power of u is equal to the neutral element in G(S). Replacing R by S, we see that 1t suffices to sl1ow this for all u E G(R). Translation by u is an invertible morphism G > G and therefore induces an Rautomo1·phis1n CT of the Ralgebra A and hence an A'automorphism, id®u of A 1 ® A. On the other hand, translation by u E G(R) C G(A) agrees with multiplication , by ·it i11 tl1e algel)ra A' ® A. Applyi11g tlris to the algebra l1omon1orphism idA in G(A), we find tha.t (id® u)(idA) = 'U · idA. Since applying a to elernents of A does not affect tl1ei1· norm to R, applying id ® u does not affect NA'. Therefore we have
Si11ce NA' (idA) is invertible in ~4', it follows that um is equal to the unit element 1 1 of A', as required.
Ren1ark . 111 this proof, tl1e elemer1t NA, (id A) of A' plays the role of the product P of all elen1e11ts of a finite group. It is well known that P is 11ot always equal to the neutral element, but its squa.re is. Similarly1 in Deligne's proof the norm NA,(idA) is in general not equal to the unit element 1' EA', but its square is. Indeed, since the coinverse morphism iA : A > A is the inverse of idA in the gro11p G(A) the same is t1,11e in the multiplica.t ive group (A' ®RA)*. It follows tl1at NA' (·iA · idA)
= 1'.
On the otl1er bancl, iA is an R .a.11tomorphism of A so that idA' ® iA is a11 A'aut.omorphism of A' ®n A == IIo1nR(A,A). It carries idA to i,4. Since A'au to1norphisn1s do not cl1ange NA', we get
It, follO\\'S that N.4' (idA) 2 = 1' as req11ired.
4
Finite group schemes over Artin rings
In tl1is sectio11 we 011tlir1e the proof given in [4] of the following res11lt.
Theorem 4.1. Let R be an Artin ring with maximal ideal m and residue field of chci1·act eri.t;tic p > 0: with t/1,e prope·rty that m11 ==pm= 0.
Rene Schoof
6
Then every finite free local group scheme G ove'r R is annihilated by its order. We first reduce to the special case that k is separ·ably closed. Indeed, by [2, 18.8.8], the st1·ict Henselization of R is a local faithfully flat Ralgebra whose maximal ideal is ge11erated by the maximal ideal of R. Therefore it is Artinian and its maximal ideal m satisfies mP == pm = 0. It follows that we may replace R by its strict Henseliza.tion and hence assume that its residue field k is separably closed. Recall that G = Spec(A), where A is a local free RHopf algeb1,a of 1~ank p 11 for some ri ~ l. Tl1eorem 4.1 says that the augmentation ideal JC A has the property that [pr"](I) = 0. If n = l, it follows from [3, Thrn. 1] that G is commutative, so that Delig11e's theorem implies that G is killed by its order. Tl1erefore we ma,y assume that n ~ 2. By the result in SGA 3, the group scheme G considered over the residue field k, is killed by pn. If it happens to already be killed by pnI, tl1en the augmentation ideal IC A has the property that (pn 1 ](1) c ml. Then Proposition 2.1 easily implies [pn](I) c mP +pm= 0 and the theo1·em follows. We are left with the local group schemes G over R of order pn whose reductions over k are killed by pn, but not by pnl. In [4] the group schemes with this property are determined. There are only two possibilities: over k they are either isomorphic to the multiplicative group µpn or to tl1e nonconnnutative matrix group scheme Mn given by 1 0
X
y
: x, y E
s satisfying xP = 0 and yPn  l
== 1 .
for every kalgebra S. If G is isomorphic to µpn over k, then it is a deformation of µPn. However, since k is separably closed, [l, Exp. X, Corollaires 2.3 and 2.4] imply that G must then be diagonalizable. Therefore we have G ~ µpri over R. In particular, G is killed by pn. If G is over k isomorpl1ic to Mn for some n > 2, then R must have characteristic p and is therefore a kalgebra. Moreover, there exists a faithfully flat Ralgebr·a R' s11ch that the base change of G to R' is isomorphic to a base change fro1n k to R' of the group scheme Mn. See (4) for more details. It follows that G is killed by pn. Tl1is completes the outline of the proof of the theorem.
Acknowledgements The author was suppo1~ted by Tor Vergata funds n. E82F16000470005.
References (1] Demazure, M. a11d Grothendieck, A.: Schemas en Groupes, clans Sero. de Geometrie Algebriql1e du Bois Marie (1962/64) SGA 3, vols. I, II ancl III, Lecture Notes in Math. 151, 152 and 153, SpringerVerlag, Berlin Heidelberg New York, 1970.
Chapter 1 Is a Finite Locally Free Group Scheme Killed by Its 01·der?
7
,
[2] Grotl1endieck A. and Dieudonne, J .: Etude local des schemas et des mor,, phismes de schemas, da.n s Eleme11t.s de Geo111etrie Algebrique IV Publ. Nlat;h. IHE/3 32 (1966). ,
[3] ·T ate, J. a11cl Oort, F.: Group schemes of p1·in1e order, Ai1n. Scient. Ee. Norm. Sup. 3 (1970), 1 21. [4] Schoof, R.: Finite Flat Gr·oup Sche111es over Local A.t·tin Rings: Compositio !v1at11en1atica 128 (2001), 1 15.
[5] Tate, J.: Finite fiat group schemes, in: Cornell, G. Silverman, J. and Stevens, G .: Modula1· Forn1s and Fermat's Last Theorem, SpringerVerlag,
New York, 1997.
,
[6] vVaterhouse, \V.: Introduction to affine group schemes, Graduate Texts in ~latl1. 66, S1)ringerVerlag, Berlin Heidelberg New York 1979.
•
Open Problen1s in Aritl1metic Alp;ebraic Geometry ALM. 46, 011. 2, })p. 959
© Higher Education Press arid Inte1·11ational P1·ess Be~jingBoston
Chapter 2 Lifting of Curves with Automorphisms Andrew Obus*
Abstract
The lifting problen1 for curves with automorphisms asks whet.h er we can lift a smooth projective cl1aracteristic p curve with a group G of automorphisms to characteristic zero. This was solved by Grothe11dieck when G acts with primetop stabilizers 1 and there has been much progress over the last few decades in the wild case. \\'e survey the techniques and obstructions for this lifting problem: aiming at a reader whose background is limited to scheme theory at the level of Hartshorne's book. Throughout, we include numerous exan1ples and clarifying remarks. We also provide a list of open questions.
1
Introduction
A flu1cla111enta] st1ccess of a1gebraic geo1netry its ability to 1·easo11 about a]geb1·aically defined objects i11 cha1·acte1·istic p using geo111etric i11tuition gleaned :fr·om ou1· experience i11 the real world. For ir1stance, 11ot only ca11 we defi11e concepts such as smooth,·ness and tangent spaces i11 character~istic p, but such concepts also tur11 out to reflect 011I· characte1·istic zero i11t111ition surprisingly well. Of cou1·se charactieristic p geon1etry does not behave exactly like geon1etry in characteristic ze1·0! T11e diffe1·ences a.re too nt1merot1s to cou11t, but let us quickly me11tion one exa111ple, whicl1 will l)e motivating for tl1e discussion to come. Consider tl1e affine line . .X = There arc certainly 110 nontrivial finite etale covers 1 Y ~ X, becal1se any s11ch co\re1· wottld gi,re a topological cover Y(C) t X(C) witl1
Ab.
University of Virginia~ 141 Cabell Dri\·e, Charlottesville, \ 1A 22904. Email: and.rewobus©gmail. com. The attthor was supported by NSF Grants Dl\,IS1265290 and Df\,1S1602054.
purposes~ a b·rat1.ched cove·r J : Y ~ X is a finite, flat, ge11erically elale morphism of geornet.rically cor111ected, 11ormal sc}1ernes. ·1r f is 1111ramificd, we say it is a11 etale cover. 1 For ottr
10
Andrew Obus
tl1e complex topology (e.g.) [Sza09, P1·01)osition 4.5.6]) 1 co11tradicting the fact tl1at X(C) ~ C is si1nply connected. Howe,,er, in cba1·acteristic p ,vc have the followi11g exan1ple.
Example 1.1. Let k 11e a field of characteristic p (say, algebraically closed to preserve the a11,1.logy as well as possible). The map from the ze1·0locus Y of yP  y = x in A~ to A} given by projecti11g to the xcoordinate is a finite etale cover. Indeed, tl1e gr·ot1p Z/p acts f1·cely 011 Y by sending (:1~, y) t, there are infinitely many linearly disjoint Z/pcovers (take the s1nootl1 projective completion of the affine cover V(yP  y  xN) i;; A 2 ~ A 1 given by projecting to tl1e xa.xis, as N ranges tl1ro11gl1 N\pN). This tells us that the n1ndamental g1·ol1p of fork a11 algeb1·aically closed field of characteristic p, is not finitely ge11erated
Al,
(si11ce it l1as an infinitely large eleme11tary a.belia11 pquotient). However" tl1e situatio11 is better whe11 we restrict our atte11tion to ta1ne covers that is, covers wl1ere p does not divide tl1e ramificatio11 indices. 111 this situatio11, G1·othendiecl{ sl1owed tl1at eacl1 lJr·anched cover iI1 characte1·istic p is the reduction of a b1·anched cover in cl1a1·acteristic zero, whicl1 is more or less 11niq11e.
Theorem 1.5. (Grothendieck's ''tame Riemann existence converse'' theorem in characteristic p) Let R be a complete discrete ·uaZ.tLation ring with cilgeb·raically clo.sed re.sid11,e fielrl k of character·i.stic p and fraction field I< with al_gtbruic closure J(. Let XR be a smooth, projective, rela,ti·ve Rcv,rve 1uith special fiber X tLnd gene'Tic fibrr X K, a,n,d let x 1 ,R, ... , x,1 ,R be pairwi.se disjoint sections of JYR . Spec R. Write Xi for the inte1'section of Xi,R 'W'ith X. If f : }I" + ~Y is a ta.n1el) 1·c1.mifiecl finite cove1~) e·tale above X\ { x1: . .. , Xn}, th,e11. there is a itniq1te fin.ite fiat branch.ed cover .fR : YR + XR, etale above XR\{x1,R, ~ .. :cn,R} u,ith the same r·aniificat·ion indices as thc1,t off, such that t/1,e special fiber· of· JR ·is j'. If f' is GGalois, th.en so is f R · 1
Proof. If f is etale the theoren1 follows from Grothendieck s tl1eory of etale lifting ([SGA03 I: Corollai1~e 8.4], corr1bined with [SGA03, III: P1~op. 7.2]). h1 ge11eral we can use Grothencliecl{'s theo1~)' of tame lifting ([SGA03, XIII, Corollaire 2.12] 1 or [v'.'ew99] for a11 expositio11). See also [Fl1169] for c:1.11 alter11ate proof. If f is a Galois cover, the tl1eorer11 also follows from the localglol,al p1"' tnciple (Tl1eoren1 3.1), along 1
with Exa,111ple 4.3.
□
Remark 1.6. Tl1e cove1· f R is called a lift of the cove1~ .f o,rer R ,. Tl111s, the tame Rjema1111 existe11cc converse ca11 be stated st1cci11ctly ,,s ta111e co,1ers lift over R ( l1t1iquel)' or1ce tl1e brancl1 loct1s iB fixed)". See §1. 2. Remark 1. 7. Asstune wiLhout loss of generality that k i8 cot111table (a11y n1orphism of ,rarieties ir1 cl1aracteristic; JJ, bei11g given 1Jy finitely many eql1a.tions~ can be definecl o,,er s0111e fi11itely gene1~ated eA.rtension of 1F1,) whose algebraic closure is tl1u. cou11tiable). If R is c:l con11)l te disc1·ct,e valt1atio11 ri11g with residue field k a11d fr' gives a bijection on finiteindex normal subgroups, thus placing the pri.metop Galois covers of Xx, etale above Ux in 11atl1ral 011etoone correspondence with the prin1etop Galois covers of X, etale above U.
Of course, we wo1tld like to know whether wild (i.e., not tame) cover·s in characteristic p come from cl1a1·acteristic ze1·0 as well. This will be our ma.in focus. ,,.
1. 2
Lifting of covers of curves
Let k be an algebraically closed field of cl1ara.c teristic p. We form11laie the lifting problem precisely:
Question 1.10. (The lifting problem for covers of curves) Let G be a finite group acti11g on a s111oot,l1, projective ct1rve Y ove1 k, so tl1at f: ·y ~ X = Y/G is a b1·a11ched Gco,,er of s1nooth, projecti,,e c111·ves ove1· k. Is tl1ere a cl1a1·acteristic zero disr1·ete valuation ring R ~rith residt1e field A.· and a branched Gcover f R : YR ~ XR. of smootl1 (i11 particular: fla.t ) 1·elative Rcur,,es "\\ritl1 SJJecial fiber f? If the answer to tl1e question abo,re is ' 0. Let be the standa.rd lift of witl1 coo1~di11a.t e T reducir1g to t. Let R be a complete characteristic ze1"0 discrete valuation 1"i11g with residue field k. We clai111 tl1at a lift f R of f can be given by taking tl1e normalization Yn of in the £11nction field
Pl
IPk
r
IC
:==
z"i 
K(T) [Z]/
IT (T 
Ai)Ci ,
i=l
where Ai is any lift of ai to R. Here K == Frac(R) and a generator of the Galois group takes Z to (nZ (note that R, being complete, contains (n)The map JR is flat by Propositjon A.3. Since the normalization of in JC contains SpecR[T, Z]/(zn_ rr:1(TAi)Ci), it is cleat" that YR XRk is birationally equivalent to Y. But we must check that it is smooth (the normal scherne YR can, in theory, have singularities in codimension 2!). The generic fiber f K :== f R x R K is branched of index ei :== 'r i/ gcd( n, ci) above Ai, so the RiemannIIurwitz formula gives that the genus g,,, of YR x R K satisfies
Ak
r
2g11
2

= 2n + L(n 
ei).
i=l
The Ra.me is true for the genus gy of Y. Si11ce f R is flat, the aritlunetic gent1s Pa.(Yn XR k) = g11 = gy (e.g., (Har77, III, Theo1·em 9.13]). Thus YR xn k is smooth (see, e.g., [IIar77, IV, Exercise 1.8]).
2.2
Wild covers
In stark contrast to the case of tame covers, tl1e lifting problen1 for· wild covers contains a great deal of mystery. Eve11 in the most basic example whe11 a wild cover lifts, writing the lift down is less straightforward than in Example 2.1.
Example 2.2. Let f : Y = IP' 1
=
lP1 be the ArtinSchreier Z/pcover over k given by the equation z 1+ zP  z, where z is a coordinate on IP'1 . The Galois action is generated by z ~ z+ 1. Let R be a complete discrete valuation ri11g with residue field k containing (p and let, A == (p  l. A lift of f to chaJ·acteristic zero is JR: YR== Pk+ XR == where JR is given by 4
X
JPk,
z 4
(1
+ AZ)P 
1
AP (note that this is defined over R, and reduces to z ~ zP  z over k). The Z/pGalois a,ction on YR is generated by Z H (pZ + 1 (which has order p  the reader should check this!), and tl1is action reduces to z ~ z + l over k.
Remark 2.3. Notice that the gene1·ic fibe1· of YR above has two ramification points (at Z == l/ A and Z == oo ), both of which specialize to the t1niql1e ra1nification point z == oo of Y. This phenomenon of a ramification point ''splitting" into several ramification points on a lift to character~istic zero happens whenever tl1e ramification point is wild (indeed, the RiemannHurwitz formula, necessitates this, as a wild 1·amification point of index e contribt1tes more than e  1 to the degree of the ramifica.t ion divisor).
Chapte1· 2
Lifting of Curves with At1ton1orphis1ns
17
The1·e a1·e ,ilso exan1ples of ,vild co,,ers t.ha.t do not lift to characteristic zero, .., t1cb as the follo~ri11g.
Example 2.4. Let Y ~ JP>l. The grol1p G = (Z/p) 11. (fo1· an)' ·n.) embeds into tl1e additive group of k, and acts on Y by tl1e additive actio11, fixing oo. Let f : }' + X ~ 1P1 l)e the ind11ced Gcove1·. If t·n : YR  t XR is a lift of f to characteristic ze1·0 a11d K == F1·ac(.R,), tl1en flatness of YR ~ SpecR irnplies that the generic fiber Y1r of YR is a gc11l1S zero ct1rve in cl1a1·acteristic zero with Ga.ction. Ho'\\rever, tl1e c1ut,01norphism group c)f YK e1nbed8 i11to PGL 2 (K), whicl1 does not contt1.in (Z/p) '" if n > 1 a11cl JJ"1 #, 4. So tl1e c;actio11 011 Y ca.n11ot lift to cha1·acteristic zero in tl1ese Cc1Ses.
Along tl1e ,s ame li11es, a11d 1nore si1nply, a cover rnigl1t 11ot lift simply becal1se the at1tomo1·1)l1ism grotIJ) is too large for cl1aracteristic zero. Tl1e following example is fi:on1 [Roq70, §4].
Example 2.5. Co11Side1' the smooth p1·ojecti,re model Y of tl1e ct1rve y 2 = xP  x over k, "rl1ere p 2:: 5. The genus g1~ of Y is (p  1)/2. Tl1e group G :== Aut(Y) is ge11erated bj' a:
X t4
T:
x
v:
i+
:r; + 1, ax y
l X
1+ 
;; '
y 1+
t7
y
~a/y (a
y~
,1.
generator of w;),
y x(P t l) /2 .
Tl1is group co11tai11s PGL 2 (p) as a11 i11clex 2 sl1bgroup ( consideri11g only the actior1 2 011 x), so G = 2p(JJ  1) . If tl1e Gcovcr f : Y + Y/ Aut(J,,,.) liftecl to characteristic ze1·0, the gene1·ic fiber of tl1c lift wot1ld be a gentL.9 9}" curve with at least IGI aL1tomorphisms . Since IGl > 84(g  1), this violates tl1e Htu·witz botmd on automorphisms of curves in chru:acteristic zero (see, e.g., [Har77, IV, Ex. 2.5J). Iridced, this is tl1e c,1.se e,,en when G is r·eplaced by its index 2 subgroup PGL2(p). Lastl)'= l1ere is an example from [Oor87, §2] of a wild cover that, is 11onliftable for reasons otl1er t11a11 t,be size of the automorphism gi·ou1J.
Example 2.6. Suppose p = 5, a11d let G be tl1e g1·011p of order 20 with presentation (a} r Ia 5 = 1, , 4 == 1, ar = Ta 1). Let f : Y  t X = Jl'D 1 be the Gco,~e1· corresponding to the e1T1bedding of frmction fields k(t) c > k(t) [x y]/(x4 t, y 5 yx 2 ), wl1ere the Gactio11 is give11 by
If Z correspo11ds to the subfield k( t, x), tl1e11 Z  t X is a Z/ 4cove1· of genus zero curves ramified at x = 0 and x = oo, and Y ~ Z is a11 ArtinSchreier Z/5cover
branched only at. x = 0. If P E Y is the point above x = 0, the11 xy 2 is a tmiforroizing parcuneter at P. and the raJuification cii,risor D of Y  t Z can be calc,ilated l)y taki11g
dx d(xy2 )
clx ?/    == x Y, .2 xydy
18
Andrew Obus
vthose divisor l1a.g Ppart 12[P]. Tl1us D == 12[P], and the Rien1annHU1·witz fo1~mula sl1ows that the ge11tL5 of Y is 2. Now, suppose tl1at f has a lift J'R : YR+ XR over a discrete valuation ring R in characteristic zero. There is an inter1nedia,te Z/5co,ter Yn  t ZR lifting Y + Z. By flatness, the generic fiber YK  t Z K over K := Frac(R) is a Z/5cover of a genus O curve by a genus 2 curve, a11cl ZK + XK is a Z/4covei.. of gen1is zero curves. Since all ramification points of Yr< ~ ZK have ramification index 5, tl1e Rierr1an o Hurwitz for111ula yields
2(2)  2 = 5(2) + 4r, where r is the n1unber of b1·ancl1. IJOi11.ts. Thl1s r == 3, a11d tl1ese tlu·ee points are permuted by the Z/4action on Z1(. Since Z1~ ~ X1< is a cover of genus zero curves this action is fi:ee apart fi'om two fixed points. So no set of three points is stable 1mder tlris action, yielding a contradictio11.
The examples alJove motivate the followi11g obstruction to lifting, kno~rn &5 the KatzGabberBert•i'fl. (or KGB) obstruction (cf. [CGHll, §1]~ " rl1ere the defi1tltion is give11 in a slightlj difl·erent context). 1
Definition 2.7. (The KGB Obstruction) If f: }'~Xis a l)ranched Gcover of smooth projective curves over k, tl1e11 tl1er·e is KGB obst1~,u1ct'io·n to lifting f if there exists no curve C in characteristic zero with faithful Gaction sucl1 that, for all H :S G, the gent1s of C / H ec111als tl1e gent1s of Y / H. If tl1ere cloes exist st1cl1 a curve, we say tl1at the KGB obstruction va11,islies.
Remark 2 .8. The KGB obstructio11 above motivates the local KGB obstruction (Definition 4.5), which will be the form we primarilj' use.
By flatness, it is clea1~ that if 1· has a. Ition known as the local lifting problem., is the key to solvi11g tl1e lifti11g problen1.
Question 4 .1. Suppose G is a finite group a11cl k[[z]J/k[[t]] is a GGalois extension. Does there exist a characteristic zero discrete valuation ring R with residue field k and a GGalois extension R[[Z]]/ R[[T]] such tl1a.t the Gac·tion on R[[Z]] reduces to tl1e given Gactio11 on k[[z]]?
If such a lift exists we say that R[[Z]]/ R[[T]] is a lift of k[[z]]/ A~[[t]] over R, or that k[[Z]]/k[[t]] lifts to cl1.ar·acteri.stic zero (or lifts over· R). R ema k 4.2. As 011e sees in §A.3> a.11y group G for whicl1 there exists a faithf11l local CJextension is of the fo1·m P >1 that is etale 011tside t E { 0, oo}, tamely I'amified of index m above t = oo, and totally rarnified above t == 0 such that the Gextension of complete local rings at t == 0 is given by k[[z]]/ k[[t]]. This is called the HarbaterKatzGabber (HKG) cover associated to k[[z]]/k[[t]]. By Theo1·em 3.1 and Example 4.3, the Gcover f : Y 4 IP'l Iifts to characteristic zero if and only if the extension k[[z]]/k[[t]] does.
Definition 4.5. In the co11text above, we say that a local Gextension has a ( 1oca~ KGB obstr·uction to lifting if the associated HKGcover does. Remark 4.6. One can also formulate the local KGB obstr11ction in a purely local way (not using HKGcovers), using the different and Proposition 6.2 below as a replacement for the fact that the genus of a lift of a curve equals the genus of the original curve. We leave this as an exercise. Remark 4. 7. The Bertin obstruction of [Ber98] is strictly weaker than the local KGB obstruction, so we do not discuss it further. Tl1at the local KGB obstr11ction is at least as strict as the Bertin obst1·uction is proven in [CGHll, Theo1·em 4.2]. An example of a local Z/3 x .Z/3extension with vanishing Berti11 obst1·uction but nonvanishing local KGB obstruction is given i11 [CGHll, Example B.2]. Clearly, if k[[z]]/ k[[t]] lifts to char·acteristic zer·o, its local KGB obstruction vanishes. If G is cyclicbyp and the local KGB obstruction vanishes for· all local Gextensions, then G is called a local KGB group for p. If the local KGB obstruction vanishes for some local Gextension, then G is called an wea.k local KGB group for p.
The followi11g classification of the local KGB groups is due to Chinburg, Guralnick, and Harbater.
Theorem 4.8 ([CGHll], Theorem 1.2). The local KGB groups fork con~sist of tl1,e cyclic groups, the dihedral group DpTI, .for any n, the group A4 (for ch,ar(k) = 2), and tlie generalized quaternion groups Q 2 m of order 2m fo'r m > 4 (for cha,r(k) =
2). Sketch of proof. We briefly outline the negative di1~ection (i.e., that there are no local KGB groups aside from the ones on the list). The first observation is that
Cl1apter 2
Lifting of Curv.e s witl1 Auton1orpl1isms
25
if G is a local KGB grol1p for p, then a11y quotient of G is as well. This is because any local G/ He.xte11sio11 ca11 be extended to a. local Gextension ([CGH08, Le1n1na 2.10]), a11d if the local G / Hexte11sio11 h~" nontrivial local KGB obst1·l1ction then the Gextension clearly bas 011e too. Tl1t1s to show that a group is not local KGB~ it suffices to sl1ow it has a c1uotient tl1at, is not. local KGB. There is ar1 explicit list of types of groups that cru1 be sl1own not to be local KGB (this list includes, for exa1111)le, Z/p x Z/7J for p odd), and it can be fm·ther shown that any cyclicbypgrottp eithe1' has a, quotie11t 011 this list, or is one of the groups i11 Theoren1 4.8 (sec [CGHll, §1112]). Tl1is con1pletes the p1·oof:. □ Fo1· exan1ples illustrating the KGB obstructio11 for Z/pn, ;>q Z/rn., and additional exa111ples foI' Z/p x Z/p, 8ee [Obu12 Propositio1is 5.8, 5.9]~ Since any loca.l Oo1't g1·oup is a local KGB g1·oup, the searcl1 for local Oo1·t g1·011ps is restricted to t,he groups in Theore111 4.8. Tl1e generalizecl q11aternion gro11ps were shown not to be local Oort groups i11 [B\V09]. The obstruction develo1)ecl in [BW09] to sl1ow tltis is called the H·u1·witz tr·ee obstru,ctio'ri, and will be disct1ssccl in §4.2 (see spcc·ifically Exan1ple 4.17).
4.1.1
Global consequences
Sl1ppose G is a11 arb itr·a1·y finite grot1p a11d fi..x: a pr·ime p. Let us revisit t,he question of V\rhetl1e1· G is ar1 Oort group (for p). By the localglobal principle, it is clear that if ,, r}r cyclicbyp subgroup of G is local Oort, the11 G is an Oort group. In fact, tl1e co11verse is tI·ue as well. 1
Proposition 4.9 ([CGH08, Theorem 2.4]). If G is a finite group, then it is an Oort .qroup for· p if a11.d 011,ly if e1Jery cyclicbyp subgro1.1,p of G it, a local Oort grO'lLp.
Proof. The iif direction follows from the localglobal principle. To prove tl1e "only if' di1·ectio11, let Is; G be a C)'Clicbyp subgi·oup, and let k[[z]]/k[[t]] be a local Iextensior1. By [CGH08, Lem1na. 2.5], tl1ere is a GGalois cove1· f: Y ~ IP1 s11cl1 that tl1e1·e is a point x E 1?1 and a pointy E Y above X £01· wl1icl1 tl1e IGalois " " exte11sion OY,·y /Opi,x is iso1norphic (as an Icxter1sion) to k[[z]]/k[[t]]. Sii1ce G is a11 Oort gToup, 1· lifts to characte1·istic zero. By the easy directio11 of the localglobal p1·inciple. k;[[z]]/ k[[t]] lifts to cl1aracte1·istic zero as well. So I is a local Oort gro11p. □
The following exa.n1ple is contai11ed ir1 [Obu16: Co1·ollru·y 1.4]. Example 4 .10. The group A 5 is an Oort group fo1· eve1·y prime. Indeed, its only cyclicbyp groups (for any p) are Z/2, Z/3, Z/5, Z/2 x Z/2, S3, A4, and D5, all of which are local Oort groups for their respective primes (see §5).
light of Proposition 4.9, classifying the Oort groups for p consists of two parts: classifying the local Oort g1·oups for p, and classifying the groups whose cyclicbyp s11bgroups are on this list. Following ChinburgGuralnickHarbater ((CGI118]), we call a. g1·011p Gan Ogroup for p if every cyclicbyp subgroup of G is eitl1er cyclic, Dp.,i, 01· A 4 if p = 2. Since tl1ese are the 011ly cyclicbyp grol1ps l11
26
Andrew OlJus
that can be local Oort groups, Proposition 4.9 shows that a group G can be an Oort group only if it is an Ogr·oup. In particular, if Dp·n. is shown to be local Oort for all p and all n, then the list of Og1·ot1ps is the same as the list. of Oort groups. The list of all Ogro11ps l1as been co1nputed by ChinburgGuralnickHarbater.
Theorem 4.11 ([CGH18, Theorems 2.4, 2.6]). If' p is an odd prime, then G is an Ogroup for p if and only if a pSylow subgroup P o.f G is cyclic an,d either
• The normalizer Nc(P) and centralize·r Zc(P) of P in G are equal, or, • Na(P)/Za(P) == 2, the order p subgroup Q s P has abelian centralizer, a,nd every element of NG(Q)\Za(Q) acts as an involution inverting Za(Q). If p == 2, then G is an Ogroup if. and only if' P is cyclic or P is dihedral, with Zo(K) == K for all elementary abelian subgro'ups K of O'rder· 4. More explicit lists of groups satisfying these criteria are given in [CGH18, Theorems 2.7, 3.8}.
4.1.2
Consequences of obstructions for weak local Oort groups
Suppose G := Z/p'n
Z/m is noncyclic (equivale11tly, nonabelian). It ttrrns out that a loca.l Gextension k[[z]]/k[[t]] whose Z/p11'subextension has first positive lower jump h has vanishing KGB obstruction precisely when /1 _ 1 (mod ,n,) If this happens, the conjugation action of Z/rri on Z/pn is faithful, OI" equivalently, G is centerfree ([Obu12, Proposition 5.9]). So gr·oups G of· this fo1·m that are neither cyclic nor centerfree caru1ot be weak local Oort. Furthermore, Green11atignon showed that if G contai11s an abelian subgroup tha.t is neither cyclic nor a pgroup, then G is not a weak local Oort group (see [Gre03, P1~oposition 3.3], which shows that no abelian group can be weak local Oort unless it is cyclic or a pgroup  it is more or less trivial to see that if a >4
gl'·oup contains a s11bgroup that is not weak local Oort, then the g1oup itself is not weak local Oo1't). For a somewhat stronger stateme11t, see [CGHll, Theorem 1.8] .
4.2
The ( differential) Hurwitz tree obstruction
Recall that the (local) KGB obstructjon comes fi,om exploiting the fact that the genus of a cu1·ve does not change when it is lifted to characteristic zero. Howeve1·, if we have a local Gextension k [[z]] / k [[t]] and a lift of its correspo11ding HK Gcover, the11 only reme111bering the genus of this cover and its subcovers means that we are actually throwing ot1t a great deal of other information. In particular, we are forgetting the padic geometry of the b1·anch locus (that is, tl1e distances of the branch points :&:om each other). It ttu·ns out that these distances satisfy subtle constraints, and the i11for·1nation about tl1ese cor1strai11ts can be pa.ckaged in ct combinatorial structure called a H'u,witz tr·ee. Tl1e local Gextcnsion k[[z]]/k[[t]] will have a Hu,rwitz tree obstruction if no Hurwitz tree exists that prope1·ly reflects the higher ramificatio11 filtration of k[[z]] / k[[t]]. Hen1·io gave the first major exposition of Htu~witz trees ([IIen00a]), dealing with tl1e case of local Z/pextensio11s. The concept was exte11ded by Bouw and
Cl1apter 2
27
Lifting of Curves with Automorpl1isms
Wewe1·s to e11compass local Z/p'>4Zj·, riextensions ([BW06]). Late1·, it was extended bj' Bre\vis ancl Wewe1~s to arlJitrar3r groups ([BW09]). A detailed overview of the construction of Hurwitz trees for Z/p XJ Z/rnactio11S is also given in [Obu12 §7.3.1 , §7.3.2], \\'e will give a much l)riefer overvic,v belov.1 , and tl1e11 we will mention how one generalizes to the case of arbii,rary local Gextensions. 4.2.1
Building a Hurwitz tree
Suppose A~[[z]]/k[[t]] is a local Gextension tl1at lifts to a Gextension R{[Z]]/ R[[T]], where R is a characteristic zero co1nplete discrete valuation ring. The key observatio11 is that we sl1ould tl1i11k of R[[Z]] as the ring of Rvalued functio11s 011 the pad,ic open unit di.c;c (since if x is c:1lgebraic over R and has posit,ive valuation. all po"rer series converge on x). Thus we will tl1ink of V := S.p ec R[[Z]] as tl1e open unit disc. By assu1nption, G acts faithfully on V b)' Rat1tomo1·phisms with no ine1·t,ia alJo,1e a u11ifor1nizer of R. (that is'\ tl1e action reduces to a faitl1ful Gaction on Spec k [[z]]). The idea of a Hurwitz tree is to 1mdersta11d the geo1netry of this actior1 in co1nl)inatorial for1n. We will write 'DK for Ll1e generic fiber of 7J, where K = Frac( R). Clea1·l)', G acts on V K. \f\Te assume R and l( are la1·ge enough for all ranillication points of 1J I< ~ DK/ G to be defined over J(. Let V' = 'D /G = Spec R[[T]], and write 1JK analogously to VK. Let Y1 ~ ... , Y.s (resp. z1, ... , Zr) be t,he bra11ch (reSJJ. ramification) points of VK ~ V'ik co1~1·espo11ding to (D; Y1, ... , Ys) (resp. (TJ'; z1, ... , Zr ) we assume r s 2:: 2, which is a.utomatic as long as G l1as r1ontrivial pSylow subgroup). This is the minimal semistable Rcurve witl1 generic fil)er 1P7< that separates the specializations of the ·y1 (resp. Zi) and oo where 'DK (resp. V~}l, and the grapl1 r is simply a point, which we call v. To build the Hurwitz t1·ee r', we add a root vertex vo and ve1·tices v 1, ... , VN+l corresponding to the branch points, with each Vi connected to v by an edge ei. The thickness t(eo) is just the valuation p/N(p  1) of ~  If i > 0, then c(ei) == 0. By definition, we have 6v0 == 0 arid ..P. Then, taking X == T/a, we have that X reduces to
Andrew Obus
30
a coordina.te a; 011 Z (the specializations of tl1e branc11 poi11t,s to x = 0 a.nd all the Nth roots of unity). Since 1 + APTN = l from [B\\T06, §3.2) (see also (HenOOa Corollaire 1.8(A)]) tha.t Wv
4.2.2
=
d(l  x N) 1  x N
= N
Z correspond to x  N it follows 1
dx x(xN  1) ·
The Hurwitz tree obstruction
In the context of a general Htu"witz tree, the way one gets a .1J obstruction is now clear. Let k[[z]]/k[[t]] be a local Gextension whose highe1· ra1nification filtration has Artin character x. We say there is a Hun..uitz tr·ee obstruction, to lifti11g if 110 Hurwitz tree for G can be constructed having Artin character x on t,he ver~tex u0 .
Example 4.17. Let G == Q 2m the generalized quate1·11io11 group of order 21n which can lJe presented as
If m == 3 this is the sta.ndaid quaternion group, a11d we assrnne 1n. > 3. Let G = G/(r 2) ~ Z/2 x Z/2. Then G is gene1·ated by the in1ages a and 7f of u ancl T. Consider tl1e Gaction on k[[z]] such tha.t
a(z)
1
= _1_+_z
anve1· a complete cl1a1·actcristic zero disc1·ete vaJt1ation ri11g R with 1·esid11e field k is a Gextensio11 M /Frac(R[[T]]) sucl1 tl1at: 1. If A is the integTal closln·e of R[[T]] in A;J, tl1en the integral closure of Ak is isomo1·pltlc to (arid ide11tified witl1) lc [[z}] (equiva]e11tly Fr~tc(Ak) rv k((z))).
2. The Gaction on A; ((z)) == F·rac(Ak) induced fi·o1n tl1at give11 Gaction on k [[z]] .
011
A restricts to tl1e
Chapter 2
Lifting of Curves with Automorpltlsms
33
In fact, Garuti has sl1own ([Gar96]) that any local Gextension ha,s a birational lift to characteristic zero. The following criterion, which saves one fro1n the effort of making explicit computations with integral closures, is extremely useful for seeing when a birational lift is actually a lift.
Proposition 6.2 (The diffe1,ent criterion, [GM98, I, 3.4]). Suppose A/ R[[T]] is
a biration,al lift of the local Gextension k[[z]]/k[[t]J. Let K = Frac(R), let 8,,, be the degree of the different V 11 of AK/R[[T]]K (i.e., the length of AK /V17 as a K module), and let 68 be the degree of the different Ds of k[[z]]/k[[t]] (i.e., the length of Ak/Vs as a kmodule). Then 8s < 877 , and equality holds if and only if A/ R[[T]] is a lift of k[[z]]/k[[t]] ( that is, A rv R[[Z]]). Remark 6.3. Replacing Rand K by finite extensions does not affect the degree of Dr, above, so we may ass11me that the ramified ideals in AK/ R[[T]]K have residue field K. Remark 6.4. Tl1e different criterion is also valid when R is an equ,icharacteristic complete discrete valuation ring (i.e., R == k[[w]]). This will be used in §6.6.
6.2
Explicit lifts
Some imes, the simplest way of lifting a local Gextension is to write down explicit equatjons. We give two examples in this section.
Example 6.5. (Z/pextensions) The following argument shows that all local Z/pextensions lift to characteristic zero. It is a simplified version of aI·guments originally from [SOS89], and can also be found in [Obu12, Theorem 6.8] . Since it is the most basic example, one would be 1~emiss not to include it here. The key observation is that any Z/pextension of a characteristic zero field containing a pth root of unity is a Kummer extension, .g iven by ext1acting a pth root. The tricl{ is then to ass11me (p E R and to find an element of Frac(R[[TJ]) such that normalizing R[[T]] in the cor1~esponding K11rnmer extension yields the original local A rtinSchreier extension. Say that an element of a field L of characteristic p is a pth, porwer if 1t is expressjble as xP  x for x E L. By ArtinSclu·eier theory, any Z/pextension of k((t)) is given by k((t))[y]/(yP  y  g(t)), and is well defined up to adding a ptl1 power to g(t). In particular, we may assu1ne that g(t) E t 1 k[t 1 ], as any element 2 u E k[[t]] can be written as ;cP  x, where x = u  uP  uP  • • • • Similarly, we may assume that g(t) l1as no terms of degree d1visible by p. If g(t) == t  Nh(t), where h(t) E k[t] has nonzet'O constant ter1n, then li(t) is an Nth power in k((t)), so replacing t with an Nth root of 1/ g(t) (which is a uniformizer), we may assume that g(t) = t  N , with pf N. Given a local Z/pextension k[[z]]/k[[t]] , we may thus ass11me without loss of generality that it is the i11tegi"al closure of k[[t]] in the ArtinSclrreier extension of k((t))[y]/(yP y  t  N) given by tN_ Let R = W(k)[(p], let A== (p  1, and let K = F1~ac(R) ♦ Then v(Ap~l + p) > 1. Consider tl1e integ1"al closure A of R[[T]] in
34
Andrew Obus
the Kummer extension of F1,ac(R([T]]) given by WP == 1 + ApyN.
(1)
Making the substitution W == 1 + AY, we obtain
(2) where o(pPI (pl)) 1·epresents terms with coefficients of valuation greater than p / (pl). This reduces to yP y = tN. So we have constructed a birational lift. The jump in the ra.mification filtration for k[[z]]/ k[[t]] occurs at N (Exercise! This can be done eXJ.Jlicitly by writing a uniformize1· in ter1ns oft and y). By (3), the degree of the different of k[[z]]/k[[t]] is (N + l)(p 1). 011 the other ba.ncl, the generic fiber of Spec A ~ Spec R[[T]] is branched at exactly N + 1 points in tl1e unit disc (at T == 0 and T == v as v ranges through the Nth roots of AP). Since the ramification is ta.,ne, the degree of the different of AK/ R[[T}] K is ( N + 1) (p1) as well. By Proposition 6.2, our birational lift is an actual lift.
Remark 6.6. It is easy to use Example 6.5 to show that all local Z/pmextensions lift to characteristic zero, when pf m. See [Obul2, Proposition 6.3]. Remark 6. 7. The case N == 1 is the local version of Example 2.2 above. Note tl1at in this ca,se, y 1 is a uniforrnizer of k[[z]], and we ca,n set z == y 1 . king Z === y 1 === ..\/(W  1), Remarl{ A.6 shows that A can be WI·itten as R[[Z]], once we verify that Z E A. Tl1is is true because expanding out the equation (1 + ..\z 1 )P = 1 + APT 1 coming from (1) and multiplying both sides by T ZP / AP gives an integral eqt1ation for Z over R[[T]]. Example 6.8. (Some Z/2 x Z/2extensions) For odd p, it is a.n open problem in general to dete1·mine exactly which local Z/p x Z/pexter1sio11S lift to characteristic zero (see Example 4.4 and also [Obu12, Proposition 5.8], which is an exposition of mate1,ial in [GM98, I, Theorem 5.1]). However, some local Z/pxZ/pextensions can lJe lifted explicitly. For example, suppose p == 2, and consider the local Z/2 x Z/21 2 2 extensiori k[[z]] of k[[t]] given lJynormalizing k[[t]] in k((t))[y,1v]/(y yt ,w w tN) for any odd N > 1. Letting R == W(JF2 ), I clai1n that normalizing R[[T]] •
lll
L == Frac( R[[TJ]) [U, V]/ (U 2  1  4T 1 , V 2

1  4T N)
gives an exte11sio11 A/R[[T]] lifting k[[z]]/k[[t]]. Si11ce tl1e field extension L/Frac(R[[T]]) is the composit11rn of two Z/2extensions, each giving rise to a lift of the con1ponent Z/2extensions of k[[t]] (see (1), and note tl1at :AP == 4), we have that L/Frac(R[(T]]) certainly gives l'ise to a birational lift. 111 order to show that it is actually a lift, we apply the different criterion (Pr·oposition 6.2). Since the upper nt1mbering is preservecl under taking q11otie11ts ( [Ser79, IV, P1~opositior1 14], the upper jtunps of· k[[z]]/k[[t]] appear at 1 and N. Tlus 1r1eans that Go = G1 == Z/2 x Z/2, and G2 == · · · == G2NI == Z/2, with the rest of the filtration trivial. By (3), t .h e degree of the different is equal to 2N + 4. On 1;he otl1e1~ l1and, since 1·an1ification groups in characte1·istic zero are cyclic, eve1~y rarnification index of tl1e Z/2 x Z/2extenBion AK/ R[[T]]K is 2, ancl tl1us
Chapter 2
Lifti11g of Curves witl1 J.b,._uton1orpl1isnis
35
eacl1 ra,mified icleal of R[[T]]K with residue fielcl K (whicl1~ after a finite extensio11 of I< , we asstune is every ra1nified ideal) cont1·ib11tes 2 to the degree of the differe11t. These ramified ideals correspond to the zeroes a11d poles of 1+4r 1 and 1+4TN in the open unit disc. Tl1ere are a total of N + 1 zeroes and 1 pole, showing that the degree of t]1e different is 2N + 4. \J\Te are fJone.
Remark 6.9. In fact every local Z/2 x Z/2extension lifts to cha.i·acteristic zero, but writing dowi1 the lift is not ge11erally as straightforward as above. For more on this, see [Pag02a] a11d [Pag021J]. Remark 6.10. Matignon ([Mat99]) has show11 that (Z/p)n is a weak local Oort group for any p and n,, by writing dow11 an explicit exa1nple and ai1d explicit lift. For an exa1nple of a local Dpexte11Sion fc}r any odcl p witl1 an explicit lift to characteristic zero, see [G:tvl99, N, Proposition 2.2.1] (01' [Obu12, Proposition ·7.3] for the same example). As in tl1e case of Z/2 x Z/2, all local Dpextensio11s lift to characteristic zero ([BW06]), bt1t it is not in general easy to write down the lift explicitly. For an exa1nple of explicit lifts of some local A 4 exteusions to cl1aract.eristic zero, see [Obu16, Pr'opositions 5.1, 5.2]. The paper (Bre08] gives examples of ex1)licit lifts of son1e local D 4 extensions.
6.3
SekiguchiSuwa theory
One potential way of obtai11ing explicit lifts for cyclic local extensions is the K11,m.rnerArti11,Schreier· Witt theory, or Sekiguch.iSui11a tlieory (developed in [S894], [S899], with [MRT14] being a nice st1rvey). Her·e, we will liniit ourselves to me11tioning tl1at KttmmerArtinSclir·eier t/1.eory, as developed by Sekiguchi, Oort, and St1v\1a in [SOS89], gives an explicit group scheme Q defined over Zp[(p] whose special fiber is G 0 , a,nd whose generic fiber is Gm . Furthermore, the theory exl1ibits t.he (more or less uniql1e) degree p isogeny on g explicitly. Any lift of a local Z/pextensio11 (whicl1 is ..~rtinScln~eie1·) to a Ku1111ner exte11sion is a torsor under the ke1·nel of tl1is isogeny, and knowi11g tl1e explicit equations cutti11g out this kernel leads one to discover tl1e I{t1mmer extension t1sed in Example 6.5. The I(urrunerArtinSchreier Witt theory gencrtilizes this sto1·y to i~ogenies of degree pn. We refc1· tl1e 1·eacler to [OlJu12, §4.8) fo1· a b1~jef exposit.ion a11d tl1e11 to [MRT14] if deeper knowledge is desired. v,re note thc:1.t Gree11 ai1d 1'latigno11 were al)le to 11se tl1e I{11mmerArtinSclrreierWitt theory to shovt tl1at Z/p2 is a local Oort group ([G1198] or [Obul2, §6.5) fo1· an o,1erv.. iew). Tl1e er111atio11s i11vol,1ed i11 I( 11111111erArti11SchreierWitt theory for Z/p11 becor11e ver·y co1nplicated ~rhen 11 > 2, and the theory has not been Sl1ccessf1tlly applied to tl1e loca.l lifti11g proble111 for these gi~ou1)s.
6.4
Hurwitz trees
Tl1e Hur"ritz trees disc11ssed i11 §4.2 l1ave been lisecl to ol tain positi,,e resttlts for t.11 :\ local lifti11g p1·oble111 in the ca..5e G = Z/p ( [Hen00a] of course: tlris is alrecldJ' p1·ovcn in Exa111ple 6.5) ctnd G == Z/p >4 Z/·n i ([B~106], [B\VZ09]). The process
36
Andrew Obus
is outlined in some detail in [Obu12, §7.3.3 a,nd §7.3.4], ai1d \\7e will not repeat it he1·e. We co11te11t ourselves with stating a (lightly pai·aphrased) version of tl1e theorem of Bouw, Wewers and Zapponi.
Theorem 6.11 ([BWZ09}, Theore1u 2.1). S1.Lppose p is a prime not dividin,g n1. A nonabelian Z/p >e of the local lifting p1½oblen1, ''nice>= will l)e related to having li1nited ramificatio11 i11 some sense. Tl1is sense might va1~y depending on the group; for Z/pn >pis not a multiple of p. We clai1n that the integ1·al closure A of k[[w, t]] •
111
is an equicharacteristic deformation. Setting ro == 0 cleai·ly yields k((z))/k((t)) after talcing fraction fields, but we must show that A ~ k [(ro, z ]] . To do this we t1se tl1e different crite1~ion (Proposition 6.2) along witl1 Re111ark 6.4. It suffices to show that the degi·ee 88 of the different of the original Gextension is equal to the degree 8,, of the differer1t of the ge11eric fiber of the deformation. We have f>8 = (N + l)(p  1) (see Example 6.5). On the generic fiber, the two ramified ideals are (t) and (t w). The function g = tP(t  w)N+p has a pole of order N  p whe11 expanded out i11 k((w))((t  ro)) , and thus the ideal (t  ro) gives a contribution of (N  p+ l)(p 1) to 6TJ. On the other hand, g has a pole of order p when expanded out in k((w))((t)). Tl1us, by replacing g with g + xP  x for some x E k((w, t)) (which doesn t change the Arti11Schreier extensio11), we may ass1101e that g l1as a pole of order less than p, and tl1us that (t) co11tributes at most p(p  1) to 677 . So 811 ::; (N + l)(p  1). By Propositio11 6.2, we in fa.c t l1a.ve equality, and tl1us A~ k[[w,z]].
Remark 6.19. Notice that the rainificatio11 j,1mps on the ge11eric fiber are s111aller than the ramification j11mps of the 01,.iginal extension. In fact, based on the example a.l)ove, it is a11 ea.c;)r exercise to sl1ow that for a local Z / pextension, one can i1.lways find an eqttlcharacte1~istic defor111at.ion s11cl1 that tl1e ra111ification j11rnps 011 the generic fiber ai·e less than p.
Chaptier 2
6.6.2
1
Lifti11g of Ctu·ves witl1 Automorphis1ns
39
Lifting via equicharacteristic deformations
order to use equicl1aracteristic deformations, we need to sl1ow that being able to lift tl1e generic fl ber of an ec1uicharacter·istic deformation to chai~acteristic zero allo¥.rs \IS to do the san1e for the 01·igi11al local Gextension. First, we must say what ~re rnean by 'being able to lift tl1e generic fiber''. Take k[[w, z]][ro 1]/k[[w, t]][w 1] a11d ter1sor over k( (w)) with the algebraic closure k( (ro)). We ol)tain a Gextension of Dedekind k( (1:iJ) )algebras, and localizing at a11y branched maximal ideal gives a Gextension of k ( ( ro)) ([ s l] for some parameter s (for insta11ce, or1e cottld have s == t ors = t ro). This is a direct sum of local H 9 eAtensions, where Hs ~ G a.nd with the field k( (w)) replacing k. V\'e say that the generic fiber Z.ifts to characteristic zero if all of the local H 8 extensions obtained tllis way lift to cl1aracteristic zero. Tl1e following tl1eoren1 says more or less that being able to lift the gene1~ic fiber of an equicha1·acte1·istic deformation in1plies being able to lift the original local Gexte11sion to characteristic ze1·0. Tl1e ai·gument comes from [Pop14] and a conversatio11 witl1 Pop, b11t was only writte11 in [Popl4) for G cyclic. The papers [Obul 7) and [Obu16] use similar· arguments, but do not directly cite (Popl4] since they deal with no11cyclic gro11ps. Our statement here is intended to be citeable for general G.
111
Theorem 6.20. Suppose that /c[[z]]/k[[t]] is a local Gextension that admits ar,, equ'ich.aracteristic deformation whose generic fiber l·ifts to characteristic zero. Theri
k [[z]]/k[[t]] lifts to characteristic zero. Proof. Let k[[ro, z]]/k[[w t]] be an equicharacteristic deformation of k[[z]]/k[[t]]. Let Y 4 W = 1Pk l)e the HKGcove1· associated to k[[z]]/ k[[t]]. Let W = IP1[[w]] with coordinate t. There is a Gcove1~ of flat relative k[[w]]curves Y ~ W = Ifl>kf[w]l such that Y xw Spec k[[cv, t]] ~ Spec k[[w, t]] co1·responds to k([w, z]]/k[[w, t]] via Spec, and that this cover is tinramified outside Spec k[[ro, t)] (this follows from Pop~s arg11ment deducing [Pop14, Theorem 3.6] from (Pop14, Theorem 3.2]). Write Y+ W for the base cha.11ge of Y+ W to th~ i11teg1·al closure of k([ro]] iJ.1 k((w)), and let Y 11 ~ W 11 be the generic fiber of Y  t W. Si11ce the generic fiber of k[[r.v, z]]/k[[w, ·t]] lifts to characteristic zero after base change to the algebraic closure by assun1ption the localglobal principle tells us that Yry ~ W 17 lifts to a cover Yo 1 + Wo 1 over some characteristic zero complete discrete valuation ring 01 with residue field k( (w)). The11, [Pop14, Lemn1a 4.3] shows that we can "glue" the covers Y  t Wand Yo 1 ~ Wo 1 along the generic fil)er of the former and the special fiber of the latter: ir1 01·der to get a. cover Yo 4 'tttro defined over a rank two characteristic zero valuation ring O with 1·esidue field k lifting Y  t W (cf. [Popl4, p. 319 >second paragraph]). We now sl1ow that Y ~ W lifts over a cl1aracteristic zero discrete val11ation 1·ing. Since the Gcover Yo 4 lif1o can be described using firtltely ma11y equations, it desce11ds to a cover YA + 111A over some s,1briI1g A ~ 0 that is finitely ge1ierated over W(k). Let m == An mo, where mo is tl1e maximal ideal of 0. Then A is a domain, a,n d A/m ~ k. Furtl1ern1ore, tl1e base change of YA 4 WA to A/m, is the origi11al cover Y ~ W. By Le1111na. A.9 tl1ere is an ideal I ~ ni ~ A Sl1cl1 that A / I is a finite exte11sion R of l f1 ( k). Base cl1anging YA + lVA to A / I gives a, lift 1
Andrew Obus
40
of Y ~ W over R. Applying the easy di1·ection of the localglobal p1·i11ciple, we obtain a lift of k[[z]]/k[[t]] over R, which concludes the proof. □
Remark 6.21. In [Popl4], Pop builds 011 tl1e method above to obtai11 a somewhat st1·onger result in the cyclic context. N an1ely, lie sl1ows that giver1 any l1igher 1"a,m ification filtration of a local Z/pnextensio11, the1·e exists a specific finite extension R/W(k) over which all local Z/pnextensions with that filtration lift. It should not lJe difficult to extend tl1is result appropriately to more ge11e1·al g1·oups. Remark 6.22. I11 tl1e notation of the proof of Theorem 6.20, 011e can alternatively i11voke the completer1ess of the theory of· chai·acteristic zero algebraically closed valued fields with r·esidue characte1·istic p ([Rob77, III]) to see that ·t he Frac( 0) is an elementary exteILc;;ion of Frac(W (k)), wbicl1 means that any firstorder sentence in the la.nguage of characteristic zero algebraically closed valued fields with residue characteristic p that is t1·t1e in Frac(O) is also true in F'rac(W(k)). Tllis can be shown to include the sentence ;tk[[zJ]/k[[t]] l1as a lift over the valuation ring". So k[[z]]/k[[t]] has a lift over the algebraic closure of W(k), which rnea.ns it has a lift over some finite extension of W(k). This is a lift over a discrete val11ation ring~ Note that this n1etl1od does not lead to ar1 effective rir1g over wl1ich lifting is possible as in Remark 6.21. 6.6.3
Consequences for specific groups
For applications to the local lifting proble1n we need to know: Ho,v nice ca.n Vle hope to make local Gexte11sions via equicharacteristic deformations? Here is the current state of knowledge. In all cases, assume k[[z]]/ k[[t)] is a local Gextension. Proposition 6.23. (i) If G == 'll/p'n: t}ien tlier·e is an equicliaracteristic deform,ation of k[[z]]/x:[[t]] 1.uhose ge1ieric fiber has no esse11,tial 1~ami,ficatio'li (i.e.) the upper jumps u.1 , ... ~ u,n at any ramified point of the ,Qen,eric fiber satis_fy Ui+1 < pui + p see Remark 6.14(i)).
If G ( ii) ,
= Z/pn
~
Z/m is cente'rfree, t/1,en there is an equicliara,cte,,.istic defor
mation of k[[z]]/k[[t]] whose gener"'ic fiber has on e brancli poi·n t with inertia group G and 11,0 essential rarriificatio1i ( in the sense of Rer,1,a1~k 6.16(iii)), and the rest of the branch points have cyclic irz,ertia groups. Pu1iliermore, the upper jv,mps of the Z/ pn subexten,sion of tlie gener··ic fiber at tlie point witli iner·tia gro11ip G are con,gruent to tlie original upper jurnJJS of this subextension 1nodulo rnp.
(iii) Suppose G == A 4 an.cl all Z/2l1ubextensio11.~'> k[[i,]J/k([u.]] of· k[[z]]/h:[[t]]. with k[[v]] f= k[[z]] 1 liaue ;~am.ifica.tio·n J 'lt'T np 11 2: 6. Theri the·re i.s an equ.ich.a1racte1·istic defo1"1n,atio1i of k[[z]]/k[[t]] wh.0s£ ge'!ie·ric fiber· Ji.as one br·a11.ch po·i nt u,ith inertia gr·o11,p G and corres11011.dirig ra·111,·ifico.t ion, .iu111,p v  G: arid a,l l of tlie other br 2 for k·[[z]]/k[[t]] (ttJe cari tli.i nk of v as the ,·:secon.d upper· j'll'rr1p ., if v.;e count jum.ps w'ith 1n ultiplicity in. case tlie ,,arriification filtra.tion .iurnps from D 4
(iv) Suppose G
Chapte1 '2
Lifti11g of Ctrrves with Automorphisms
41
st,~a,ig/1,t to Z/2). If v > ·1 : th,e·n tliere i.s an eq·uich,a,racte1istic defom1rat·i on of k([z]]/h~[[t]] such that the branch points of the generic fiber with, inertia group G have correspondirig upper jump less tlian v, and all of t/1,e other branch point., Jiave inertia group Z/4 or Z/2 x Z/2.
Proof. Parts (i) (ii), (iii) and (i,r) are 3 ([Obul2, P1·oposition 5.9]). By Tl1corem 7.12, Y ~ Y/G cannot even lift to characteristic pn. for any n > 1.
Remark 7.15. In the exainple above~ if p bou11cl to ,show that lifting is impossible.
~
41, then one can also use the HUI·witz
;{ext we give a result on whether mi11iversal deformation rings are in fact uni11ersa.l. U11surprisi11gly) the strongest 1·es11lts are i11 the weakly ra,m ified case.
Proposition 7.16 ([BC09, Theorems 011 p. 879], [BCK12]). Let k[[z]]/k[[t]] be a local Gextension with, mini1;er·scil defor·r,1,ation ring Rzoc an,d equichar·act,eristic m,i'niversal deformation ri·ng Szoc == Rzoc/p.
(i) If k[[z]]/k[(t]] ·is u1ea.kly ramified, th.eri Rtoc is not universal if and only if 2 char·(k) == 2 an 1 but does not lift to cl1aracteristic zero'? Tl1at is: can the nli11i \"e1·sal deformation ring of a local Gextension have characteristic other than 0 or p?
As we have seen in Theorem 7 .12, tl1is is not possilJle £01· a iueakly ramified local Gexte11sion. We can also ask about obstructions to lifting to nor1pri1r1e cl1aracteristic. Question 8 . 8. Is it possible to write down any general obstr11ctio11 to lifting a local Gextensiou to characteristic p 7,, (fo1, some r,, > 1), in the spirit of the KGB or Hurwitz tree obstrt1ctions'? Of coU1·se, if the answer to Questio11 8. 7 is negative, ther1 obstructions to lifting to cl1aracteristic zero worl< ()qt1ally well as obstructions to liftil1g to cl1a1. acteristic pn for n > l. In another directio11, Vv?e have seen in Exa111ple 7. 7 that the relationslnp bet,wee11 ArtinSchreie1· equa.tions, Kumme1· equations, and deformation rings of local Z/7;actior1s is not straightforward (although Bertin and ~,fezard's proof of Theoren1 7.4(iv) is based on a. l)artial understa.nding). Having a. I(1unme1· eqt1ation for a lift of a Z/pexten.sion a.llows one t.o write dov{n its Hurwitz tree and tl1e geometr , of its branch loct1s.
Question 8.9. In tl1e ca,se where G = Z/p, can one give a11 explicit link betwe011 deformation parameters and Hurwitz trees of lifts? To w11at exte11t can Htrrwit;z t1·ees be used Lo distinguish isomorpl1ism classes of lifts? Cai1 Hur"witz t1·ees be t1sed to identify whether· or 11ot a clefor1r1aLion lies in the OortSekiguchiSuwa con1ponent (Re1nark 7.5)?
Ai1 answer to the following question would reclt1ce the ct1rrent inexplicitness 01· lifts achieved tl11·ough Theorem 6. 20. Question 8.10. Su1Jpose a local Gext.e11sion is shown to lift to chai·acte1·istic zero using a specific eciuicharacteristic defo1·mation. Can 011e say anything about the geomet1·y of tl1e b1,anch locus of tl1e 1·cs11lti11g lift? In particula1', ca.n one sa,j ' anything about tl1e resulting Hurwitz tree·? ~
It would be natural to attack Quest,io11 8.10 b)r first, a.ssu1nir1g G ~ Z/p wl1ere the st1·uctU1·e of HUI·witz trees is welltu1derstood. Saidi has called tl1e following conjectt1re t]1e ''Oort co11jecture revisited,, ([Sa112]).
Conjecture 8.11. Local cyclic exte·n.sions are liftable i1i tower·s. That is, giveri ci local Gextension W'itli k[[z]]/k[[t]] witli G cyclic arid a lift R.[[S]]/ R[[T]] of a sitbexter1,sio11, k[[s]]/k[[t]] to cha.racteristic ze·r·o there is a lift c>J k[[z]]/A:[[t]] to cliar·acle1istic zero co11tainirig R[[S]]/ R[[T]J a., a S'l1.bextensio11,.
TL.e p1·oof· of th~ Oo1·t, co11jectu1·e i11 [OvV14] already proceed. by induction) a11d one could in theory J)rove Conjectu1·e 8.11 by making tl1e i11cl11ctjo11 p1·oces' more
Lifting of Cur,,es with Automo1,.phis1ns
Chapter 2
51
flexible. In particular, tl1e induction a1gt1r11ent of [OW14] only works if one can lift a subextension so that the branch poir1ts all l1ave high e11ough valuation, see [OW14, Tl1eore1r1 3.4(i)]. If 011e could remove tl1e valuation restriction, Conjecture 8.11 would follow ( a11d one would additionally get a p1,.oof of the Oort conjecture witl1011t usi11g equicl1aracteristic deforn1ation). Removing this restriction directly see1ns n1ore promisi11g than t1·ying to 11se deformation theory in towers as in [Bysl 1] (see discussion before Theorem 7.17). 111 a11y case thougl1, it would be interesting 11 to u11derstand the defor1nation tl1eory of loca.l Z/p actions for n > I.
8 .4
Nonalgebraically closed residue fields
Throughout this entire paper, we have conside1·ed the local lifting problem over an algebraically closed field of characteristic p. Tl1ere is no reason that one cannot co118ider local Gextensions k[[z]]/k[[t]] where k is a,n arbitrary field of characteristic p , a11d try to lift them to a characteristic zero 111ixed characteristic loca.l Gj and Gi 2 GJ. The subgro11p Gi (resp. Gi) is lmown as tl1e ith h,igher rarnification group for the lower nttrribering ( resp. th,e upper numberi1ig). One k11ows that Go = G0 = G, a.nd that G1 = G¼. == P (in particular, ifr p = 0 then G 1 is trivial). For sufficiently large i, Gi == Gi == { id}. Any i such that Gi ;2 G'i+E for all € > 0 is called an upper jump of the extension L / F, Likewise, if Gi ;2 Gi+€ fort > 0, ther1 i is called a lower jump of L/F. If i is a lower· (resp. upper) jump, i > 0, anrl t > 0 is sufficiently s1nall, then Gi/Gi+e (resp. Gi /Gi+E) is a.n eleme11tary abelian pgroup. Tl1e lower jumps are clearly all integers. The HasseAr£ theorem says tl1at the upper jumps are integers whenever G is abelian (in general, the upper jumps need only be rational). The extension L/F is called tamely r·amified if G1 == {id} (equivalently, G ~ 7l/m), and wildly r·arriified otherwise.
Exainple A.7. Suppose L/ Fis a GGalois extension as above, with residue field of characteristic p > 0. Let M be the subextension corresponding to P ~ G. By the definition of the lower numbering, we l1ave Pi == Gi for i > 0. By the definition of the upper nl1ro bering, we have pi = Qi/·m . In par~ticular, if P is abelian then tl1e upper j111nps for L/ F lie in
iZ.
The degree J of the different of a Gextension L / F is given by the formula 00
8
=
L( Gi
(3)
 1).
i=O
Note that b ra1nified.
> IG  1,
with strict inequality holding if and only if
L/ Fis wildly
Example A.8. Suppose k[[z]]/k[[t]] is a Z/p·nextension with upper jumps u1 < · · · < 'Un. For i > 1, we have Ui 2:: pui1, with pf u,; wheneve1· strict ineqt1ality holds (see, e.g., [Gar02, Theore1n 1.1]). Now> let f : Y ~ X be a degreed b1·anched cover of curves over· an algebraically closed field k. The cove1" is called tamely (resp. wildly) ramified at a point y if the corresponding exte11sion OY.y/Ox,J(y ) is. Tl1e RiemannHitrwitz for'mula ([Har77, IV, §2]) states that "
A
2_gy  2
= d(2gx 
2)
+L
by,
yEY
where 8y is tl1e degree of the clifl·erent of 6Y,y/Ox,J(y), a11d gy (resp. gx) is the genus of Y (resp. X). The Riema.n n~Hurwitz formula, combined with (3), shows
Andrew Obus
54
that if Y ~ X is a wildly ramified cover of curves over k, the11 the genus of Y is higher than it would be if the cover had tl1e same ramification I)Oints and indices but was in characteristic zero.
A.4
Miscellaneous
The followi11g le1nn1a is 11sed in the proof of Tl1eorem 6.20.
Lemma A. 9. Let k be a··n algebraically closed field of cha1'acte·ristic p 1 a.nd let A be a firiitely generated 'ttV ( k )algeb1~a th.at is a dom,ai11. If m ~ A is a1i ideal s·uch tha'l A/m ~ k as a W(k)algeb ra, tlie1i tliere is an ideal I~ m ~ A s11,cli tJiat A/ I is a finite extension of W(k). 1
Proof. Embed Spec A into AW(k) ~ ?~' (k) for some n and let Z be the projective closure of Spec ..:4 in JP>W(k ) . Let x E Spec A ~ Z be the point correspo11ding to m. Since A is a do111ain, x is in the closure of the generic fiber of Spec A, and thus of Z. By [Mu1n99, II, §8, Theo1·em 1], x is the specialization of son1e geomet1~ic point on tl1e generic fiber of Let y E Z be tl1e image of this point. Since x does not lie in the l1ype1"plane at infuli ty, neither does y. So y is in the ge11eric fi her of Spec A. Since the clostu"e {y} of y is firut.e over Spec W(k) and co11tains :z;> taking I = I ( {y}) ~ A gives the desired ideal. □
z.
1
AcknowledgeII1ents I tl1a11k Frans Oort £01~ guidance and help in preparing this chapter i and Florian Pop fo1· useful co11versations.
References (Al)l157]
Slrreer·am A\Jhyankar. Cove1·i11gs of alge]Jraic curves . ..4mer. J. Math.,
#
79:825 856, 1957.
[BC09]
J akub Byszewski and Guntl1er Cornelissen. Whicl1 weakly r·amified group actions admit a u11iversal fo1·mal deformation? Ann. lnst. Fourie·r {Grenoble), 59(3):877 902, 2009.
[BCK12] .Jakub B3rszewski, Gunther Cornelissen: a.nd Fumiharu Kato. Un anneau de deforn1ation unive1,sel e11 ronducteur superie11r. Proc. Japan Acad. Ser. A Math,. Sci., 88(2):25 27, 2012. (Bel79]
G. V. Bely1. Galois extensions of a maximal cyclotomic field. Izv. Akad. Nauk; .f3SSR Ser. Mat. , 43(2):267 276 479, 1979.
[Be1·98]
Jose Be1·tin. Obst1·uctio11S locales at1 releve1uent de revete111ents galoisier1s de courbes lisses. C. R. Acad. Sci. Pa1is S er. I Math., 326(1):5558, 1998. 1
Chapte1· 2
Lifti11g of Cw·ves with At1tomo1·phisms
55
[Bl\100]
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Jose Be1·tin ai1d Aria11e Mezard. Deformations for·melles de revete111ents: un pri11cipe localglobal. Israel J. Math., 155:281 307 1 2006.
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Louis Hugo Brewis. Liftable D4cove1·s. Manuscr·ipta Math.: 126(3):293313, 2008. Louis Brewis. Ra1nification theory of the ~adic ope11 disc and the lifting problen1. Ph.D. Thesis, available at http://webcloc.sub.gwdg.de/ebook/dissts/ Ul1n/Brewis2009.pdf, 2009.
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Lo11js Hugo Brewis and Stefa11 Wewers. A1'tin cl1aracters, Hurwitz trees and the lifting p1·oblen1. Math. Ann., 345(3):711 730, 2009.
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G11ntl1e1· Co1·nelissen a11d Fumihar11 l(ato. Equivariant deformation of Ml1mford curve~ a11d of ordinary curves in positive cl1aracteristic. Duke Math. J. 1 116(3):431 470 2003.
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[I 2 there are 011ly finitely 1nany torsion poi11ts of J 011 C c J( C). Jvlore generally, 0I1e ca.n work with an arbitrary abelia11 variety A and a11 arl)i t.raI'Y sul)variety V. A translate b + B of a nonzer·o abelia11 subvar·iet)' B C ..4 by a torsion point b E Tors(A) contains infinitely torsio11 points. \1/e refer to such sub varieties as special subvarieties. It tur11s out that also the converse holds:
Theorem 1.1 (the I\1aninMumford conjecture) . Suppose k is an algebr·aically closed field of character·istic zero, A is an abelian variety over k, and S C A is a subvariety such that S n Tors( A) is Zariski dense in S. Then the1·e exist a tor·sion point b E Tors( A) and an abelian s'itbva.riety B C ...4 s·ucli thltt S = b + B. There l1ave been n1any?' proofs of tllis tl1eorem. A partial result was pro\'e11 by Bogomolov [7] . The first ftill proof was then given b)' Ra)rnaucl, see (47] 1 (48]. Later proofs and generalizations we1·e given by ma11y people, e.g. Hind1·y [19], [20], Hrushovski [21], a.n d Pi11k and Roessler [41]. We make a rernark on the origi11al proof by Raynaud. Clearly any point on a.n abelian variety C over the algebraic closure of a .finite field is a torsio11 point. He11ce £01· c\ny subvariety Sc Cover we see that SnToI·s(B) = S(i1) is dense in S ( a.n d we an easy analogue of the theoren1 does not hold in positive characteristic). Hence it seen1s that !'recluctjon 1nodulo pn does not lead to a, proof of the t.l1eoren1. However in (47] Raynaud's idea is to red11ce mod p, a.nd then study whether the lifting to the tl1e 1·eduction 1nocl p 2 lies 011 " S mod p 2 ". This ca11 be app1·oached
n
n
Cl1apter 3
Tl1e AndreOort Co11jectu1'e
63
algebraically by lifting the Frobe11ius automorphism to an automorphis1n modulo p 2 a.nd characterizing the torsion points as tl1e fixed points. This iclea con1es back several times i11 history: In [10] in 1986 Dwork and OgtLS use this iclea to sl1ow that the cano11ical lifting of a general ordinar)' Jacobian of dimension g ~ 4 in chai,acteristic p is not a .Jaocobian (11ote that 011e needs g ~ 4 for dimension reasons). In [9] in 1987 Deligne a11d illusie use this idea to study De Rl1am col1omology. Moreover, Be11 Moone11 (32] used tllis san1e idea to deduce partial results on the AndreOort conjecture. At present a 1·ather elementru:y proof of the :Niani11Mumford conjecture, a proof using basic n1ethods of algebraic geometry, is available. For example see [49].
2
The AndreOort conjecture
In generalizing the Ma.n ini\ l11mford conjecture, we start with a Shimura va1·iety S and a1·gue by a11alogy. As a special case, one can consider S = X (1 )n, the nth power of the rnodular cm·ve, or S == A 9 , the moduli space of principally polarized abelian varieties. We define a point x E S(C) to be special if x lies on a 0din1ensional subvariety of Hodge type in tl1e sense of [31]. Sinrilar1y, we defi11e a subvariety V c S to be a special subvariety if V is a subvariety of Hodge type. It is clear that special subva1·ieties have a Zariski dense (in fact dense in the a1·chin1 .1deaJ1 topology) subset of special poir1ts, and the AndreOort conjecture asserts that the conve1·se is true: 1
Conjecture 2.1 (The AndreOort conjecture). Suppose k is an algebraically closed field of characteristic zero, S is an Shimura variety over k 1 and V C S is a subvariety whicli contains a Zariski dense subset of special points. Does this imply v is a special subvariety? Note the formal analogy that can be made between the AndreOort and MaoinM11mford conjectures. In fact, these two conjectu.I·es can be con1bined by studying Mixed Shimura Varieties  whicl1 can be thot1ght of as Moduli of mixed Hodge structures. A typical case to keep in mind is the universal abelian variety X 9 over A 9 , and a special point is a torsion point in abelian variety fibered over a CM point of A 9 . Thus, the corresponding conjecture formally encapsulates both AndreOort and Ma11i11Mlunford fo1~ CM abelian varieties. 111 recent work, Gao [17] has reduced this co11jectU1'e to a statement about, Galois orbits on pure Shimura Vru:ieties.
For questions arou11d the Colema11 conjectt1re and special subvarieties in the Torelli locl1s see [38].
64
Frans Oort and Jacob Tsi1ne1·1nan
3
Special subvarieties are linear in SerreTate canonical coordinates
3.1. Suppose ~ =:) JFp is a perfect field and suppose Ao is an ordinary abelian variety of dimension g over ~. The deformation (lifting) theory of Ao is described in SerreTate canonical coordinates: Def(Ao) ~ Spf(W00 (~)[[ti,j 1 < i,j ~ g]]), such that the canonical lifting is identified with origin) see [22]. Let us write xo E A 9 ® IFP for the corresponding moduli point for some polarization on Ao. There is an isomorphism
In this description quasicanonical liftings (CM liftings, abelian varieties isogenous with the canonical lifting) are torsion points in this group structure. Theorem 3.2 (Noot, Moonen). Let S C A 9 ,zp be an algebraic subscheme, a'n d consider 5/xo =: S 1 c Ag 0 Woo(K,).
With this assumption S is a special subvariety ( a Shimura variety) if and only if S' is the translation by a torsion point of a formal subtorus in the Sere Tate coordinates. See [33, 2.2.3], [34, Thm. 3.7], [30, Thm. 5.2], [31, 5.5] for more details.
The AndreOort conjecture: recent results
4 4.1
Conditional on GRH
Edixhoven in [11], [12] gave a proof of the AndreOort conjectu1'e for products of modular curves conditional on the Generalized Riemann Hypothesis (GRH), and this stategy was used to give a proof general case assuming GRH by Klingler, Ullmo, and Yafaev [54], (25]. The idea runs through intersection theory and proceeds roughly as follows:
(1) By GRH, CM fields have small split primes, and thus there are low degree Heckeoperator which have CM poil1ts as fixed points.
(2) The Galois orbits of CM points are related to class groups, and by GRH, one can show that these Galois orbits are large. Thus if a variety V co11tains one CM point, it necessa1~ily contains 1nany such points.
(3) Putting (1) and (2) together gives that V
n Tp V
contains too many points
for Bezout 's theorem, so tl1ey do sl1are a co1npo11ent.
(4) A topological argument sl1ows that if V c Tp V then V is a special va1'iety.
Cha1Jte1, 3
4.2
The AndreOort Conjecture
65
Unconditional results: The PilaZannier method
In [43] Pila and Zannier gave a new p1·oof of the ManinMumford conjecture wl1ich p1·oceeded tl1rough model theory · specifically, ominimality. Crucial to thei1· approach was the PilaWilkie theoren1 (42] which. says that transcendental varieties cannot contain ma11y rational poi11ts witl1out containing algebraic subvarieties as ~"ell. The way transcende11ce theo1·y enters the sto1"y is via the uniformization map, which is transcendental but which pulls back special varieties to algeb1·aic varieties. Based on this a.p proach, Pila [44] gave an t111conditional proof of the AndreOort conjecture for· products of n1odttlar curves. Since then, this strategy has been used (44], (45], (24), (52] to give an unconditio11al proof for the case of A 9the moduli space of pri11cipallypolarized abelian varieties of dimension g. The two main ingreclients are as follows:
(1) A t1,ansce11dence resu]ts that shows that besides (weakly) special varieties, the uniformization n1ap from Siegel uppe1~ half space to A 9 is very transcendental: tl1at is, the Zariski closure of the image of a.n y algebraic subvariety of Siegel space is weal 0, depending only on S. Fo1.. mo1·e on tl1is proble1n and its relation to torsion in class groups, see [51], [54], [52].
3. (Heigl1t B01111ds) Let h, be a logarith1uic Weilheight on 11 ,ve know tl1e list of cow1terexa1nples is cornplete. For othe1~ families of Galois covers we do 11ot la1ow whetl1er the lists are complete.
76
FJ:ans Oort
Note that we have no co11nter examples to the Colemai.1 conjecture i11 all other cases along these li11es: we do not. know a11y exa1nple of a family C  t S clefuti11g an (ST) variety (for g > 3) such that the generic fiber is not a Galois cover (say, a fa.mily with tri,,ial automorphism group for the geometr·ic gene1·ic fiber). I have no idea wl1ethe1~ this is just lack of insight, or whetl1er finally this will be the true a11d correct status of tlie pro ble1n.
4
Weyl CM fields
This new ter1ninology was introduced in [12], altho11gl1 the concept already existed before.
4.1. Let L be a CM field of degree 2g over Q. Let L 0 :J (Q be the Galois closure of the maximal totally real subfield Lo :) (Q and let L"' :J Q be tl1e Galois closure of L => Q. Easy fact: where we write S 9 for the symmetric group on .9 letters.
Definition. A C1'I field of degree 2g is called a Weyl CI\1 field if
if tllis is the case, we have:
N
~
(Z/2) 9
is normal in
G := Gal(Lrv /Q),
and
G/ N
~
S9 .
We say A is a Weyl CM abelian variety, if E11d0 (A) is a Weyl Cl\,1 field . We say C is a Weyl CM cu1've if J(C) is a Weyl CM abelian va1·icty. We write WCM(A9 ) I·espectively WCM(M 9 ) = WCM(Tg) for tl1e set of isomorphism classes of principally polarized Weyl CM abelian varieties, respectively vVeyl C~1 Ja.cobia11s. Note that if for a stable curve Cits Jacobian J(C) is a vVeyl CM abelian variety, then C is irreducible, i.e. WC1v1(½0 ) = WC1'1(7g). Note that a Weyl CM abelian variety over C is simple; because of tl1is choice of the definition of a Weyl CM abelia,11 variety we see tl1at a.11 abelian va1·iety isogenous to a product of a.t least two factors, e.g. all of tl1en1 Weyl CNI, we do not call a Weyl Ci\~ 1 abelia11 variety. Note tl1at for g = l every CNI field is a \1/eyl CM field. Ho\\1eve1· >fo1· ever·y g > 1 there are many CM fields that are not a Weyl CNI field; e.g. every cycloto1nic field of degree at least 4 is not a Weyl CM field.
4.2. Theorem (C.L.Cl1ai FO) . For evenJ g E Z ~ 4J
See [12], 3.7.
Chapter 4
Special Subvarieties in the Torelli Locus
77
4.3. uMost Cl\11 fields are We)'l CM fields',· this can l)e ma,d e precise, either by comp11ting densities in the set of all Ci\1 fields, see (42], also [40], [41], [29], or by considering this as density,. of closed ''Weyl points'' in a family over a finite field with large monod1·01ny, see (14] . We see the appare11t cont1·adiction that for '
Lo
S9 1 >
L0
~ 7l'ite G :== Aut(Lo I 1\10), Then
Aut(Lo / Lo) ~
Sg1
CG C Aut(Lo /Q)
~
Sg.
By the previous lemma we conclude
either Aut(L0 /Lo)= G or Aut(£ 0 / Lo) ~ G ~ 5'9 .
If Aut(L0 /Lo)= G 1 we arrive at l\1o = Lo, l1ence A1 = L. If G = S 9 this in1plies 1YJ0 = Q; we conclude tha.t L equals tl1e compositt11n L = Lo· ..Alf beca.use Af is imaginar·y quadratic and ror1tai11ed in L. This i1nplies Lrv == L 0 l\.1 is of degree 2·(g!), hence ,q = 1, and M = L. □
Let C be a c11ir·ve of genus at lea.st 2. Suppose Aut(C) i={1} (in case C i.s rio12.hyper'elliptic) , ·espectively A ut( C') f:= { ±1} (in case C is hyperelliptic). S1.1,ppose J (C) is absol1.Ltely simple. Then J (C) is not a T,Veyl Ci\1.1 abelian variety, i.e. C 'is not a lVeyl C1'1 c·urue. In other terms:
4.6. Corollary.
78
Frai1s Oort
a Weyl Cl\1 curve does 11ot have 11011trivial at1ton1orpl1is1ns.
Proof. Suppose J(C) is absolutely simple. Let ±1 =/
0 allou1s a finite sepa1·able mo·rphisrri C + IP1 branched in one point (even if the field of definition of C is of lugh tra11scendence degree over lFp), see [37]~ pp. 91/92 and (91], Corollary 3 on page 715. See [66]. _ We k11ow that a CM abelia11 va1,iety i11 cha1·acte1·istic zero is dcfi11ed ove1· Q; however, as G1·otbendieck showed, a CM abelian vai·iety in positive characteristic is isoge11ous to an abelian variety defi11ecl ove1~ a finite field, see [62), (90]; there are 1r1an)r exan1ples of C'.Nl a.belia11 'uarietie.s in positive clla1·acterit;tic not defined over a finite field. Tl1is leads to interesti11g, natural stratifications of A 9 ® IFP that l1ave no cot1nterpar·t i11 cha1·acte1·istic zero. It is as if we a1·e in a completely different we .rlcJ.s in zero cba1"acte1·istic versus positive characteristic. Tl1e fruitful method of ''recluction modulo p" has some disappointments, but also some ve1·y positive results (on CNI liftings of· abelian varieties) as we see in
the next sectio11.
6
Jacobians in Inixed characteristic
For an abelian va1·iety Ao in positive characteristic tl1at is ''or·dinary'' Serre and Tate defined a canonical lifting (78] and canonical coordinates arotmd sucl1 a n1oduli point in n1ixed cl1aracteristic, see [36]. Does this allow tis to const1·t1ct n1any CivI curves? Just take an algebraic curve Co with ordinary Jacobian Jo, a11d asl< whether tl1e ca1101lical lifting of that ,Jacobia11 (with the canonical polarization) is tl1e Jacobian of ru1 algebraic curve in characteristic; if so, we encotu1ter a CNI curve i11 characteristic zero. For all ctrrves with g = genus(C) < 3 this indeed is the case. Howe,1c1·, tl1is app1·oach/question originally suggested by Katz l1as a negative a11swe1~ for .CJ > 3, see (20] [71]:
for ever·y g ~ 4 ther·c exist ordi1iary curves Co over finite .field s~tch that the canoriical lift of .Jo = J( Co) is riot a Jacobia.n in c/1,ar·acteristic zero . The method de,reloped b)' D,¥ork and Ogus: (20] shows that most canonical liftings of (ordinar)r) Jacobians ove1· a fir1ite field do not give a a Cl\1 J acobia11 in characte1·istic zero. This 1·es11lt has bee11 successfully 11sed for givn1g aJ1swers to 2.3 in several ca~es, sec [34], [53], [45].
80
F1. a11s Oort
\1/e now have a reasona.b le tmderstanding of CI\1I liftjngs of abelia.n ,,a1:ieties il1 general see [13], but this does not gi,re results we would like to have for Cl\1 Iifti11gs of algebraic curves.
I do not know a method in positi'lJe cha·ra.cteristic enablin,g us to prove existe1ice of infinitely m,ariy C11I curves in characteristic zero. Again we see differences between properties of a curve on the 011e l1ancl and properties of its Jacobian as a.1J abelian variety 011 the other ha.nd.
7
Appendix: CM abelian varieties
1.1. We say that L is a. CM field, if L is a nlunber field, say of degr·ee [L : Q] = 2g, which contains a su l)field Lo C L such tl1at Lo is totally real, i.e. every embedding of L into C sends Lo to a subfield of JR c C 1 with [L : Lo] = 2 and L is totally complex, i.e. no e1nl)edding of L int.a C sends L onto a subfield of JR. Equivalently: under every embedding cp : L ~ C complex conjugation on (C leaves Gi+l} over K to the smooth Zpsheaf Tp( G) = lim . Gi over ( i K. In the case when K is a perfect K and p E Kx, Tp(G) is determined by the Gal(K / K)module lim. Gi(K). ( i However if p is the cl1aracteristic of the base field K we should avoid using tl1e notion of a "padic Tate module", but the notion of a pdivisible group remains useful. In this case the notion of a pdivisible group is much more than ''just a Galois representation". Over an arbitrary base scheme S one can define the notion of a pdivisible group over S; for details see [37]. For an abelian scheme A ~ S indeed X = A[p00 ] is a pdivisible group over S. This enables us to use this notion in mixed characteristic, even when the residue cha1·acteristic is p. An isogeny of pdivisible groups 'l/J : X ~ Y over a base scheme S is a faithfully fiat Shomomorphism with finite locally free kernel; such an isogeny 't/; induces an isomorphism X /Ker( 1/;) rv > Y. A nontrivial pdivisible group X over a field K is said to be simple if for every pdivisible group subgroup Y C X over K we have either Y == 0 01~ Y == X. 2
1.3. A basic tool for studying pdivisible g1,oups over a perfect field ~ of characteristic p is the tl1eory of Dieudonne modules, which gives an equivalence of categories from tl1e exact category of pdivisible gro11ps over K to a suitable category of plinear algebra data; the latter is collectively known as "Dieudonne modules''. There are many Diedonne theories, each with a distinct fla.v or and range of validity. We refer to [50], [66], [9] and [18] for contravariant variant Dieudonne theories, to [46], [52], [112], [115], [113] for covariant Dieudonne theories, and to [2, App. B3] for a disc11ssion of the vario11s approacl1es. 2A
simple pdivisible group X has many no11trivial subgroup schemes (such as xtpn]); here "simple" refers to either the category of pdivisible groups, or the category of pdivisible groups up to isogeny.
Chapter 5
Moduli of Abelian Varieties
101
We use the covariant theory i11 this article; see (3, 479 485] for a summary. For a perfect field K. :) lFP, let A == A 00 ( "') be the ring of padic Witt vectoi·s 3 with entries in K., and let u E At1tring(A00 (K.)) be the natural lift to A of tl1e Frobenius auto1norphisn1 xo r+ Xb on K on /{;. We have tl1e Dieudonne 1 ing RK, which conta.ins tl1e ring A a.nd two elements :F, V, such that the follows relations are satisfied: 1
F· V
=p=
V·:F,
F·x =
Xu ·:F,
X· V
== V·Xu,
Vx EA.
vVe define R"' to be the noncom1nutative ring generated by A(1t), :F and V and the relations given in the previous sentence. (The completion of R for the linea.r topology attached to the filtration by right ideals vn.R, n E N, is the Cartier ring Cartp(K) fo1·"' used in Cartie1· theory.) The ring RK is commutative if and only if "'1
== lFp.
To every finite commutative Kgroup scheme N (respectively every pdivisible groLtp G) over K,, there is a left RK,module ][l)(N) (respectively a left R"'module fil( G)) functorially attacl1ed to N (respectively X). The Amodule 11nderlying the Dieudo11ne module Il))(G) attached to a pdivisible group G over "' is a, free Amodule whose rank is equal to the height ht( G) of G. Conversely every left R1tmodule whose underlyi11g Amodule is free of finite rank is the Dieudonne 1nodule of a pdivisible group over K.. The Amodule u11derl3ring the Dieudonne module ID>(N) of a commutative finite group scheme over K. is of finite length over A and the le11gth of ]IJ)(N) is equal to the order of N. Every RKmodule whose und rJying Amoclule is of finite lengtl1 is the Diedonne module of a commutative fi1ute group scheme N over ""· Moreover the map from homomorphisms between z>divisible g1·oups over "' (respectively comml1tative finite group scl1emes over K) to R~n1odule l1omomo1·phisms between Dieudonne 1nodules is bijective, so that the Diet1donne functor defines equivalence of categories botl1 for pdivisible groups a11d for conunutative finite group schemes over Ji,. In (50] the contravariant theo1·y is defined, t1sed and developed. The covariant theor)' is easie1· i11 a nu1nber of situations, especially ,vitl1 respect to Cartier theory a11d the theor·y of displays. Up to duality the covariant and contravariant theories are essentiallj.. eq11ivalent, so it does not make n1t1cb difference wltlch is used to prove desil·ed results. Tl1e covariant Dieudonne theory commutes with exte11sion of perfect base fields: if '""  ? ~1 is a ring l1omomorphism between pe1·fect fields of characteristic p, then there is a functorial ison1orphisn1 ID>(..tY) ®A (") A00 (K 1 ) .r,>, Il))(X"'1 ) for e,..,ery pdivisilJle g1·oup X over Ii, whe1~e X".1 := X Xspec(~) Spec(tt 1 ). In partict1lar we have a functorial isomorphism ID(X
) c"' l[))(X) ®A,u A, where X (P) is the pullback of X b)' the absolt1te Ftobe11i,1s morphisn1 FrK : Spec(K) ~ Spec(K). Note tl1at tl1e relative Frobeni11s homon1orphisms Fi:a;K. : G ? Q(P) ancl tl1e cano11ical isorr1orphisn1 ID>(X(P)) (X) ®A)o A i11d11ces the a 1linear operator V 011 the cova1·ia.1Jt Die11do1111e module KD(G). Similarly tl1e Ve1·scl1iebung Vera;K : G(P) ~ G indt1ces the olinear operator F on JD)( G). For this reaso11 we use different fonts to ciist.i11g11ish tl1e morphisms F and V (on gi·oltp schemes) from the operator V and ;F ( on co,rariant Dieudo11ne mod11les). Fo1~ a11 explanation see (76, 15.3]. 3The usual notation for t}1e Witt ring is W. However we ha,,e ,1sed the letter W to denote
Newton polygon strata (see below), l1ence the notation A for the Witt ring.
,
102
ChingLi Chai and Frans Oort
Some examples. (i) The Dieudonne module of the pdivisible group Qp/Zp over a perfect field r;; :) 1Fp is a rankone fi..ee Amodule with a free generator e such that V·e == e and :F•e == pe. (ii) The Dieudonne module of the pdivisible group [l.Jpoo over a perfect field
=> 1FP corresponds to a free rankone free Amodule with a free generator e such that V e == pe and F·e = e.
Ki
(iii) For a pdivisible group X of dimension dover"", the quotient ID>(X)/(V ·
]])(X)) is a ddi1nensional vector space ovet·
K:
Lie(X) ~ ]])(X)/(V · Il))(X)).
In particular dim(X) == dim~(Il))(X)/(V · ~(X))). For coprime nonnegative integers m, n E Z>o we der1ote by Gm n the '/>' divisible group over ]FP whose Dieudonne module is
]I))(Gm,n)
= RwP/ RwP·(Vn  :rn)
It is easily checked that dimJFP ( RwP / (V · Rwp + RIF p • (Vn  ;:m))) == m and that 1 ]I))( Gm,n) is a free Amodule with the image of 1, V ' ... 'vn' F 1 ' ... ';:ml as a set of free Agenerato1~s. In other words dim(G1n,n) == m and ht(Gm,n) = m + n. One can check that pdivisible group Gm,n remains simple wl1en base cha11~ cl to any field of characteristic p, or equivalently to any algebraically closed field of characteristic p. We will ,vrite Gm,n instead of Gm,n ® K if the base field K ::> IFp is understood. 1.4. Theorem (Dieudonne, Manin [50, p. 35]). Suppose that k = k ::> IFp is an algebraically closed field. For any pdivisible group X over k there exist coprime pairs of non . . negative inte_qers (m1, n1), ... , (mr, nr) with m1
mr
 lF P assigns the isomorphis1n class of the polarized pdivisible g1·oup (A[p00 ], µ[p 00]). This invariant gives rise to a foliation structuI·e on A 9 ~Fp, called the central foliation. For an algebraically closed field k :) JFP and a kpoir1t xo == [(Ao, µo)] E A 9 ( k), the central leaf C(xo) with 1educed structure is a locally closed reduced subscheme of Ag ® k such that for every algebraically closed field k' containing k, the set of all k'points of C (xo) is the set of all isomorphism classes [(A, µ)] E A 9 ( k') such that the polarized pdivisible gro11p ( A [p 00 ], JJ, [p00 ]) is isomorphic to (Ao [p 00 ], µo [p00 ]) x Spec(k) Spec(k'). The existence of a locally closed subscheme 1
C(xo) of Ag ®wk witl1 the above properties is a basic fact for the central foliation p structure of Ag ® IFP. See [79] and also [82]. We mention several facts. See §4 for more information.
• The whole ordinary locus Wp 9 , 29 in the moduli space A 9 ,1 ® JFP of pri11cipally gdimensional abelia11 varieties is a single central leaf. • Let ~1 == NP((g  1) * (0, 1) + (1, 1)) be the symmetric Newton polygon of dimension g and !rank g  l. Then the Newton polygon stratun1 W~ 1 , sometimes called the "almost ordinary locus", is a single leaf.
• An open Newton polygon stratum
contains an infinite numbe1· of central leaves. • Every central leaf in the supersingular· Newton polygon stratum Wug (A9 ,d ® Fp) of the moduli space Ag,d ® IFp is finite.
2.2. Remarks. (i) Isogeny correspondences in characteristic p over a Newton polygon strat11m We(Ag ® Fp) ir1volves blowup and blowdown in a rather wild patte1·n in general. When restricted to central leaves such isogeny correspondences become finitetofinite above the source cent1~a1 leaf and the target central leaf. (ii) Unlike the case of central leaves, tl1e dimension of Newton polygon strata i11 Ag,d ® lFP associated to the same Newton polygon ~ in general depends on the degree of the polarization in consideration. Similarly the dimension in A 9 ,d ® 1FP of EO strata associated to a fixed isomorphism class of BT1 g1~011p generally depends on the polariza.t ion degree. (iii) The padic inva.r iant (A,µ)~ J(N(A[p 00 ],µ[p 00 ])) defines a. stratification of Ag)d ® lFp by the p1·ank, which is coarser than the Newton polygon stra.t ifica.tion. For any integer f with 0 < f ::; g, Every irreducible compo11e11t of the stratun1 of Ag 0 jFp witl1 prank .f is equal to g(g  1)/2 + f; see [63, Theo1·e1n 4.1]. (iv) The dimension of an u~reducible component of a central leaf contained in a given Newton polygon str·atmn W€ ,d c A 9 ,d ® lFp is determined by~ a11d does not depend on the polarization degreed; see [79], [82], 4.13 ancl 4.5(ii).
Chapter 5
Nloduli of Abelian Varieties
109
(v) l11 (79] we also fi11d the definitior1 of isogeriy leave.s, wl1ere it is sl1own that e,rcry irreducible component of We(A 9 0 lFp) is the i111age of a finite morphism wl1ose source is a finite locally free cover of an irreducible component of the p1,.odt1ct of a central leaf a11d isoge11y leaf; see (79, 5.3]. A wa.1·ning: 1""he name ~'isogeny leaf" may seen1 to suggest that isogeny leaves in A 9 ,d ® lFP are defined with the same paradigm in 2.1, using a s1titable ir1varia11t for polarized pdivisible groups. Tlus optimistic hope does not hold, but there is a11 invaria.11t \\•hicl1 is closely· related to isogeny leaves. Namely, to every polarized abelian variety (A,µ.) in chru.~acteristic p one associate its aiRogeny class· two polarized (A 1 ~ µ 2), (A 2, J.t.2 ) are said to be in the sarr1c aisogeny class if tl1ere exists a11 aquaa~iisoge11y 1j; : A 1  ~ A2 such that 'lj;* (µ2) = µ1 . See 4. 7 .1 for the definitior1 of aqttasiisogeny and 4. 7.2 fol' the definition of isogeny leaves. (v·i) Clearly every central leaf C([(Ao, µo)]) in A 9 ® Fp is contained in the open Newton ~lygon stratum W{ wl1ere € == N(A 0 [p 00 ]). Similarly every central leaf in A 9 ® 1Fp is co11tained in a single EOstratum. Tl1e question whether an EOstratU111 is equal to a central leaf is answered by tl1e tl1eory of 111inimal pdivisilJle groups; see 1.5. The incidence relation between Newton polygon strata and EOst1'"ata is 1nore complicated. A NPstratum may intersect, se,1 eral EOst1·ata, and a.n EOstratun1 111ay intersect several NPstrata; see 10.3. 2.3, Gene1·all)r tlie st1·ata or leaves of st1persing11lar abelian varieties in rnoduli spi1t:es of abelian varieties bel1ave quite clifferently from strata 01· leaves for 11onsupe1·si11gular abelian varieties. What has been proved 110w inclt1de:
supersingular NPstrata and EOstrata iri Ag,d ® 1Fp ar·e reducible for p >> 0 (Hasse, Deuring, Eichler, Igusa for elliptic curves, Kats11raOo1·t, Li Oort, Hashlmoto Ibtikiyama, Ha1·ashita. for l1igl1er di111ensional abelian varieties), 1
e.g. see [35]:
[31], (34] [40]: [48] and man)' other references. Bt1t
nonsupersingular NPstrata an,d EOstrata. in A 9 , 1 ® Fp, and nonsupersing,ular cent1·al leaves in Ag,d®IFp are irreducible (Oort~ EkedahlVa11 der Geer, Chai Oort). 1
Note that we l1avc abuse the adjectives 'tsupersingular" and "11onsupersingular'' al)ove. StrictlJr spealIFp. • For an separated scheme T of finite type over a field K we write Ilo(T) for the set of irreducible components of Tk, where k is an algebraically closed field conta.ining K. This definition is i11dependent of the choice of the algebraically closed field k => K.
3.1. Let S be a scheme in characteristic p. For any abelia11 scl1eme A ~ S (or respectively any pdivisible group X  t S) and a symmetric Newton polygon ( (respectively a Newton polygon (), we define
W~(A ~ S) := {x ES N(A[p
00 ])
= ~}, W
S) := {x ES I N(A[p
00
])
:$~}
and
W~(X ~ S)
:==
{x ES N(X) = l}, W:s S is gfc. Does tl1e sa1ne hold for· tJ1c extension T 1/ 5'' '?
Example 2. Let x: be an algebraically closed field and let S 1 beakscheme of finite type. Suppose T 1 / S' is a smooth prope1· 1norpl1is1n wl1ose fibers are geomet1·ically connected algebraic curves of genus g ~ 2, S 1 is 1·educed, ai1d the restriction T' x s S + S of T' + S' to a dense open subscl1eme S c > S' is gfc over /3. Then T 1 / S' is gfc: the assumption implies that the moduli map S' ~ M 9 to the coarse moduli space M 9 is co11stant when restricted to S. Since S 1 is reduced and S c > S 1 de11se. so the rnoduli n1ap 011 S' J M 9 is consta.n t. The san1e holds for a.n y other situation v.rhere there a1·e good coarse n1od11li spaces, e.g. fo1· polarized abelian ,,a,1·jeties. 1
Example 3. Finite flat group schemes. Here is a gfc fan1il)r where specializations gives a nonisomorphic fiber. We will use for an arbitrary prime nun1ll r p tl1e description in [100] of finite loci1.lly fi·ee group schemes of orde1~ p. Not ~ tl1at for the special c~asc p = 2 there is an easier description that wo1·ks very well. We choose S' = Spec(K[t]) => S = Spec(K[t, l/t]) with I 1Fp and consider a== 0, b = t, hence ab=  p = 0 E K[t]. Tl1e11 there is a finite, fl,it grottp scheme Ta.b ==
N' ~ S' == Spec(K[t])
of rank p. The scl1cme structure is give11 by Spec(J([t, r])/(rP  ar). The comultiplication on tl1e bialgebra (the grottl) struct11re) is dete1·n1inecl lJy b == t, see [100]. Fo1· p = 2 it is give11 by
s(r) = r ® l
+ 1 ® , + b·r ® r.
We see for any val11e t r+ t 0 E K \Ve have tl1at to =I= 0 in1plies To~to ® St ~ Jlp• and To,o ~ ap. We see ··geometrically Jt,p moves in a fa111ilJ,:· and it 8pecializes to a:.p. Alt'early for surl1 cases tl1e 11otion gfc is interesting. Many mo1~e examples can be given (e.g. i11 the theoIJ' of EO strata: maximal gfc families of BT 1 group scl1emes). In general it is not easy to decide which clegeneratio11s car1 hapJ)en at boundary points of gfc fa111ilies of fi11ite group scl1e111es.
Example 4.. pdivisible groups. We will defir1e "ce11t1·al leaves" as "maximal gfc fa.milies'' over a reduced base. We will see interesti11g strt1ctures are desc1·ibed i11 this way. Here is a first inst1·t1ctive case: tak · a fa111ily £ > S of ordinary elliptic curves in characteristic p with nonconsta11t jinvariant. The associa,tecl pclivisible group X = E [p 00 ]  t S is gfc · we see alreadJ' here a.n interesting n1ix of aspects of Galois theory and of otl1er properties.
Cha.pte1· 5
115
11oduli of Abelian Va1·ieties
St1ppose E 0 is ai1 ordi11ary elliptic Cl1rve over K, a11d let E ~ S == Spf(I (A, ;\)[a]r1 whicl1 c,:1,11 be lifted ctale locally over 11 to ison1orpl1is111s (A, A)[ai]T, ""') (A, A)[ai]rz\ fo1· all i ~ 1. Let U2 : T? T1
be tl1e stal)ilized Ison1scl1eme Iso1n;µ1 ((A A)[pri] ., (A, A)[p"·]) over T 1 , wl1ich is a. st to1~sor over T 1 for tl1e stabilized A11tgro11p scl1en1e Aut ((A, A)[p',,]). Here (A, A) de11otes the unive1·sal polarized al)elia11 sche1ne over A.qfd1 . The composition u := ·u2 o u1 : T + C( .4 A)[voo] (A9,d) is a finite locally free cover and over T we have a iHon1orpl1is1n from to (A A)[apn]T to (A, ..\)[apr1 ]r. Tl1e morphism v : T + C(B.µ)[p cc] (Ag.d 2 0 ~) is defined by an isogeny 1/J1 : (A, ,\)r + (B, µ) of polarized alJelian schen1es ove1~ T whose lteruel is cor1tained in A[ap11.], sucl1 tl1at tl1e quasiisogeny 1/J = (bp111 )  11/; 1 respects the polarizations? and there exists a, geomet1ic poi11t [ of T over [(A, A)] sucl1 tl1at "Pl coi11cides witl1 'tj;. (ii) The invariant c(oint xo E A 9 .d(1Fq), a11d q == pr is a power of p. Assume for· simplicity that the pr'ank of Ax 0 is 0) i.e. neither O nor l is a slope
o.f Ax0 [p
00 ].
The central leaf C(xo)d is a sn1ooth locally closed subva1·iety of Ag ,cl ® lFq· Let x = [(Ax, J.lx)] E A 9,d ( n) be a geon1etric point lying over a 1naximal point 'T/ of tl1e IFqscl1eme C(xo)cl, where n iH an algeb1·aicaliJ' closed field which contai11S the function field IF q(rJ). Clearl)' for every ni E N, the base cha11ge of [(Ax, µx )] 1n1der tl1e absol1.1te F1:obenius t ;"7 tqm of f2 is also a geometric point x(qTn) :== [(A1qm) µ~qm))] of C(xo)d. Tl1e set of all these points x 1}.
• minsd(~) :== Min{dim(T)
• maxsd(~)
4.16.2. Theorem. (dimensions of open Newton polygon strata) For any symmetric Newton polygon we liave
e
minsd(e)
= sdim~ = # (~( 1) we have: (W r.r ( a = 2)) Zar == W o ( a 2:: 2) , and (vl' ,_,. (a. = 2) )Zar is connected a11d of codi111en.'iion one in HIer; see [26] fo1 this and more general results on strata in W17 given by the anumber; also see [27], (29], (30] for more information on EO strata in Newton polygon strata.
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[92) M. R.aynaud  Courbes .sur 1.tne variete abelienne et points de torsion. Invent. :rviatl1. 71 (1983),
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.
Open Problems in Arithmetic Alge1Jraic Geometry ALl\146, Ch. 6, pp. 179 207
© Higher Education Press and International Press BeijingBoston
Chapter 6 Current Results on Newton Polygons of Curves Rachel Pries*
Abstract
Tl1ere are open q11estions abo11t which Newton polygons and EkedaWOort types occur for Jacobians of smooth curves of genus gin positive characteristic p. In this chapter, I survey the current state of knowledge about these questions. I include a 11ew result, joi11t with Karen1aker , wlLich verifies, _for each odd prime p, that there exist supersingular curves of genus g defined over Fp for infinitely 1nany new values of g. 1 sketch a proof of Faber and van der Geer's theorem about the geometry of the prank stratification of the moduli space of curves. The chapter ends with a new theoren1 1 in which I prove tl1at questions about the georr1etry of the Newton polygon and EkedahlOort strata can be reduced to the case of prank 0.
The author would like to thank the organizers of the conference 11oduli Spaces and Arithmetic Geometry (Leiden, November 2015) and thank Achter, Kare1naker, Oort, and the 1·eferee fo1" helpful conversations. The author was partially supported by NSF grant DMS1502227.
1
Introduct io n
Suppose X is a smooth projective connected c11rve defined ove1· a11 algebraically closed field of p1·i111e characteristic p. The Newton polygon and EkedahlOort type are invariants of the Jacobian of X. I11 gene1·al, it is not known whicl1 Newton pol}..gons and Ekedal1lOort types occ11r for Jacobians of smooth curves. More • Raebel Pries, Colorado State University, Fort Collins, CO, 80521 . Email: rachelpries ~gmail.com, http://wwv.math.colostate.edu/ pries/
180
Rachel Pries
generally, it is not know11 how the open Torelli locl1S intersects the Newton polygor1 and EkedahlOo1·t strata in the moduli space A 9 of principally polarized abelia11 varieties of din1ension g. In thig paper I survey the current state of knowledge a11d prove two new res11lts alJout, these 2 defined by the affine equation yq + y xq+I. Because Hq is a smooth plane curve, the genus of Hq is g = q(q  1)/2.
=
Proposition 2.3. [Sti09, VI 4.4], [Han92, Proposition 3.3) The He1·111itian curve Hq is maxim.al over ]F q2. Also L(Hq/Fq, T) = (I+ qT 2 ) 9 and Hq is supersing'ular.
Proof By definition, tl1e ctl.fve Hq is n1aximal ove1~ 1Fq2 when tl1e n11111ber of points realizes the upper bo11od in the Hasse Weil bound. So it suffices to show
#Hq(Fq2)
= q2 + 1 + 2gq == q3 + 1.
"rhe1·e the last equality uses that g = q( q 1) /2. Since Hq has one point at infinity, it suffices to show that there a1·e q3 pairs (x, y) E (JFq2 ) 2 such that yq + y = x 9+1 . Let Tr : ]Fq2 ~ IF q ancl N : IF~2 ~ lF~ denote the t1·ace and norm maps respectively. If y E IF q2, then Tr(y) = yq+y E lF q; also Tr is a surjective qto1 homomorphism. If x E 1F;2 , ther1 N(x) = xq+I E lFq; also N is a surjective (q+ 1)to1 homomorphism. For x = 0, there a1'e q values of y E IF q2 sl1ch that yq + y == 0 . Given z E tl1e nl1mber of pairs (x, y) E lF~2 such that z = yq + y = xq+l is q(q + 1). Tl1is yields q + q( q  1) ( q + l) = q3 affine points, which cornpletes the proof of the first statement. Since Hq/IFq2 is maximal, the 1·eciprocal roots ai of L(Hq/Fq2, T) all equal ytq. This implies tl1at the reciprocal roots of L(Hq/Fq,T) are ±.j=g with multipli ity g. It follows that L(Hq/JFq T) = (1 + qT2 ) 9 . Since q ==pr, for O ~ i :S 2g, one can check with the binomial theorem that Vi/r 2: i/2, where Vi is the padic valuation of tl1e coefficient of Ti i11 L(Hq/Fq,T). Thus H 9 is supersingular. □
JF;,
2.5
Supersingular ArtinSclrreier curves
To begin, given a prune p, consider tl1e following curve:
Yo : y 2

y
= x 3 when p == 2 and Yo : yP 
y
= x 2 when p is odd.
Whe11 p = 2, Yo is a supersingula1' elliptic curve by Proposition 2.3. W11en p is odd, Yo is a hyperelliptic curve with genus (p  1)/2 by the RiemannHurwitz formula. In fact , Yo is supe1"singular wl1er1 p is odd. This is due to work of Hasse and Davenport [DH35) who proved that the eigenvalues of Frobenius for Yo are Gauss sums and so the cl1aracteristic poly11omial of Frobenius is a (p l)st power of a li11ear polynomial. An alternative way to prove that Yo is supersingular is to note that it is a quotient of the He1'mitia11 curve Hp2. A polynomial R(x) E k[x] is additive if R(x1 + x2) = R(x 1 ) + R(x2). This is equivalent to R( x) being a linear combination of monomials xd where d is a power of p. S11ppose that R(x) is additive and consider the ArtinSchI·eier curve
Y : yP  y
= xR(x).
Then one can show that Jac(Y) is isogenous to a p1·oduct of Jacobians of ArtinSchreier c1rrves arising fro111 additive polyno1nials of smaller degree. Working inducti'1"ely1this shows tl1at Jac(Y) is isogenous to a product of supersingula.i· curves (isomorphic to Yo). This yields the followi11g result.
·
184
Rachel Pries
Theorem 2.4. [vdGvdV92, Theorem 13.7], [Bla12, Corollary 3.7(ii)], [BHM+16, Proposition 1.8.5] If"lFq is a finite field of characteristic p and R(x) E Fq[x] is an additive polynomial of degree ph, then Y : yP  y = xR( x) is supersingular with genus ph(p 1)/2. More generally, the £polynomial of Y is determined in [BHM+16, Theorem 1.8.4]. In [vdGvdV95], when p = 2, the authors use Theorem 2.4 to prove that there exists a supersingular curve of every genus.
Theorem 2.5. [vdGvdV95, Theorem 2.1] If p = 2 arid g EN, then there exists a supersingular curve Y 9 of genus g defined over a finite field of characteristic 2. In fact, the authors prove that Y9 can be defined over F 2 [vdGvdV95, Theore1n 3.5].
2.6
New result about supersingular curves
In this section, we generalize Theorem 2.5 to odd characteristic using the techniques of [vdGvdV95]. For every odd prime p, Corollary 2.6 ve1·ifies that there are infinitely many new values of g for which there exists a supersingular ct1 rve of genus g in characteristic p.
Corollary 2.6. (Karemaker/Pries) Let p be prime. Let {JEN be such that O and 1 are the only coefficients in the basep expansion of 8. If g = {Jp(p1) 2 /2, then there exists a supersingular curve of genus g defi'n ed over a finite field of characteristic
p. Remark: When p = 2, then Corollary 2.6 is the same as Theorem 2.5 because the condition on 8 is vacuous and g = 8 .
. Proof. The condition on 1 for all f EL  {0}. By [KR89, Theorem BJ, Jac(Y) is isogenous to E0f#0Jac(C1). By Theorem 2.4, Jac(Cf) is supersingular for each f and so Jac(Y) is supersingular. The genus of Y is gy = L f #O go1 . If f E L is such that it has a nonzero cont1,ibution from Li, but not from Lj for j > i, then gc1 == pui (p 1)/2. There are pdi 1 nonzero polynomials in Li. The number off E L which l1ave a nonzero
c, :
Chapter 6
Current Results on Newton Polygons of Curves
contribt1tion from Li, but not from Lj for j > i is (pdi  1) t
gy
=
rr~:i pdj.
185 So
i1
E(pdi  l)
II pdi
pv., (p  1)/2
i=l
t
= L (P _
l)(pr· + ... + l)pE}~ (rJ+l)puilp(p
_
l)/ 2
i=l (,
==
L(pri
+ ... + l)psip(p 
1)2 /2
= ~p(p  1)2 /2.
t=l
□
Newton polygons and EkedahlOort types of
3
curves Suppose X is a smooth projecti,,e curve of genus g defined over an algeb1·aically closed fielcl k of chai"acteristic p. Then X can be classified l)y se,"eral inva.r iants of its .Jacobia11, sucl1 as tl1e prank, the Newton polygo11, and the El 4. The dimension of M 9 is 3g  3. There are si11gular curves that a1~e ordinary, 11amely chains of g 01·dinary elliptic curves. Since Mg is irreducible ai1d the prank is lower semicontinuous, the generic geometric point of M 9 is ordinary, with prank g . Let S be a component of M~. The length of tl1e chain which co11nects the orclina1'Y Newton polygon v 9 to the largest Newton tJolygon ha,v ing (f, 0) as a break point is g  f. Using purity of the Newton polygon stratification [dJO00), dim(S)
>
(3g  3)  (g  f')
= 2g 
3 + f.
By [FvdG04, Lemma 2.5], S intersects D..i for each 1 ~ i ~ g  I. Si11ce codim(~i,Mg) = 1 1 it follows that dim(S) ~ dim(S n ~ i) + 1 from [Vis89. page 614].
C11apter 6
Ctu·rent Resttlts on Nevtton Polygons of Cur,,..es
199
The 7J1·ank of a singular c11rve of compact type is the sum of the prai1ks of its co1npo11ents, [BLR90, Example 8, page 246]. As seen in [AP08, P1,opositio11 3.4] , one can restrict the clt1tching morphism to the prank strata:
This 1neans that dim(S n L'!i.i) is bounded above by dim(M{ 1) + dim(M~:__i;l), for some pair (f1 , f 2) such that f 3 + f 2 = f. Adding a. marked point adds one to the dimension. By the inductive hypothesis (01· an explicit computation when 1
= 1,g  1), one checks that din1(M{;~) == 2i  3 + /1 + 1 and dim(M~:_i;i) = 2(gi)3+ f2+l. It follows that dim(Sn~i) ~ 2g4+ f. Thus di1n(S) ::; 2g3f f,
i
which co1npletes the proof.
6
□
An inductive result about Newton polygon and
EkedahlOort strata This sectio11 contains the main result of tl1e paper·: star·ting with a Newton polygon ~ that can be realized for a smooth curve of genus g, the goal is to prove that any sy1nmetric Newto11 polygon wl1ich is formed b3r adjoir1ing slopes of O and 1 to an also be realized for a sn1ooth curve (of larger genus and prank). In the n1ain r ult, I show this is possible under a geometric condition on the stratum of M 9 with Newton polygon The importa11ce of· this result is that it allows 11s to restrict to tl1e case of prank 0 in Question 5.1. This type of inductive process can be found in earlier work, e.g., [FvdG04, Theorem 2.3], [AP08, Section 3], [Pri09, P1·oposition 3. 7], and [AP14, P1,,oposition 5.4]. However Theorem 6.4 is stronger than these results because it allows for more flexibility ,,rith the Newton polygon and EkedahlOort type.
e
e.
6.1
The main result
First, we fix some notation about Newto11 polygons and BT 1 group schemes.
Notation 6.1. Let { denote a symmetric Newton polygon (or a symmetric BT1 group scheme) occurri11g for principally pola1·ized abelian va1·ieties in dimension g. Let A 9 [€] be the st1·atum in A 9 whose geo111etric points represent principally polarized abelian varieties of dimension g and type~ Let cd{ = codim(A9 [e],A9 ). Let M 9 [~] be the strat1.1m in Mg whose geometric points represe11t smooth projective curves of genus g and type ~.
Notation 6.2. In tl1e case that~ denotes a symmetric Newto11 polygon occurring in dimension g: fore E N, let ~+e be the symmetric Newton polygon in dimension g + e s11ch that the clifference between the m1tltiplicity of tl1e slope A il1 ~+e and the multiplicit)' of the slope .,\ in eis 0 if .A (/. {0, 1} and is e if.,\ E {O, 1 }.
Racl1el Pries
200
Notation 6.3. In the case that
edenotes a symn1etric BT1 group scheme occu1·
ring i11 di1nension g: for e E N, let, e+e be the synnnetric BT1 group scheme in dime11sion g + e give11 by ~+e :;::::; Le EB ~, where L = Z/p EB µp. If [v1 , ... , v 9 ] is the Ekedal1lOort type of~' then
Xx s Ti of pdivisible groups for all i E J.
The above terminology is based on the notion of IFp is perfect. A pdivisible group X over K is strongly Ji;Sustained if and only if the statement (6B) (i) hold.
We are now in a position to give a schemetheoretic definition of central leaves in terms of sustained pdivisible groups.
1.6. Definition (a schematic definition of central leaves). Let n 2: 3 be a positive integer relatively prime top. Let K:) IFp be a field.
(i) Let xo = [(Ao, Ao)] E d 9 ,n(~) lJe a Krational poi11t of dg ,n· The central leaf lf(xo) in d 9 ,n Xspec(Z[µn)l / nJPPec(~) passing through xo is the maximal member aimong the family of all locally closed Sl1bscl1eme of .rdg ,11 Xspec(Z[µ. l1/ n)) Spec("') such that the principally polarized pdivisible group attached to the restriction to Cf! ( xa) of tl1e universal a belian sche1ne is strongly Ksustained modeled on (Ao[p00 }, .Xo[p00 ]). (This definition depends on the fact that there exist a mernber 1f(x0 ) in this family which contains every member of this 11
family.)
(ii) A central leaf over "" is a maximal
member
CC in the family of all locally
closed subscheme of J.?19 ,n Xspec(Z[µn,l / n]) Spec(~) such that the pri11cipally polarized pdivisible gi~oup attacl1ed to the 1,estriction to ~ of' the universal abelian scheme is Ksustained.
Chapter 7
Sustained pdivisible Groups
215
Of course there is a backward compatibility isst1e: we have to show that tl1e above defirntions coi11cide with the definition in [24) of central leaves when the field K is algelJ1·aically closed; in particular '"t'(x 0 ) is reduced (and smooth over K).
1. 7. The rest of tl1is article is organized as fallows. • In §2 we define the stabilized Hom schemes ~M st (X[pn], Y[pn]) and the st pdivisible group Yt'oM~iv(X, Y) = li1T:n ~ M (X[pn], Y[pn]) associated to pdivisible groups X , Y over a field ~
• In §3 we eAl)lain descent related to sustained pdivisible groups. In particular a strongly sustained pclivisible group X over a ~scheme modeled on a'])divisible group Xo over a field K :) 1Fp corresponds to projective system of right torsors for the stabilized Aut group schemes .~(Xo[pn]). Properties of the stabilized i\ut group schemes quickly lead to the existence of a natural slope filtration on every sustained pdivisible group; see 3.2. • The main theorerr1 4+3 of §4 shows that the definition of central leaves in 1.6 is compatible with the definition in [24] using the notion of geometrically fiberwise constant pdivisible groups.
• Defor111atior1 of sustained pdivisible groups is clisc11ssed in §5. Isoclli1ic sustained pdivisible groups a1~e Galois twists of constant isoclinic pdivisible groups a.n d do not deforn1, hence evel'Y ,s ustained deformation of a pdivisible g1:oup X 0 is obtained b)' deforming its slope filtration. This leads to an assembly of maps between s11stained defor111ations spaces of subquotients of X 0 with 1~espect to its slope filtratio11. These maps shows that the space Vef (Xo)sus of sustained deformations of Xo can be "assembled'' from the space of s11stained deforn1ations of pdivisible groups with two slopes. Such structural description of Vef (Xo)sus ca11 be regarded as a generaliza.tion of tl1e Ser1·eTate coordinates for ordinary pdivisible groups and abelia11 varieties. ¥.'hen X 0 has exactly two slopes whose slope filtration corresponds to a short exact sequence O + Z ~ X 0 ? 1r ; O, where Y a.n d Z are isoclinic pdivisible grot1ps wit11 slope(Y) < slope(Z), the space Vef (Xo)sus of sustained deforn1ations of Xo is a torsor for the pdivisible fo1~1nal gro11p £oM~iv(Y, Z); see 5.6. Tl1e full description of the structure of the space of sustained deformations of a pdivisible group in the general case is somewhat technical. A full doc111nentation will appear ir1 a planned monograph [8] on Hecke orbits.
2
Stabilized Horn schemes for truncations of pdivisible groups
2 .1. !11 this sectio11 we discuss a fundamental stabilization pheno111e11on fo1· the Hom schemes and Isom schemes between the truncated Barsotti Tate groups attached to two pdivisible groups over the san1e base field. These stabilized Hom.,
216
ChingLi Chai and Frans Oo1~t
Isom and Aut schemes enter the theory of sustained pdivisible groups in (at least) two ways. (a) The category of strongly sustained pdivisible groups over a. base Kscheme S modeled on a pdivisible group Xo over a field K is equivalent to the categOI'Y of projective families of torsors over S for tl1e stabilized Aut scl1emes for Xo[pn], indexed by n EN. (b) Given two pdivisible groups X, Y over a field K => IFp, the direct limit of stabilized Hom schemes from X[pn] to Y[pn] is a pdivisil)le group, denoted by &oMdiv(X, Y). Such pdivisible group appear as "building blocks" of formal completions of central leaves. It is te1npting to think of tl1e bifunctor (X, Y) vt £oM~iv(X, Y) as an "internal Hom" for the category of pdivisible gI'Oups. However the Hom functor between two pdivisible groups is the projective limit of the Hom functors between their truncations, not an inductive limit. For many purposes it is better to consider (X, Y) ~ ~Mdiv(X, Y) as a disguised Ext functor. The relation between &oM~iv(X, Y) and the functor £xtder(X, Y) of "deforming X x Y by forming extensions of X by Y over Artinian ~algebras" is given in 2.10 2.11.
2.2. An outline of the basic stabilization phenomenon. Let
Ii
:J lF11 be
a field. Let X, Y be pdivisible groups over K. Let ~M(X[pn], Y[pn]) be the scheme of homomorphisms from Xfpn] to Yfp""] over K,schemes; see 2.3. It is a group scheme of finite type over ""·
(i) For each n, the image in &oM(X[pn], Y[pn]) of the restriction homomorphism
stabilizes as i ~ oo, to a finite group scheme &oM st (X[pn], Y[pn]) ove1' "'·
(ii) Besides the restriction maps Hn+i
~ Hn used in
(i) above, there are 11atu
rally defined monomorphisms
which induce monomorphisms
st The inductive system (£oM (X[pn], Y[pn]))nEN is apdivisible gioup, denoted by &oMdjv(X, Y). The precise definitions and statements about the stabilization process can be found in 2.3 2.6; the key stabilization statemer1t is Lemma 2.4. The importance of this stabilization phenomenon is twofold.
Chapter· 7
217
Sustained pdivisible Groups
• A stro11gly Ksustained pdivisible g1·ol1p X ~ S over a Kscheme S is esse11tially a projective systeni of torsor over S for the stabilized Aut groups JOUTSt (XO[pn]).
• The sustained locus of the characte1·isticp deformation space of a pdivisible group over a perfect field K is built up from pdivisible groups isogenous to pdivisible groups of the form t.7CtJM~iv(Y, Z), where Y, Z ai·e isoclinic pdivisible g1:ot1ps over~ Note tl1at the 7rdivisil)le group ~M~iv(Y, Z) is an induct'i'ue system of stabilized Hom schemes YrbM 6 t(Y(pn], z(pn']. 2.3. Definition. Let
IFP be a field and let X, Y be pdivisible groups over ~
K :)
(1) For every n E N "''e l1ave a comm11tative affi11e algebraic grot1p
of finite type over "' whicl1 represents the functor
s I+ Horns(X[p''l] Xspec(K) s, Y[p
11 ']
Xspec(1t) S)
the category of all Kscl1emes S. We often shorten £oM(X[pn] Y[pn]) to Hn.(X, Y)~ and sometin1es we will sl1orten it fu1·ther to Hn if there is 110 dange1~ of confusio11. 011
(2)
ror any n,, i
E N, let. Jx ,n+i.n : X[pn] * X[pn+i] be the inclusion homomorl)llism, and let 7rX ,n ,n +i : x[pn+i] ~ x(pn] be the faithfully flat hon1omorplusm such tl1a,t [pi]X[pn+i ] == j ~, ;n+i,n O 1fX.n.n.+i . Define 1r,~.n ,Y1 + i and )y n + i ,n si1nila1·ly.
(3) For all ri, i E N, denote by
the ho11101norphism such that
for any Kscheme S and any homomorphism a : X[pn+i]s _. Y[pn+iJ 5 . In othe1" words rn,n+i is defined by restricting l1omomo1"phis1ns £1,om X[pn+i] to Y[p11 + 1] to the subgroup scheme X[p1"] c X[pn+i].
(4) For a.11 n i EN, denote by
the homomorphism such that
for any Kscl1en1e S and any homomorphis111 homomorpl1is1n ln+i.n is a closed embedding . •
/3 : X[pn]s
~
Y[pn]s. This
218
ChingLi Chai and Frans Oort
2.4. Lemma. Let X, Y be pdivisible groups over a field
(1) The homomorphisms (rn,m) m_n, > m,n tative group schemes
ER.l 1'1
K,:)
lFp.
form a projective system of commu
indexed by N. Similarly homomorphisms system of commutative group scliemes
(in ,m) m_n,m,n < E~r form 1'1
a inductive
indexed by N. Moreover we have
fo'r all n, i EN.
(2) For n, i, j E N we have a commutative diagram

with exact rows. This diagram applied twice, the second time with j induces monomorphisms
whose compositio'n is equal to
= 0,
Vn,;i+i.
(3) There exists a positive integer n 0 such that the monomorphism
is an isomorphism of commutative finite group schem,es over integer n 2: no.
K
for e11ery
2.5. Definition. Let X, Y be pdivisible groups over a field K => JFp. Let rt,o be a positive integer such that 2.4 (3) holds, i.e. the monomorphism Vn;l,2 is an isomorphism for every n > no.
(1) Define commutative finite g1,oup scl1emes Gn == Gri(X, Y) ove1~
(2) Define monomorphisms Vri: Gn Vn;io is defined in 2.4 (2).
>4
Hn by
Vn
==
Vn;io
K,
n EN, by
for all n E N, where
Chapter 7
219
Sustained pdivisible Groups
(3) For every n, i EN, denote by
.in+i,n : Gn. the homomorphisms over
K,
Gn+i
induced by ln+i+no: Hn+no ➔ Hn+nu+i·
(4) For every n, i EN, denote by
the homomo1·phism induced by Tn1na,n+no+i : Hn+no+i ➔ Hnino · 1
2 .6. Definition. (1) Denote by £bM (X, Y) the inductive system of commutative group scl1emes Hn(X, Y) of finite type over"', with t1·a.n sition maps
(The superscript ' in Yt£>M (X ~ Y) is meant to indicated that the arrows in the projective system Yti!>M'(X. Y) of dt'bMschemes Hi(X: Y) = Jt'oM(Xfpi], Y[pi]) are reversed, giving rise to an inductive system instead.) 1
(2) We will write
£0M st (X 1
Y)n for the grot1p scheme Gn (X, Y) over the base fi eld "'· We will call it the stabilized r.YtbM scheme at truncation level n. (3) Denote by ~OM~iv{X, Y) the pdivisible group
111
other words £bM~iv(X, Y)[pn]
= £bM st (X, Y)n for
all
11,
EN.
(4) Let v : YfloM~iv(X, Y) > ➔ Ya!JM'(X, Y) be the n1onomorpl1isn1 defined by the compatible family of monon1orphisms
Remark. The li1nit of the ind11ctive system ,Yt'oM'(X, Y) is canonically identified with cXbM(Tp(X), Y), the shea.fified £bM (or the internal Hom) in the category of sheaves of abelian groups, fro1n the projective system
to the inductive .system
Tl1e group scheme £bM(X[pi] , Y[pi]) is t.he kernel of the e11don1orphism ''multiplication by pn" of this inductive limit:
220
ChingLi Chai and Frans Oort
2.7. Corollary. (a) The Jor,nation of dt6M'(X Y) arid ~M~iv(XJ }') commute with extension of base fields: f O'r every ho·momorpl1ism of fields K ~ K.1 , th.e ·natural m.aps
and are isomorphisms.
(b) The rrior1,0111,orphism v : YtbM~iv ( X, Y) YfbM' ( X, Y) identifie~c; th,e ind1tctive system .Jr'oM~iv(X, Y) as th,e 1naximal pdivisible s·ubgroup of ~M'(X: Y) , wliich satisfies the fallowing univer·sal property: for every pdivisible group Z over a Kscheme S and every Shomo11iorph,i.s1n )?
there exists a unique homomorph.is·m g: Z ~ c.l'tbM~iv(X Y)s ove·r S s·uclt that f = v o g.
2.8. Proposition. Let X, Y be pdivisible groups over a field "' that X, Y are both isocli·nic and let AX and Ay be their slopes.
~
JFP.
uppo.se
(a) If Ax < Ay, then the pdivisible gro11,p ~M~iv(X, Y) 'is isoclinic of slope Ay  Ax.
(b) If .Xx > Ay , tlien £oM~iv(X, Y) = 0. (c) If AX= Ay, tlien ~Mdiv(X, Y) is an etale pdivisible gro1Lp.
2.9. Corollary. Let X, Y be pdiuisible groups over a field Ii. If every slope of X is strictly bigger than every slope of Y then £oM~iv(X Y) = 0.
We end this section with an exainple which illustrates a ren1ark in the last paragraph of 2.1, that ~M~iv•(X, Y) is l)etter thought of part of the :'divisible part' of the sheaf CxT(X, Y) of extensions of X by Y, in the set,ting of defo1·mations of X Xspec(~) Y via extensions.
2.10. Definition. Let
K
be pdivisible groups over
~
lFp be a perfect field of cha.r acteristic p
> 0. Let X, Y
K°J.
(a) Denote by 21tt~ the ca.tegory of augmented comn1utative Artinia11 local algebras over "'· Objects of 2ltt! are pairs
(R, j: k  t R,
E :
R
t
k)
where
 R is a commutative Artir1ian local algebra over  k is a perfect field containing
K,
K,
Chapter· 7
221
Sustained pdivisible Groups
 j a1ncl E are ~lli1ea1· ri11g homomorphisms st1c·h that 
€
oj
= idk.
R / 1n > R is a bo111omo1·plusn1s of KaJgelJ1~as s11(~h. that the composit.io11 p1~ o t off with tl1e quotie11t 1nap pr : R t k is ec1ual to idk. f. :
Morphis111,s from (R1, J1: k1 ~ R1, t1 : R1 t k1) to (R2, j2: k2 ~ R2, c2: R2  t k2) are K,li11ear ho111011101·phisms f1·om R1 to R2 which are compatible with the augme11tations €1 and E2.
(b) Define £xtdet(X, Y) : 21tf~ > 6ets to be the functo1· from 21tt! to the categoI·y of sets wl1ich se11ds any object (R, j: k ~ R, E: R ~ k) of 2ltt! to the set of isomorphisni classes of
( o  YR_, E  xR  o, (: Ek ~, xk
x
Yi.)
where O  t YR  t E ~ X R ~ 0 is a shor·t exact sequence of pdivisible groups over R, and ( is an iso1norpl1isn1 fi·o1n tl1e closed fiber of the extensio11 E to the split extension O t Yk  t Xk x Yk ~ Xk  t O which induces the identity maps idxk and idyk on both Xk and Yk. The Baer su1n co11struction gives &tder(X, Y)(R) a natural st1'ucture as an abelian group, for every object (R, c) i11 21rt;, so "'re ca,n promote &tder(X~ Y) to a functor h'OID 2!tt~ to the category of abelian groups.
Tlle n1nctor &tder(X, Y) can be identified as a sl1bfunctor of the deformation fuo c or of the pdivisible group X Xspec(I() Y; it is the largest closed formal subscl1eme of Vef(X Xspoc(K) Y) over which the defor~mation of X Xspec(K) Y is an extension of X by Y. 2 .11. Proposition. Let X, Y be pdivisible groups over a field
(i) Ther·e is a nat·1Lral isomorphi.c;m
K°J :=)
JFp.
com,111,·zttati'lJe sm,ooth f orn1.al gro'lLps
8 : ftbM' (X, Y) ""') &tder(X, Y) over
K.
o.f dimension dim(Y) • dim(Xt).
(ii) Tlie i.somorph·i sm 6 : YtbM' (X, Y) ➔ t:ridef(X, Y) in (i) induce.s an isomorph,i.~m c5: YfbJv1~iv (X: Y) &z:tdef(X, Y)div, l"V
f'V •
wh.ere &i;tdef( X, 1r)div is t/ie maximal pdivisible subgroup of Extdef( X, Y).
3
Descent of sustained pdivisible groups, torsors for stabilized Isom schemes and the slope filtration
In this sectio11 we discuss son1e basic properties of susta,ined pdivisible groups related to descent; notable arr1ong them are the to1~s01· interpretation 3.1 a11d tl1e
222
ChingLi Cl1ai a11d Frans OoI·t
existence of slope filtration 3.2. We also compare sustained pd.i,,isible groups with completely pdivisible groups, and unravel what it means to Sa)r that a p,divisjble group over a field R"' is sustained over a subfield K of K. Proposition 3.1 below says that a strongly Ksustained pdivisible group X  t S modeled on a pdivisible group Xo over K gives rise to a co1npatible family of torsors over S for tl1e stal)ilized Aut group JliUT st (X0 [pn]), and the strongly sustair1ed pdivisible group X can be recovered from this family of torsors. It is a consequence of the stabilization results in §2, tl1e defi1lition of strongly sustained pdivisible g1,oups and desce11t.
3.1. Proposition. Let "' => 1Fp be a field ar,,d let X 0 be a pdi11isible group o·uer "'· Let X  t S be a strongly ~sustairied pdivisible group over S rr,.odeled on Xo. Let fsoMs(Xo[p 7i] X[pn]) be tlie Sscheme of isomorphisms from X0 fpn] to X[pn] over Sschemes.
(1) There exists a 11,atural number n 0 s·uch that the image of the restrictio11, mo·rphisms rn ,n+i : JfSVMs(Xo[pn+i] x[pn.+i]) + fs0Ms(Xo[p 1i] , X[pn]) and Tn ,n +i: JISOMs(Xo[pn+i+l],x[pn+i+l])  t Js0Ms(Xo [pn] ,X[pn]) are equctl as fppf sh1eaves, for all i ~ no. 1
(2) The stabilized image in fsoMs(X0 [pn], X[pn]) is represented by a closed subscheme fsoM~( ..X"o[pn] X[pn]), which is a right torsor for (the base change to S of) tlie finite group scheme J22Urst (X0 [pn]) o'ver K.. Denote the torsor JsoM1(Xo[p1t],X[pn]) by Tn , and let Gn := ~ st (Xo[pn]) for each n EN. (3) The natural transition maps Tn+I t Tn are faithfully fiat and compatible with the t1"'ansition maps G,,.,,4 1 t Gn. (4) There is a compatible family of natural isomorphis1ns O'.n :
Tn
X Gn
wliere Tn x 0 n Xo[pn] = Tn Gntorsor with Xo[pn].
X .. Am.1, if Ai 2: A > Ai1 , 1 Si :5 m  1, if ..\ :5 .Xo
for any real n11mber .,\ E [O, 1].
4
Pointwise criterion for sustained pdivisible groups
The main result of this sectio11 is Tl1eo1·e111 4.3: for a pdivisible group ,. ~ S over a reduced base ~scheme to be strongly ~sustained 1nodeled on a given pdivisib]e group Xo, it is necessary a11d sufficie11t that every filJer Xs of X ~ S is stror1gly ~sustained modeled on Xa. It follows that when the base field "' :, IFp is algebraically closed, the latter condition is equivalent to the notio11 of geometr·ically fiberwise co1istarit pdivisible groups in [24]. The proof of 4.3 is tcch11ical; see [8] for details. We will fo1·mulate the technical lemn1as which give son1e indication of the intermediate steps, but ornit all proofs.
lFp is an algebraically closed field and S is a + S be a corripletety slope divisible p P x s P is an isomorphis1ri of S,scheme~,. 1
Cl1apter 7
225
Sustained pdivisible Groups
(a) The quotien,t P/G of the Srpprsheaf P by the Sfpprsheaf G is representable by an Sscheme T. (b) The morpliism P
~
T gives P a structure as a right Gtorsor for the site
Sfppf· •
I
T ~ S factors as a composition T ~ T' 1r > S of morpliisms of schemes, where 1r' is a finite morphism of finite presenta,tio·n and j is a quasicompact open immersion.
(c) The structural morph,ism
1r :
T ~ S' is a universal monomorphism and induces a continuous injectiori from ITI to SI, 'Where ITI and ISi are the topological spaces underlying T and S respectively.
(d) The morphism
(e) For every t E T,
1r' :
induces an isomorphism between the residue field of 1r' (s) and the residu.e field of t. In pa1ticular 1r' is unramified. 1r'
(f) Let t E T be a. poirit of T such that
ffS.1t'(t) ·i s reduced.
 If 1r' is etale at t, then 1r' induces an isomorphi.sm of schemes between a·n open neighborhood oft and an open neighborhood of 1r1 (t).  If 1r1 is not etale at t, then 1r' (Spec( tJr,t)) is contained in a closed subset of Spec(t>'s,1r'(t)) which is not equal to Spec(tYs.,r'(t))· In particular there exists a point s E S whose closure in S contains 1r' ( t) such that s =/ 1r'(t1) for any t 1 ET wliose closure in T contains t.
(g) S1tppose that S is reduced and 1r' : ITI ~ ISi is a bijection such that every specializatio1i in S comes from a .specialization in T. Tlien 1r' is an isomor·phism. In other words P ~ S i.c; faithfully fiat 1 making P a Gtorsor over
s. (h) Suppo.se that S ·is reduced artd locally N oetherian) and 1r' (T) is Zariski dense in S. Tlien the1'e exists a der1se open subset U of S such that 1r1 ind1.tces an 1 isomorphism of schemes from 1r' (U) to U. In particular P xs U+ U is a torsor for G Xs U. 4.3. Tl1eorem. Let S be a reduced Ktscheme, where r;, :) IFp is a field. Let X + S be a pdivisible 9ro11,p, and let Xo be a pdivisible group over K,. Suppose that for
every point s E S, the geometric fiber Xs Xs s i.s isomorphic to Xo Xspec(K) s; equivalently Xs is strongly K,sustained modeled on X 0 for every s E S. Then X + S is strongly Kisustained modeled on Xo. Remark. Tl1eore1n 4.3 says that a strongly geometrically fiberwise coristant pdivisible gi·ol1p relative to K over a reduced Iischeme is strongly Ksustained. So over a. reduced Kscheme, being strongly x:sustained is equivalent to being strongly geometrically fiber\vise consta11t relative to "· The latter notio11 was introduced i11 (24] to define tl1e centra.l foliation 011 Siegel n1odular vai·ieties.
226
Cl1ingLi Chai and Frans Oort
4.4. Corollary. Let S be a reduced scheme over a field
lF11 • Let X
~
S be a pdivisible group, and let X 0 be a pdivisible group over· K. Assume that Newton polygon of every fiber of X + S is equal to the Newton polygon of Xo, and there is a dense open. subset U C S such that X x s U ~ U is strongly K,sustained modeled on Xo. Then X t S is strongly ,.,sustained modeled on Xo& K :)
Remark. Corollary 4.4 is a generalization of a result in [24], where it is shown that every central leaf C in d 9 is closed in the open Newton polygon strat11n1 containing C. 4 .. 5. Lemma. Let S be a reduced noetherian scheme over a field
1Fp. Let X ~ S be a pdivisible group, arid let Xo be a pdivisible group over K. Suppose that there is a Zariski dense subset B C ISi such that Xb is strongly K,sustained modeled on Xo for every element b E B, where ISi denotes the topological space underlying S. ThJen there exists a Zariski dense open subscheme U C S such tliat the pdivisible group X xs U+ U is strongly 1"'sustained modeled on Xo. K :)
4.6. Lemma. Let S be a Noetherian scheme over a field K. :) 1Fp. Let X  t S be a pdivisible group over S, and let X 0 be a pdivisible group over K. Let ( be the
Newton polygon of Xo.
(a) The subset of S consisting of all points s ES such that the fiber Xs of X  t S at s whose Netuton polygon is equal toe is a locally closed s·ubset IS{ of ISi.
(b) The subset of S consisting of all points s E S such that the fiber X3 of X ~ S at s is strorigly Ksustained modeld on Xo is a closed subset ISx0 I of IS~l 
(c) Denote by Sx0 the locally closed reduced subscheme of S whose underlying set is Sx0 I. The pdivisible group X Xs Sx0 4 Sx0 is strongly 1"'sustained modeled on X o.
5
Deforniation of sustained pdivisible groups and local structure of central leaves
In this section we consider defor1nations of a pdivisible group Xo over a field K :) IFp which are K.sustained, and the structure of the space of such sustained deformations V cf (Xo)sus· The general phenomenon is that Vef (Xo)sus is "bliiltup', from pclivisible groups via a syste1n of fib1. ations, with pdivisible groups as fibers. This ''bigpicture descriptio11'' is literally t1.. ue when Xo is a product of isoclinic pdivisible groups. Otherwise the descriptio11 needs to be taken with a grain of salt, because the fibers may be an e>..'tension of a connected finite group scheme b)1 a pdivisible group. The pdivisible groups which appear as the 'buildi11g blocks' of Vef (Xo)sus aJ."'e of tl1e forn1 £oM~iv(Y, Z), where Y, Z are isoclinic pdivisiblc groups over ~ with slope(Y) < slope(Z). Their Dieudonne modules a1·e given in 5.3.
5.1. Theorem. Let K:) 1Fp be a perfect field. Let Xo be a pdivisible group over""· Let V ef (Xo) be the equicharacteristicp deformation space of Xo, isomorphic to the fonnal spectr11,m, of a po'luer series rin,g over K in di111(Xo) •di1n(X6) variables.
Cl1apter 7
Sustained pdivisible Groups
227
(1) There exists a closed formal sttbscheme Vef(Xo)sus of Vef(Xo), uniquely characterized by the following p·roperties.
The pdivisible _group ~univ Xvef(Xo)
Vef (Xo)sus + Vef (Xo)sus
is strongly ksustained modeled on Xo.  If Z is a closed formal subscheme of Vef (Xo) such the pdivisible group ~univ Xvef(Xo)
Z
>
Z
is stro'ngly ksustained modeled on X 0 , then Z is a sub.scheme of Vef (Xo)sus•
(2) The formal scheme Vef (Xo)sus is formally smooth over k. (3) Suppose that Xo is isogenous to a product
where ~ is an isoclinic pdivisible groups over "' with slopes Ai for each i, and O ~ A1 < A2 < · · · < Ar ~ 1. Then dim(Vef (Xo)sus)
=
L
(Aj  Ai) · ht(J'i) · ht(Yj).
lSi IFp is a nonperfect field and X, Y are pdivisible groups over l'C. How to descril)e the display of the pdivisible group YtbM~iv(X, Y) in terms of the displays of X and Y?
References [l] C.L. Chai, Every ordinary symplectic isogeny class
positive characteristic is dense in the moduli space. Invent. Math. 121 (1995), 439479.
[2] C.L. Chai, Hecke orbits
i?1
Siegel modula,r varieties. Progress in Mathematics 235, Birkhat1ser, 2004, pp. 71 107. 011,
[3] C.L. Chai, Monodr·omy of Heckeinvariant subvarieties. Pure Appl. Math. Quaterly 1 (2005) (Special issue: in memory of Armand Borel), 291 303.
[4] C.L. Chai, A rigidity result for pdivisible formal groups. Asian J. Math. 12 (2008), 193 202. [5] C.1. Chai, Hecke orbits as Sliimura 'Varieties in positive cliaracteristic. Proc. Ii1teru. Cong. Math., 11adrid, Aug. 22 30, 2006, vol. Il, 295 312. [6] C.L. Chai & F. Oort, Hypersymmetric abelian varieties. Quaterly J. Pure Applied Math. 2 (2006), 1 27.
[7] C.L. Chai & F. Oort, A1onodromy and irreducibility of leaves. Ann. J\1ath. 173 (2011), 13591396.
232
ChingLi Chai and Frans Oort
[8] C.L. Chai & F. Oort, Hecx;e orbits. (monograph in preparation) [9] M. H. Hedayatzadeh, Multilinear theory of commutative group schemes. Master thesis, ETH, 2006. (10] M. H. Hedayatzadeh, Exterior powers of Barsotti Tate gr·oups. Ph.D. Dissertation, ETH Zurich, 2010.
(11] M. H. Hedayatzadeh, Exterior powers of 'ffdivisible groups over fields. preprint. [12] M. H. Hedayatzadeh, Exterior powers of Lubin Tate groups. preprint. [13] L. Illuise. Complexe cotangent et deform,ations I, II Lecture Notes in Math. 239,283, SpringrVerlag, 1972. [14] N. Katz, Slope filtration of Fc'rystals. In Journees de Geometrie Algebrique de Rennes (Rennes, 1978), Vol. I. Asterisque 63 (1979), 113 164. [15] Y. Mani11, The theory of commutative formal gro'ups over fields of finite characteristic. Usp. Math. 18 (1963), 3 90; Russ. Math. Surveys 18 (1963), 1 80.
[16] H. Matsumura, Commutative algebra~ Benjamin, 1970; second edition, 1980. [I 7] D. Mumford, Geometric invariant theory. Fir·st printing 1965, Ergebn.isse Math. und ihre Grensgebiete, SpringerVerlag; 3rd enlai~ged edition by D. Murnford, J. Fogarthy & F. Kirwan, SpringerVerlag, 1994.
[18] D. Mumford, Biextensions of formal groups, In Algebraic Geometry, (Intl. Colloq., Tata Inst. Fund. Res., Bombay, 1968), Oxford Univ. Press, 1969, 307 322. [19] D. Mumford, Abelian 'uariet,ies. Tata Inst. Fund Research Studies in Mathematics 5. First edition, 1970; second edition TIFR a11d Oxford Univ. Press, 1974; corrected reprint TIFR a11d Hindustan Book Agency, 2008. [20] P. Norn1a11 & F. Oort, Moduli of abelian varieties. A11n. Math. 112 (1980), 413 439. [21] F. Oort, Newton polygons and formal groups: conjectures by Manin and Grothendieck. An11. Math. 152 (2000), 183 206.
[22] F. Oort, A stratification of' a mod,uli space o.f polarized abelian 'varieties. In Mod'uli of Abelian Varieties. (Ed. C. Faber, G. van der Geer, F. Oort). Progress Math. 195, Bi1·khauser Verlag, 2001; pp. 345 416.
[23] F. Oort, Newton polygon str·ata i'n the moduli space of abelian varieties. In Moduli of Abelian Varieties. (Ed. C. Faber, G. van der Geer, F. Oort). Progress Math. 195, Birkhauser Verlag, 2001; pp. 417440. [24] F. Oort, Foliations in m,oduli spaces of abelian var·ieties. J. Amer. Math. Soc. 17 (2004), 267 296.
Chapter 7
Sustai11ed pdivisible Groups
233
(25] F. Oort, Mi·ni1nal pdivisible 91,oups. Aru1als of lVIath 161 (2005), 1021 1036.
(26) F. Oort & T. Zink, Families of pdivisible groups with constant Newton polygon. Docume11ta Math. 7 (2002), 183 201. [27) M. Rapoport & Th. Zink, Period spaces for pdivisible groups. Ann. Math. Studies 141, Princeton University P1·ess, 1996.
(28] The Stacksproject Autl1ors, col 11mbia. edu, 2015.
The Stacksproject,
http:// stacks . math.
[29] T~ Zi11k, Cartiertheorie komm1ttative forrnaler Gruppen,. Teubner, 1984. [30] T. Zink, On the slope filtration. Duke Matl1. Jot1rn. 109 (2001), 7995.
[31} T. Zink, De JongOort pitrity for pdivisible groups. In Algebra, Arithmetic and Geometry  Volume II: In Honor of Yu . I. Manin (Manin Festschrift; Y. Tschinkel & Yu. Zarhin, ed.). Progress in Ma.thematics 270, Birk.ha.user, 2009; pp. 693 701.
Open Problems in Arithmetic Algebraic Geometry ALM 46, Ch. 8, pp. 235262
© Higher Educa.tion Press and International Press Beijir1gBoston
Chapter 8 The Hecke Orbit Conjecture: A Survey and Outlook ChingLi Chai*t and Frans Oortt
Abstract
Every Hecke orbit in the moduli space of gdimensional principally polarized abelian varieties A 9 over a field of characteristic zero is dense. However a Hecke orbit rl·x of a poi11t x in A 9 ® 1Fp is contained in the open Newton polygon stratum NP:v containing x. Co11jecture 15 that rl·x is Za.riski dense in the Newton polygon stratum NP x has been proved. This chapter gives a survey of the methods developed for the Hecke orbit problem and related open problems.
1
Introduction
Entry 15 of [41], Hecke orbits, contains the following p1·ediction.
Given any IFppo int x == [(Ax, Ax)] in the moduli space dg of gdimensio·nal principally polarized abelia·n varieties in characteristic p > 0, the Hecke orbit of x, wliich, consists of a.ll points [(B, v)] E .flfg related to [(Ax, Ax)] by symplectic isogenies, is Zarisk'i dense in the Newtori polygo1i strat1.tm of d 9 which contains x. 1
"'C.L. Cl1ai was partially supported by NSF grant DMS 1200271 and a Simons fellowship. t ChingLi Cl1ai, Dept. of :t\1atl1ematics, Univ. of Pennsylvania, Philar the local structt11·e of central leaves. See 3.2 for a precise definition and (11], (13] for more information. (3) Generalized Serre Tate local coordinates on central leaves . In seve1"al as1:> , ·ts central leaves in a 1nodt1lar variety M of PEL type ove1· JFp deserve to be tl1ot1ght of as a11alogs of Shlmt1ra varieties in positive characteristic p. A prominent property of a central leaf C is that the formal completion c/xo of C at a point x 0 E C(IFp) is "built up'' fron1 a syste1n of fibrations whose fibers are pdivisible £01~mal groups. We would like to thjnk of such generalized SerreTa.t e coordi11a,t es on the formal completions cJ:vo as a sort of Tatelinear str·u.cture of Cat the infinitesin1al neighborhood of each closed 1)oi11L of a central leaf C. Here "being Tatelinear" means 'resembles a pdivisible groups,,. \f\Te emphasi~e that the resulting '·coordinates" live e11ti1~e1y in.5ide a central leaf C, in characteristic p. Let's illustrate the Tatelinear structure on c/xo with two easy examples. (i) Tl1e first exai.nple is the Ser1·eTate coordi11ates for the or·di11ary locus of 0rd rd ~ , ·w here the whole ordinary loc11s d ° is a si11gle leaf and the fo1·mal com9 l)letion at a closed point . 0, ai1d let S be a scheme over Ki. Let (X+ S,i: O'B> E11cls(X) 1 ..\: X) Xt)
be a polarized lTBlinea1' pdivisible grouJ) over S. (a) A pola1~ized URlinear pdivisible gr·oup
(X ts, l:
f.jB +
Ends(X), A: X+ xt)
over S is st1·01igly Ksustained modeled on an tlalinea,r polarized pdt risible group
(Xo, lo, ..\o) over
is faithfully flat over S
Ii
if the Sschen1e
£01·
every n E N.
(b) An OBlinear polarized pdivisible g1·oup (X + S, JsOM sx Spec (x:)s
i,
A) is Ksustairied if
(pri(X[p'L], i[pn·]: A[pn]), pr;(x[pn] l[p"], ,\(pn]))iSXspec(l'i.'' 1) + (1,2) + (1 .3)) in a Hilbert 1nodular variety associated to a totally real number field F with (F : Q] == 14. We refer to [56] for a complete solution of the existence p1·oblem of hypersymmetric poiI1ts
8
011
modular varieties of PEL type.
The Hecke orbit conjecture for dg
8.1. In this sectio11 we ot1tline a proof of the Hecke orbit conjecture for the moduli spa .r:19 of gdimensional principally polarized abelian varieties over 1Fp. A more detailed sketch of the proof of the Hecke orbit conjecture fo1· J?.19 can be found in
(3}; see also [4]. The proof uses a special p1·operty of Si19 , that every Fppoint of .fl19 is contained in a Hilbert modular subvariety of d 9 ; see 8.3. It is a consequence of the fact that the endo1norpllism algeb1·a (A, A) of ev·e1·y polarized abelian variety over iFp contains a. product of totally real fields F 1 x · · · x Fr fixed by the Rosati involution, with [F1 : Q] +···+[Fr : Q] = dim(A). The same t1·ain of thougl1t, together witl1 the consideration of the action of local stabilizer subgroups, leads to the trick of ''splitting at supe1·singular point',; see 8.2.
8.2. Proposition. Let ·n 2:: 3 be a posit,ive integer pr'i1ne top. Let xo = [(Ao, .Ao)] E d 9 ,n.,1Fr (Fp) be an Fp point of dg,n,jfp . Let C(xo) be the central leaf in d 9 ,n,Fp containing xo. Tliere exist
(i) an Fppoint
Xi
= [(A1, A1)]
in tlie Zariski closur·e in C(xo) of the primetop
Hecke orbit of xo,
(ii) totall'y 1~eal number fields F 1 , . .. , Fr ·wit'h [F1 : Q] that Fi©Q(Qp is a field for i
=
+ · · · + [~. : Q] = g
such
1, ... r, and 0
(iii) a subring of the endomorphism algebra End (A1) := End(A1)®z Q of fixed by the Rosati involution attached to A1 ) which is isomorphic to F 1 x · · • x Fr.
ChingLi Chai and Fra.n s Oort
256
Clearly if we can show that the p1·imetop Hecke orbit of x1 is dense in C(x1), then the pri1netop rlecke orbit of x 0 is dense in C(x 0 ).
8.3. Proposition. We keep the notation as in 8.2. There exist
(i) a positive integer m
~ 3
prime top)
(ii) a Heckeequivariant finite morphism
with respect to the embeddin,g SL2(F1) x · · · x SL2(Fr)  t Sp29 of algebraic groups, where MFi,m is the Hilbert modular variety with levelm structure attached to Fi, (iii) Heckeequivariant finite morphisms
with respect to the embedding SL2 (Fi)
t
Sp 2 [Fi :Q) for i == l, ... , r,
(iii) an Eppoint (z1, ... , Zr) of M ~,m,~  x ··· x M
~,m,~
with
such that for any rtuple of points Y1, ... , Yr of Mp1 >m ,.IC'p rrF"', ••• , Mp m corr, ,.IC'p respondin,q to {fpi linear [Fi : Q]dimensional abelian varieties Bi, ... , Br, the abelian variety corr·esponding to f 1 (Y1, ... , Yr) is (Fi x · · · x Fr )linearly isogenous to B 1 x · · · x Br, and the abelian variety corresponding to hi (Yi) is isogenous to Bi for i = 1, ... , r. Tl.'
The general linearization method implies quickly that the Zariski closure of the primetop Hecke orbit of Zi on the Hilbert modular variety MF· m iE contains p an open subset of the central leaf Cp(zi) in Mp. m IE passing through Zi for i == 1.' ' p 1, ... ,r, because Fi®QQp is a field. The irreducibility result [53] of C.F. Yu says that Cp(zi) is irreducible unless Zi is supersingular, in which case Cp(zi) is a finite set and the primetop Hecke correspondences opera.tes transitively on Cp(zi) Note that there exist hypersymmetric points in Cp(z,i ) for every i. The statement 8.4 follows. i'
'
8.4. Corollary. Notation as in 8.3. Ther·e exists an °iFppoint x2 == [(A2, A2)] E C(x 1 ) which lies in the Zariski clos'ure tlie primetop Hecke orbit of X1 such that A 2 is isogenous to a product of hype rsymmetric abelian varieties. 1
8.5. Theorem. Notation as in 8.2 ancl 8.3. The pr~imetop Hecke orbit 1{_(P) ·X1 of every f;point xi of dg ,n,~ is Zariski dense in the central leaf C(x1) in J.11'9 ,n,,~ .
As we have mentioned earliet', the combination of the local stabilizer principle and the generalized SerreTate theory for central leaves shows that the Zar·iski closure of the primetop Hecke orbit of the hypersymmetric point hi(zi) in the
Cl1apter 8
dr
The Hecke Orbit Conjecture
257
Siegel modular variety F.i :Q) m,jfp is dense .in C(hi ( Zi)). Another application of this method plus easy representation theory allows us conclude that the Zariski closu1'e of the primetop fIecke orl)it of x 1 contains an open subset of C(x1). However we k11ow that the cent1'al leaf C(x 1 ) is irreducible unless x 1 is supersingular, so we are done.
9
t
Open questions
9.1. In this section we list several open questions l'elated to the Hecke orbit conjecture. We have not attempted to put these questions in the most gene1·al setting possible. Instead we have formulated them in relatively simple cases where essential aspects of the difficulties are preserved. • Conjectures 9.2 and 9.3 are samples of strong local rigidity predictions in the direction of 5.4 (d). To make progress on them, the first step would be developing a theory of Qp such that geometric generic fiber V Xspec(OL) Spec(Qp) of Vis a Shimura subvai·iety of the moduli space dg,n Xspec(W(Fp)) Spec(Q°p)(ii) The scheme Z is an irreducible component of the closed fiber V Spec(Fp) of V.
Xspec(t7L)
9.5. Problem. Let Z be an irreducible closed subscheme of the Siegel moduli scheme ~ n,Fp :== dg,nXspec(W(ifp))Spec(iFp) over iFp which is li11ear at an ordinary point as in 9.4; i.e. there exists a closed point zo == [(Ao, Ao)] E d 9 )n (Fp) witl1 Ao ordinary such that Z I zo is a formal subtorus of 91 1zoF". Let A  4 d. n E be the 9 1
9,n1ll:p
'
l
p
11niversal abelian scheme over d 9 .n,Fp and let Zord be the largest open subset of Z such that the abelian scheme A x .rug ,n ,.=. Zord  t Zord over Z is ordinary. l'p
(i) Show that the Zariski closure of the image of the Galois representation attached to tl1e maximal etale quotient (AxP!I9,n.,Ep Zord) [p00 ]et of the pdivisiblc group (AX Jlf'g.n .iirj; Zord) [p ] over Zord is a reductive sul)gro11p Q Z,naivelzo of GL 9 (Vp(Ao[p00 ]et)). Here Ao[p00 ]et is the maximal etale quotient of tl1e pdivisible group A 0 [p 00 , a11d Vp(A 0 fp 00 ]et) is the padic Ta.te module of Ao [p00 ]et, noncanonically ison1orphic to Qi. 00
(ii) Let Tz .d be the Qplinear Tan11akia11 subcategory generated by the isocrystal 00 attached the pdivisible group Ax .fdg ,Fp _ Zord [p ] in the Tannakian category of all overconvergent isocrystals over Zord . 01
.fl
(a) Sho,v that the Galois group Gal(Tzord,zo) attached to the Tannakia11 category Tzord and the fiber functor at zo is reductive. (b) Show that tl1e1·e exists a parabolic subgroup P of Gal(Tzord ,zo) st1ch tl1at the unipote11t U radical of P is naturally isomorphic to the cocharaeter · group of the £01·rnal torus z !zo, a.n d tl1e reductive quotient of P/U is naturally isomorphic to Gz,naive:zu .
Chapter 8 The Hecl{e Orbit ConjectUI·e
259
let C be
9.6. Problem. Let C be a ~11tral leaf~ Pig over IFp, the Zariski closu1~e of C in d 9 , and let s 0 be an IFppoint of C, C. Analyze the structure of the formal /so

 / so
completion C of C at s 0 , and relate the structure of C structures of c/z as z varies over points of C.
to the family of linear •
References [1] C.L. Chai  Every ordinary symplectic isogeny class in positive characteristic is dense in the moduli space. Inv. Math. 121 (1995), 439 479. [2] C.1. Chai  A1onodromy of Heckeinvaria,n t subvarieties. Pure Appl. Math. Quarterly 1 (2005) (Special issue: in 1nemory of Armand Borel), 291 303. [3] C.L. Chai  Hecke orbits on Siegel modular va1ieties. Geometric 1'1ethods in Algebra. a.n d Nu1nber theory, ed. F. Bogomolov & Y. Tschinkel, Progress in Mathematics 235, Birkhauser, 2004, pp. 71 107.
[4] C.L. Chai  Hecke orbits as Sh.i mura varieties in positive characteristic. Inte1·national Congress of Mathematicians, Vol. II, pp. 295 312, Europ. Math. Soc., Ziirich, 2006. [5] C.L. Chai  Methods for· padic monodromy. J. Inst . 11ath. Jussieu 7 (2008), 2 7 268.
[6] C.L. Chai  A rigidity result for pd·i visible fomial groups. Asian J. Math. 12 (2008), no. 2, 193 202.
[7] C.L. Chai and F. Oort  Hypersymmetric abelian varieties. Pure Appl. Math. Quarterly 2 (2006) 110. 1, part 1 1 27. [8] C.L. Cl1ai and F. Oort  Moduli of abelian varieties and pdivisible groups: density of Hecke orbits, and a coriject·ur·e of Grothendieck. Arithmetic Geometry, ProceediI1g of Clay Mathematics l11stitt1te 2006 S11n1mer School on Arithmetic Geometry, Clay Ivlathematics Proceedings 8, Editors H. Darmo11, D. Ellwood, B. flassett, Y. Tschinkel, 2009, pp. 441 536. [9] C.L. Chai and F. Oort  Jvlonodromy arid irreducibility of leaves. Ann. of Math. 173 (2011), 1359 1396. [10] C.L. Cl1ai and F. Oort  Moduli of abelian ·varieties~ [This volume]
[11] C.L. Cl1ai a11d F. Oort  Sustained pdivisible grou,ps: a foliation retraced. [This ,,0J11n1e]
(12] C.L. Chai and F. Oort  Rigidity for biexte·nsio'ns of pdivisible formal groups. [Draft of a cl1apter of [13], available from www. math. upenn. edu/ chai]
[13] C.L. Chai and F. Oort  Hecke orbits. [Book project in preparation]
ChingLi Chai a.n d Frans Oort
260
[14] G. Faltings and C.L. Chai  Degeneration of abelian varieties. With an appendix by David Mumford. Ergebnisse Bd 22, Springer, 1990. (15] A. Grothendieck  Groupes de Barsotti Ta,te et cristaux de Dieudonne. Sem. Math. Sup. 45, Presses de l'Univ. de Mont1,eal, 1970.
[16] E. Hecke  Neuere Fortschritte in der Theorie der elliptischen Modulfunktionen. Proc. ICM Oslo, 1936, 140 156. (= Mathematische Werke, 627 643.) ••
[17] E. Hecke  Uber Modulfunktionen und die Dirich,letschen Reihen mit Eulerscher Produktentwicklung. I. Math. Annale11 114, 1 28. ( = Mathematische Werke, 644671.) ••
(18] E. Hecke  Uber Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. II. Math. Annalen 114, 316351. (= Mathematische Werke, 672 707.) [19] H. Hida  The Iwasawa µinvariant of padic Hecke Lfunctions. Ann. Math. 172 (2010), 41 137. [20] H. Hida  Irreducibility of the Igusa tower. Acta Mathematica Sinica English Series 25 (2009), 1 20. (21] H. Hida  Irreducibility of the Igusa tower over unitary Shimura varieties. Clay Mathematics Proceedings 13 (2011), 187 203.
[22] L. lliusie  Deformations de groupes de Barsotti Tate. Seminaire sm· les pi11ceaux arithmetiques: la conjecture de Mordell (ed. L. Szpiro, Paris 1983/84), Asterisque 127 (1985), Soc. Math. France, 151 198.
(23] N. Katz  Slope filtration of Fcrystals. Journees de Geometrie Algebrique de Rennes (Rennes, 1978), Vol. I, pp. 113 163, Asterisque, 63, Soc. Math. France, Paris, 1979.
[24] N. Katz  Appendix to Expose V. In: Surfaces algebriques (Editors: J. Giraud, L. Illusie, M. Raynaud). Lecture Notes in Math. 868, Springer, Berlin, 1981; pp. 127 137.
[25] N. Katz  Serre Tate local moduli. In: Algebraic surfaces (Orsay, 1976 78) , pp. 138 202, Lecture Notes in Math. 868, Springer, BerlinNew York, 1981. [26] R. E. Kottwitz  Points on som,e Shimura varieties over finite fields. J. Amer. Math. Soc. 2 (1992), 373 444.
[27] R. E. Kottwitz  lsocrystals with add itional structures. Compositio Math. 56 1
(1985), 201 220. [28] L.W. LanArithmetic compactificatioris of PELtype Shimura varieties. London Math. Soc. Monographs 36, Pri11ceton Univ. Press, 2013.
Chapter 8
The Hecke Orbit Conjecture
261
(29] J. Lubin, J .P. Ser1~e and J. Tate  Elliptic curves and formal groups. In: Lecture notes prepared in connection with the seminars held at the Summer Institute on Algebraic Geometry, Whitney Estate~ Woods Hole, Massachusetts, July 6July 31, 1964; availal)le fromwww.jmilne.org/math/Documents/. [30] Yu. Manin  The theory of commutative formal groups over fields of finite characteristic. Usp. Math. 18 (1963), 3 90; Russ. Math. Surveys 18 (1963), 1 80. [31] W. 1\'1essi11g  The crystals associated to Barsotti Tate groups: with applications to abelian schemes. Lecture Notes in Math. 264, SpringerVerlag, 1972. [32] B. Moonen  Serre Tate theory for moduli spaces of PEL type. Aim. Scient. Ee. Norm. Sup. 4e Serie 37 (2004), 223 269. [33) D. Mumford ~ Biextensions of fonnal g1·oups. In: Algebraic Geometry, P1♦0ceedings of Internat. Coll. Bombay, 1968, Oxford Univ. Press, 1969, 307 322. [34] D. Mumford  Abelian 'Varieties. Tata Institute of Fundamental Research Studies in Mathematics No. 5, Oxford Univ. Press, 1970. [35] D. 1vlumford  Geometric invariant theory (3rd ed.). SpringerVerlag, 1965, 1982, (3rd ed, with J. Fogarthy and F, Kirwan) 1994. [36] ti.H. Nicole and A. Vasiu  Traverso 's isogeny conjecture for pdivisible g1~oups. Rend. Semin. Mat. U. Padova 118 (2008), 73 83. (37] P. Norman  Lifting abelian varieties. Inv~ Math. 64 (1981), 431 443.
[38] P. Norman and F. Oort  Moduli of abelian varieties. Ann. of Math. 112 (1980), 413 439. [39] T . Oda and F. Oort  Supersingular abelian varieties. Itl. Sympos. Algebr. Geo1n. Kyoto 1977 (Editor: M. Nagata). Kinokuniya Bookstore 1978, pp. 595 621. [40] F. Oo1·t  Cornmuta.tive gr·oup schernes. Lectm·e Notes in Math. 15, SpringerVerlag, 1966.
(41] F. Oort  Some questions in algebraic geometry. Unpublished manuscript, Jru1e 1995. Available at http://www.math.uu.nl/people/oort/.
http://igiturarchive.library.uu.nl/math/20061208201315/oort_
95_some_questions_in.pdf. Available at http://www. math. uu. nl/people/ oort/.
[42) F. Oo1·t  Newton polygons and formal gro11,ps: conject11.1·es by Manin and Gr'Otliendieck. Ann. of i\1ath. 152 (2000) 183 206. (43] F. Oort  A stratification of a moduli space of polar'ized abelian varieties. In: Mod11li of alJelian varieties. (Texel Island, 1999, Editors: C. Faber, G. van der Geer, F. Oort). Progress Math. 195, Birkl1auser Verlag, 2001; pp. 345 416.
262
ChingLi Chai and Frans Oort
[44] F. Oort  Newton polygon strata in the moduli space of abelian varieties. In: Moduli of abelian varieties. (Editors: C. Faber, G. van de1· Geer, F. Oort). Progress Math. 195, Birkhauser Verlag, 2001; pp. 417 440. [45] F. Oort  Foliations in moduli spaces of abelian varieties. J. Amer. Math. Soc. 17 (2004), 267296. [46] F. Oort  Minirrial pd'ivisible groups. Ann. of Math. 161 (2005), 1 16. [47] F. Oort and T. Zink  Families of pdivisible groups with consta nt Newton polygon. Doc11menta Math. 7 (2002), 183 201. 1
[48] J. Tate  Endomorpfiis'ms of' abelian varieties over finite fields. Inv. Math. 2, 1966, 134144. [49] J. Tate  Class d'isogenie des varietes abeliennes sur un corps fini (d'ap1~es T. Honda). Semina.ire Bourbaki, 1968/69, No. 352. Lecture Notes in Math. 179, SpringerVerlag, 1971, 95 110. [50] A. Vasiu  Deformation subspaces of pdivisible groups as fonnal Lie groups associated topdivisible groups. J. Algebraic Geom. 20 (2011), 1 45. arXi v: math/0607508.
[51] E. Viehmann  Moduli spaces of pdivisible groups. Journ. Algebr. G 01n. 17 (2008), 341 374. [52] E. Viehma,n n  The global structure of m,oduli spaces of polarized pdivisible groups. Documenta Math. 13 (2008), 825 852. {53] C.F. Yu  Discrete Hecke orbit problems for HilbertBlumenthal varieties, preprint 2004. [54] T. Zink  The display of' a formal pdivisible group. Cohomologies padiques et applications arithmetiques, I. Asterisque No. 278, Societe Mathematique de France, 2002, pp. 127 248. [55] T. Zink  On the slope filtration. Duke Math. J. 109 (2001), 79 95. [56] Y. Zong  On hypersymmetric abelian varieties. Ph.D. dissertation, University of Pennsylvania, 2008.
Open Problems in Arithmetic Algebraic Geomet1y ALM 46, A. 1, pp. 263283
© Higher Education Press and International Press BeijingBoston
Appendix 1 Some Questions in Algebraic Geometry Frans Oort (tJriginal versio11, J tine 1995)
aone fool can ask n1ore questions than ten vtrise men (:an answer.'' An old Dutcli sayi:n.g.
"Ee11 zot ka.n mccr vragcn clan tie11 wijzen k1111nen hea11twoor 7. For a pri111c 11un1l)er > 7, tl1is gives ct oncdimensio11al f,11nily in j(M 9 ) c Ag,1 but. tl1e s111allest Sl1i11111ra variety contai11111g it l1as 11di11g fiber ha.".; ct .J ac:r>ia11 witl1 8tnC!vI i8 fiuite.
e,
1
e
5
Frans Oort
268
5.3. By the way, 11ote tl1at for a prime 1111mber JJ, tl1e set M 9 (Fp) is infinite. and e,.rery poi11t corresponds witl1 a JacolJian v,.rhich admits s111Cl\l (l)_y a tbeoren1 b:y Tate 1 see [63]). \''e st1e that the characteristic p analog of the c1,1cstion by Coleman gi,~cs an io611itc set for e,~cry p a11d every g > 0.
w;;,
5.4. One could for111ulate the ''local Colen1ari probleni ··': fix a11 algebraic curve C10 o, cr and consiclcr the set £( Co) of iso111orpliisn1 cl~IBses of c11. . c11rves clefi11ed o,rer C ~rlucl1 111odtLlo p recluc·e to Co (one ha.~ to give proper an(l prera11k strata arc part,iculc1r cases of the NPstrata. For any .9 and an~r .f witl1 0 ~ f ~ .9 there is a unique NP 113 sucl1 that e,,er:y· NP ljing above /3 has prank at 111ost f, i11 partic11lar ~3
Ii1 fa 2):
clin1(V1 n (:Jg ® 1Fp))
?
== 3g 
:,  g + f.
Orie col1ld l1ope tl1at intersections as above give effective cycles in Mg ® 1F p of whic;l1 tl1e Cl1c1w cl&;ses ca11 be c.:on1puted. Tl1is 111ight give insigl1t in tl1e Cl1ow ri11g of M 9 .
9.2. We see a kind of question wl1icl1 in ge11eral is diffic:ult. Consider two sl1bsets of a rnocl11li space, eac~l1 cl1aracterized by certain properties of t11e object8 we classify. Try to cletern1ir1e propertie8 of tl1e ir1tersection of tl1ese sets. If tl1e pro1)erties are difficult to compare, such a c1t1estion s~eru8 difficult in ge11eral. Example are tl1e following: consider CM.JacolJiaris (see 5) 1 or Ja,eol)ians in p(lsitive characterLc.;tic witl1 a giver1 Newto111>olygor1 as above, c)r .Ja('obia11s with a given e11do111orpl1ism ring [5], iniersectir1g JVIurr1ford Cllrves witl1 the Torelli loctts as in (7B), or study 1noduli spaces of hyperelliptic supersi11gular curves [48].
10. Algebraic computation of fundamental groups.
For fill algebraic: variety Grothenraic c11rve i11 c11ara.c teristic zerc) the Rtruct11re of its fu11clan1ental gTOUl) is deter11iiuetl witl1 the help of co1nparison with tl1e topological .fl1n R  t k is a re:;idt1e class n1ap, Xo ~ X ® k with char(k) = p > 0, tl1e11 tl1e natural 111ap E11d(X) ~ Eud(Xo) is injective, a11d tl1e i11dex Note tl1at if X
t
[E11d0 (X)
n End(Xo)
: End(X)]
is a power of p.
12A. Q t1estion: Describe lio·w the Tatepgroups sclienie o,f an abelian variety X i·n. chara,ctcristic zero reduces to a submodule of the Dieudonrie rnodule of' X 0 ; use this to show tliat in certain cases C'Mliftin._qs do not exist. I.ii particular it n1igl1t be used to a11swer tl1e followi11g questions:
12
Appendix 1 Some Questions in Algebraic Geometry
275
12B . Question: Let E be a supersirtgti.lar C'ltrve rdinate~·>, Hee (31], see [24], Cl1c1,p ter !5, see (33]. Nc>te that the torsion J)oints iu 11l1e for111al group
(A g,l )" :r.o
~
((Grn )A)g(g+l)/2
correspond \\rith tibe ·'q11asicanonical liftings': of Xo ~ i.e. tl1e Cl\1liftings (but End(X) need 11ot be a maximRl order i11 End 0 (X) = End0 (Xo)). ,,re po:e t.l1e q,1e~tio11 "~l1etheT s11ch a '"cano11ical': paran1etrizatio11 is possible in tl1e nonorclinaT}r case.
13 . Question: SupposP ,gi'ven. an, abcl1.an 11ariety .1Yo u1itl1. a polariza.tion, .Xo over a. fin,ite field and r1, C!vll'ifting (X, .X). Can, 111e define ''canor,,iral ClJordin,n,tes)' ori (Ay,l )~? Variol1s attempt~ have beer1 made in tl1e past; see (30]. (16], ((i8] [2], (67]. Note tl1at different chc)ices of a Cl\1flifti11g of ...¥0 belo11gi11g to different crvr Stll)algebras of E11d0(Xo) 1r1ay gi,re quite differe11t coordi11ate s~ sterns. This question shot1ld be made n1t1ch more fJrC\cise before it ca.11 l)e t,,lcen seriously. i
1
13.1 . Remark: Tl1e terrninolog~' "car1onical lifti11g:! 1nigl1t cause co11fusion. I inte11d to use tltis J)hr,lS only" in c,1s0 ..X0 is an ordinary abclia11 variety, Some autl1ors use this concept for an arl)it rary abclia11 ,,a,riety over a finite fielcl requiring that the geometric Frol)enius can also l1c lifted; for a11 ordi11ary Xo we do get t l1c rigJ1t co11ccpt, b11t for no11orclinary al)elia11 vilrietioF t:l1erc n1ay l)c r11ai1y lifti11gs such tl1at tl1c geon1ctric Frobcnius lifts alo11g (e.g. a .11persj11g1.1Iar elliptic c1.1rve E 11 12 lE) Also for abelia.11 varieties o,rer a 11011finite field in positive characteristic \\'e can define a ca11or1ical lift,i11g for an ordinarj" Xo in tl1e SerreTai.e tl1eory: l1ov.re,.,.er iu thcit c~~e ther is no geon1et,ric Frol>eni11s.
14. Comp1ete sttbvarieties. 1~his rnateria l is partly ta.ken fro111 (51]. We o sucl1 tha.t cp*(A) ::::: rn,·Jt,. This set Q(x) is called the Hecke orbit of x in A. II we co11sider only isogenies with degree prime to p we write
the Hecke1>rirnet()p c>rl)it. Tl1e fr>ower Hecl{e orl)it 9e(x) is the set wl1ere we cc)nsider 011ly isogenies where the cleg( :.p) is a power of the pri1r1e 1111rr1ber e. Clearly
if€/= p. The8e defi11itio11s car1 lJe founc1 i11 [41, a11d i11 that paper by Cbai we find (Th. 2): Theorem: S ·t1,ppose e is a p'r •irne ·nti·rr1,be·r difj'e•rerit fr·oillt with ir1t witl1 a== 1, HO in the first case wfl see that g(p)(x) c V0 (a == 2); if the co11jecture is correct tl1is sl1ot1.ld be a de11se subset. 111 case a,(X) == 1 we can show tl1at Q(P)(:r;) iH not er fon11ulc1tio~5 of this ('onjecture: e.g. st:=> (44], SecLio11 3.
19.3. S,1ppose (ST\\1) does ltc)lcl. \\7e can st11d)" tl1e following ho1mdedness couc1ition (,,rhere 'n E Z>o): (Bn)? : For e·ue1·y ell'i1Jtic c·urve E o·uer· Q ~n·ith co·r1,d1.tctor N : cond11cto1·(E) tliere ex·ists a p(1·r·a1r1,et,,~;zatio1z, (11,o·rico1i:;tc1,rit rnorph.i~1n} ooer· Q: cp : Xo(1V) ~ E
·wit/1,
ectatio11/ c111estio11 lik~ (Bn)? is rea5onable. 16
Appendix 1
S01ne Qt1estions in Algebraic Geon1etry
279
19A 7 5Tit,pposc C'on_jecture (STW) is correct, and suppc,se tliat there exists a, positive inleger ·n. such that the boundedness Bn holds. Then there exists o E JR sucli that (MO;a) holds. This seems to be rea..sonable, 011ce deep res11lts like (STW)? and (Bn.)? are settlecl.
Acknowledgments: During many years I l1ave prc>fited on a large scale ft· 2.
5. Abelian varieties isogenous to a Jacobian. (Frans Oort) See [12], 8.0.1, pp. 165 168. This problem has been solved. In [6], 1.3 and 3.1 we find that under AO a11y proper subset
and any x = [(A,A)] E A9,1(Q) the Hecke
01..bit
1l(x) of x has an empty intersection with X ?t(x) n X
= 0.
In [28] we find a proof for this fact without assuming AO; also we know that AO holds for A 9 in characteristic zero, hence both :references solve a generalization of the problem asked. Also see (31] where the ainalogous problem is discussed (but not sol,1ed) over fields of positive characteristic.
6. Newton polygons and the prank. (Frans Oort) See [12], 8.5.1  8.5.6, pp. 168 172. For 1nore information and references, see (12] Section 5, and [24].
6.1. Question. For which pair (p, ~) is there an algebraic curve in cliaracteristic p havi·ng as (symm,etr·ic) Newton Polygon?
e
6.2. Example (private communication Regis Blacl1e, 2009). For p g == 11 the hyper·elliptic curve
=2
and for
has Newton slopes 5/11 and 6/11. Her1ce we see that [12], 8.5.3 does 11ot hold for p be said or decided £01~ other vall1es p > 2.
= 2.
I have no idea wl1at can
290
Editors: Gunther Cornelissen and Frans Oort
6.3. Example (see [24], 5.3). The curve Z(Y
2
+Y + X + X 25
9
) C
Ai,2
C
P:2
has Newton slopes 5/12 and 7/12.
Note that we do not know whether the question above is (in)dependent of the choice of p; it might very well be that there exist (p 1 , e) that does appear and (P2, () that does not.
7. Deformations of a stable curve with constant Newton polygon. (Frans Oort)
e'
See [12], 8.5.7, p. 172. Suppose and~,, are Newton polygons. We write c;'LJ~'' for the Newton polygon consisting of all slopes of~' and ('' with their multiplicities. 7.1. Conjecture. Suppose C' and C'' are curves ( absolutely irreducible, reduced, complete) over a field k == k :) lFp with Newton polygons ~' and~''. Write g == g' + g'' = genus( C') + genus( C''). We conjecture there exist nonempty, closed sets S (C', C'') C C' respectively S (C'', C') C C'' such that for every P E S (C', C'') (k) and Q E S (C'', C') (k) the stable curve
Co= C'
UP=Q
C''
obtained by attaching the components via a normal crossing has the property that Co can be deformed {in equicharacteristic p) to a curve with Newton polygon (' U We expect these closed sets can be chosen as the maximal ones having this property.
e''.
We note that there exist situations where S (C', C'') can be equal to C', and also situations where S( C', C'') is a finite set.
7.2. Corollary of the conjecture. For every g and every prime number p there exists a supersingular curve of genus g in characteristic p. This is known in special cases, see [24] for results and references. We see that the conjecture would imply that every Newton polygon (arbitrary genus) with all slopes having a small denomina,tor appears: 1/2 and (1/3) + (2/3) do occur, hence unions of these do appear if the conjecture holds.
8. The maximal number of automorphisms of curves with a • given genus. (Frans Oort) See [12], 9.3 and 9.5, p. 173. W11ile we have a good insight about the achieved maximum of number of automorphism of an algebraic curve of given genus in characteristic zero, it seems that 1nuch work still has to be done for the analogous problem in positive characteristic; partial 1·esults are known, e.g. see (16].
Appendix 2
Automorphisms of Curves2005 Collection
291
References [1] K. Azdorf  Semistable red1tction of cyclic covers of prime power degr. ee. PhD thesis) Leibniz University Hannover (2012). (2) K. Arzdorf and S. Wewers  Another proof of the semistable reduction theorem. arXiv:1211.4624 (2012). (3] L.H. Brewis and S. Wewers  Artin characters, Hurwitz trees and the lifting problem. Math. Ann. 345 (2009), 711 730.
[4] J. Byszewski, G. Cornelissen and F. Kato  Un anneatt de deformation universel en conducteur superieur. Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 2, 25 27.
[5] J. Byszewski and G. Cornelissen  Which weakly ramified group actions admit a universalfo1nial deformation? Ann. Inst. Fourier (Grenoble) 59 (2009), no. 3, 877 902.
[6] C.L. Chai a.nd F. Oort  Abelian varieties isogenous to a Jacobian. Ann. of Math. (2) 176 (2012), 589635. [7] T. Chinburg, R. Guralnick and D. Harbater,  The local lifting problem for actions of finite groups on curves. Ann. Sci. Ee. Norm. Super. (4) 44 (2011), 110. 4, 537 605.
[8] P. Chretien a11d M. Matignon  Big action with a non abelian derived subgroup. arXi v: 1206. 446 (2012). [9] P. Chretien a11d M. Matignon  Maximal wild monodromy in unequal characteristic. J. Number Theory 133 (2013), no. 4, 1389 1408.
(10] P. Chretien  Lifting ArtinSchreier covers with maximal wild monodromy. Manuscripta I\ lath. 143 (2014), no. 12, 253 271~ 1
(11] G. Co1·nelissen a.n d A. Mezard  Relevements des revetements de courbes faiblement ramifies. Math. Z. 254 (2006), no. 2 , 239 255. [12] G. Cornelissen and F. Oort (editors)  Problems from the workshop on automorphisms of curves. Leiden, August 2004. Rend. Sem. Math. Univ. Padua 113 (2005), 130 177.
https://doi.org/10.1016/S00074497(00)010757 http://Janl.arxiv.org/abs/math/0411059 [13] T. Dol 0 munies d'un gros pgroupe d'automorphismes. These Universite de Bordeaux, 2008. See https://www.math.ubordeaux.fr/roroatigno/RocherThese.pdf
[26] M. R,ocher  Large pgroups actions with a pelementary abelian second ramification group. Journ. Alg. 321 (2009), 704 740. (27] A. Silverberg and Y. Zarhin  Jn.ertia .91~oups arid abelian surfaces. Joun. N11mber Theory 110 (2005), no. 1, 178198. [28] J. Tsimcrrr1a11  The existence of' an abelian var·iety over Q isogenoul, to rio Jacobian. Ann. of Math. (2) 176 (2012), no. 1, 637  650.
(29] D. Turcbettj  These Unive1·site cle Ve1~sailles (2014). See http://YTYw.theses.fr/2014VERS0022 [30] D. Tur·chetti  Equidistan1t liftings of elementa·ry abelian Galois covers of curves .. arXi v: 1510. 06453v2.
Appendix 2 Automorphisms of Curves2005 Collection
293
(31] A. Shankar and J. Tsimerman  Unlikely intersections in finite characteristic. https://arxiv.org/abs/1610.03552 [32] B. Weaver  The Local Lifting Problem for D4. arXi v: 1706. 03751.
•
Open Problems in Arithn1etic Algebraic Geomet1'.Y
© Hig·J1er Education Press and International Press
AL1VI46, A. 3 pp. 295 33'1
Be{jingBoston
Appendix 3 Questions in Arithmetic Algebraic Geometry Editor:
F)~a.11s 001,t*
In 913 November 2015 several mathematicians did meet in Leiden for a conference
Moduli Spaces and Arithmetic Geometry. This collection of problemf,, open questions and conjectures is written for the occa5ion of that conference.
At the BIRS conference "Lifting Proble1ns and Galois Theory'', Banff, 17 21 August 2015, we l1ad a11 inspi1'ing P1,oblem Sectio11. Questions presented there a1'e incl11ded (i11 revised forn1) below. I thank the organizers of tha.t conference for inspiration>a.nd J eroen Sijsling for editing those proble,m s. Later proble1ns were updated, a.n d some new p1·oblems were added.
I thank all colleagues who contributed to this collection cooperation.
£01·
their inspiring
1. Points on Hecke translates of a divisor. (Jeff Achter) Lets( q) denote the probal)ility that a rando1nlychosen principally polarized abelian surface over 1Fq is IF qisogeno11s to a product of elliptic curves. It ttirns out that •Frans Oort, lviatl1ematical Institute, University of Utrecht. Email: [email protected]
295
296
Editor: Frans Oort
s(q) varies like 1//q. In fact, one can show that tl1e1·e are positive constants c1 and c2 such tl1at
see [2] for details. From a geometric perspective, it might be more natural to consider s* ( q), the probability that a randomlychosen principally polarized abelia11 surface is absolutely isogenous to a product of elliptic curves.
Expectation 1.1. The func'tion s* (q) satisfies bounds like those in (l). In fact, consider the diviso1· A 1 x A 1 inside A 2 . Then s*(q) is the normalized cl1aracteristic function of tl1e JF qpoi11ts of the in1age of this divisor under all Hecke correspondences. Now let S be a ShimUI·a variety, and let D c S be a divisor which is not stable 1mde1· Hecke correspondences. (Tht1s, in the case of fixed cha1~acteristic, we specifically forbid the non1,1,ordinary locus.) Let ?lD be tl1e 11nion of the images of V under all IIecke co1~respondences.
Question 1.2. Does the function #HD(Fq)/#S(Fq) vary like
1/ Jq?
It seems that, each time one has an affirmative a11Swer to Question 1.2~ a co11jecture of LangTrotter type is plausible ((2, Sec. 2]).
Remark 1.3. Some prelimi11a1·y results a.r e available in the case where the ambient, Shimura variety is the space of principally polarized abelian surfaces and the divisor is a HilbertB]11menthal modular surface. Pairs (V, S) where V is also a Shimura variety are esset1tjally classified in [44].
Remark 1.4. In [114, Prob. 21], Oort and Moonen ask for information on the size of an isogeny class of principally polarized abelian varieties, when ave1. aged over abeliar1 varieties with a given Newton polygon. If one restricts their philosophy to the case of ordinary abelian varieties, and then generalizes their ideas to arbitrary Shimura varieties, one fi11ds a broad compatibility l)etween their expectation and Question 1. 2.
2. Eigenvarieties at the boundary. (Fabrizio Andreatta) Let p be a prime 11umber. Robert Coleman noticed in (24] that the characteristic series Pk(X) :== det(l  XUp) of the Up operator on the eigencurve [25] has coefficients in the Iwasawa algebra A= Zp[z;] and, hence, admits reduction P(T) modulo p . In private, 11npublished notes he formula.t ed the following conjecture:
Conjecture 2.1. There exist a Banach space in char·acteristic p and a compact operator whose Fredholm series is precisely P(T).
Appendix 3
Questions in Arithn1etic Algel)raic Geometry
297
Recently in collaboration with Adrian Iovita and ViJ.1cent Pilloni [4] we establisl1ed such co11ject1u. e l)roving that tl1e eige11curve ca11 be extended to H11ber's adic versio11 of weight space. Tl1is differs n·o1n the classical one considered in (25] exa,ctly by the presence of an extra, characteristic p point (the point at infinity). I{. Buzza1·d and L. Kilford [10] have prove11 that= for p == 2, the eigencurve has a) very uniform beha,rior when one app1·oaches the bou11dary. Mo1·e pI·ecisely t.hey sl1o"red that if t.he weigl1t ~ is padically close enough to 1, namely v(K,(5)1) < 3, tl1e11 the slopes of the UP operator on tl1e overco11vergent fo1·ms of weight K are O, t, 2t, 3t, .... I11spil"ed lJy this work Colernan also conject11red tl1e following: Conjecture 2.2. Let (nj, m i ) with .i 2 0 and ni+l > '"i be the breaking points of the Newton polygon of P(T). The·n for every padic we·ight Ki satisfying vp(K(l + q)  1) < ·vp(q) the break·in.g po·i11,ts of Pfi(X) are (n,.i , vp(Ki(l + q)  l)1n.i) (her~ q = p if p is odd a.n d q = 4 if p = 2).
J. Bergdall and R. Pollack ha:\'e also proven that this conjecture implies that tl1e slopes of the Newto11 polygon of P(X) a1~e the union of finitely many arithmetic progressions. Fo1· qtlaternionic overconver·gent forms Conjecture 2.2 has been established by R. Li11, D. Wan and L. Xiao in recent worl{ (with some explicit bounds, wltlch a1·e riot as good as those p1·esc1·ilJed by Coleman). They also give an independent proof of the result of Bergdall and Pollack in this setting. The constrt1ction in [4] is quite geometric and is inspired by I{atz· construction of convergent forms using the Igusa tower.
Question 2.3. Does ther·e exists a geometric explanation j'or ·this characteristic p pheriomenon of aritlimetic pr·ogression? Question 2.4. What car, we say about th.e geometry of the rnap from the adic eigencurve of [4] to the ad'ic weight space, close to oo?
The works of Bt1zzard and Kilford and b)i Liu, WaJ1 and Xiao i1nply that tl1e Conjecture 2.2 and tl1e first question are very strongly 1·elatred and suggest that the adic eigencurve close to the boundary should be the disjoint. m1ion of infinitely n1any connected components each one finite and flat over (an open) of the weight space. The second question amounts to asking for a geo1netric p1·oof/reason of this behavior.
3. Irreducible polynomials with unit coefficients. (Lior BarySoroker) Let Sn be the set of polynomials of the form
f (x)
== xn
± xn.1 ± ... ± 1
(2)
tl1at are irreducible. Question 3.1. What proportion of polynom,ials of the for,n (2) are irred1.tcible? In other words: what is the value of
. #Sn 7 l 1m   .
rt+(X)
2n
(3)
298
Editor: Frans Oort The limit is conjectured to be 1. Known results:
1. Poonen [103] considered polynomials with coefficients in {O, 1} and showed that the liminf is at least 1/n.
2. Konyagin [59] improved this bound to a (nonzero) multiple 1/log('n ). We can also ask these questions modulo large p. Large sieve might help to lift results modulo large primes to Z, in a similar fashion to Cohen's (22] p1·oof of the Hilbert irreducibility theorem. Modulo p = 2, I= 1, so all polynomials in Sn reduce to xn + xn l + •••+ 1. Using Artin's conjecture on primitive roots (which is known to follow from the generalized Rien1ann hypothesis) we have infinitely many n, those of the form n == p  1 with 2 is primitive modulo p, such that all polynomials in Sn are irreducible modulo 2; hence irreducible in Z[X]. This question was posed by «some guy on the street" [sic] on MathOverflow, see [103].
4 . Construction of Hodgetype line bundles in characteristic
p. (Bhargav Bhatt) Let S be a smooth scheme over a perfect field k of characteristic p. Let E be an Fcrystal on S, i.e., E is a crystal of coherent sheaves on the crystalline site of S relative to W :== W(k), and one is given a map ¢E : ¢ 8E ~ E of crystals that is an isomorphism after inverting p. To such an E, one can associate a line b11nd1e LE E Pic(S) that, roughly speaking, measures the variation of the coker11el of Frobenius ¢E• This construction is recalled in the next paragraph, and is extracted from [9]. ~ If S = Spec(A) is affine and A is a deformation of A to W equipped with a map ¢ A lifting Frobenius on A, then evaluation of E gives a finitely generated ~ Amodule M equipped with a map ¢M : ¢JM ► M which is a.n isomorphism after inverting p. The twoterm complex ( 1>1M ► M) defines a point of the Ktheory space KA(A) parameterizing perfect complexes of Amodules which are ~ acyclic after inverting p. By Quillen's theorem, since A is regular, inclusion defines an equivalence K(A) ~ KA(A). Thus, we obtain a point of K(A). Applying determinants defines a line bundle L(M) E Pic(A). (More informally: one filters the isogeny ¢Min such a way that the graded pieces are killed by p, and then simply tensors together the determinants of the graded pieces.) One can show that this construction is canonically co1npatible with localization, a.rid thus globalizes to define a lir1e b11ndle LE E Pic(S). Now let f : X ~ S be a proper (log) smooth morphism of F pschemes. Then for each cohomological degree i, one obtains an Fcrystal Ei :== Ri f*Ox,crys, with C. . is in addition projective: under sorne extra assun1ptions it is proved in [37] tl1at the question has a positive a11Swer.
10. Good, ordinary reduction.
(Folklore / Frans Oort) Question 10.1. Let A be ari abelian variety over. a nurnbe1· field I(. Doe.s tlie1·e e:i;isi a prim,e of K where A ha.s good ordinary r·eduction? See [113], Question 11.
In fact, Serre niade the conjecture that tl1e set of such pri1nes has de1isity 011e in some exte1ision L => K · this conjecture is rnore precise: L sl1oulcl be . chosen in such a way tl1at the image of the Galois rep1·csentation 011 each of the Tate groups of A1..1 is co1111ected. See [98], Vol. II, page 710. For elli ptj c c11rves Serre showed this to be t1·11e. In case rliln (A) S 2 tl1e answer is affirmative, as proved on pp. 370372 of [72]; also see [87], [93]. For a (potentially) C1'1I abelian variety the Newton polygons of the reductions can be dete1·minecl and in this case primes of good, ordinary reduction do exist. For· abelia11 va.1:ieties ' 0 and where n == ord(N) is a power of p. In this case the answer to 11.1 is shown to be affirmative by Rene Schoof if: • If (mR)P
=0
a,nd p·mR
== 0.
This includes the case ( mR) 2
See [94], Theorem 1.1.
= 0, and
it includes the case
mR ==
0, i.e. R is a
field. It seems the answer to Question 11.1 in other cases in ge11eral is not known. I thank Rene Schoof f'or discussions on this problem. See Cl1apter 1 of this volun1e for a further details.
Appendix 3
Questions in Aritllilletic Algebraic Geometry
307
12. Rational points on curves? Not very likely. (Dick Gross) Question 12.1. Sho'lLJ th,at n1,ost cur1Jes X of positi·ve gerius over· Q have no rational points, even a,ssuming tliat they lia1Je local points at all places. Everi stronger, sliow that for· most curves X of positive genu.r;, Pic 1 (X) is a nontrivial homoge0 neous space for tlie abelian va1·iety Pic ( X). There are results in this direction fo1· hype1·elliptic curves, due to Bhargava, and others. See [7], [8] .
13. Questions on families of pdivisible groups. (Shushi Harashita) Let R be a discrete val11ation ring of characteristic p > 0. This question may cont1·ibute to the classification of isoge11y classes of pdivisible groups ove1· R. Let X be a pdivisible g1·ot1p over R. Let ( and ~ be the Newton polygo11s of Xk a11cl XK 1·espectively, where k is the residue field of R and K is the fraction field of R. We call the pair ((,e) the NPtype of X, ai1d especially sa.y that Xis JVPcoristant if ( = ( a11d that X is NPsaturated if ( < ~ is saturatecl 1 i.e., the1·e is no Newton polygo11 p1·operly betwee11 (and~ We know:
(1) Any NPconstant pdivisible gro11p over R is isogenous to a con1pletely slope divisible pdivisible group over R. See Zink [112, Theorem 7] and OortZink (85, Theorem 2.1]. (2) An.y NPsaturated pdivisible group over R is isogenous to a. fil)erwise n1inirnal pdivisible group. See (45, Corollary 1.1]. A pdivisible group is said to be jiber'wise miriimal if its geometrical fi1Je1·s are all mi11i1nal. See (79] for tl1e definitio11 of rr1inimal pclivisible groups. This result can be regarded as an anc:ilogue of ( 1): since the assertion of ( 1) still holds if we reJ)lace "completely slope divisible:: by '•fiberwise miniroal11 ( cf. [45 Proposition 4.1 ]) .
W((k)
These facts lead us to ask the follow_!_ng. Let denote the set of ison1orphism classes of pdivisible g1·ot1ps over k of Newton polygon (.
Question 13.1. Let ( and~ be Newton polygons with ( < (. Find a min'imal s·ubset P of W((k) such that for any pdi11isible group X over R of NPtype ((, ~) tliere exi.si,s an isogeny from X to a pdivisible ,qroup Y s1Lch that ~ belongs to P and YK is a minimal pdivisible group.
e
The above facts say t.l1at we can tal 1 such that the higher ramification groups G 1 , ... , G1 are all equal (see (97, Chapter IV] for the definition of these groups). Equivalently, j is maximal such that xj+I divides g(x)  x for all g E G of ppower order. We say that a cyclicbyp group G is an alniost local Oort group for p if the1~e is a positive integer N such that for every algebraically closed field k of characte1,istic p, every GGalois cover of k[[x]] whose fir·st wild jump is greater than N lifts to a GGalois extension of R[[t]] for some complete disc1~ete valuation ring R of characteristic zero having residue field k. Question 14.2. Which cyclicbyp groups are almost local Oort groups for p?
It seems likely tl1at the almost local Oort groups are the same as the local Oort gro11ps. On the other hand, for p > 2 and n ~ 2, (Z/pz)n is a weak local Oo1·t
Appendix 3
Q11estio11s i11 Aritl1metic Algebraic Geon1etry
309
g1·011p (sec [62]), bl1t not a local Oort group. If G contai11s a Sl1bgroup of tl1e form (Z/pZ) 2 x 'll/'1r1Z with rri > l 1Jri111e top, then G is not a weak local Oort group. A11d 'll/pnz > 0 and many cases with m(A) (/. Z do exist.
Question 22.2. Si1.ppose given, o, positive ra.tio·nal ·n itmber J1, E Q>o and a pr..ime num.ber p. Does there exist a simple abel1:an variety A over an algebraically closed field k =, lFp s ucli tliat ni(A) = µ? 1
I expect every positi,re rational number does appear· in this way in eve1·y positive characteristic. 11ai1)' exc1n1ples of 11011integral rat.ional numbe1· appea1·ing this way ca11 be given; everyµ= 1/d with d E Z> 1 does aJ)I)ear. Question 22.3. Suppose giveri a positiue rational nu·mber a d == µ E Q >o
with gcd(a, d) == 1 a,nd d > 1. There exists an A with, µ(A) == 1/d. Let B == Aa., say 1.uith diagonal polarizatiori ..X. arid diagonal action i : D :== End(A) c > E11d(B). Does ther·e exist a deforrriation of (B, ,,\, t.) sucli tha.t the generic fiber has eridom,orph,ism algebra. isomorp/1,ic to D ? •
If so, we l1ave µ(B) this case.
= a/b
and ,ve see the question has a positive answer in
Editor: Frans Oort
316
23. MumfordShimura curves in the Torelli locus? (Frans Oort) 23.1. Let ~ = At, 1 ® C be the moduli space of principally polarized abelian varieties over C. Mumford constructed Shimura varieties (special subvarieties) of dimension 1 in A 4 in [67]. Any geometric generic point corresponds to an abelian variety with endomorphism ring Z. We refer to these curves in At ® C by calling them MumfordShimura curves. The terminology Mumford curve is sometimes used in a different co11text; this might explain the terminology in our case. Sometimes a curve of genus 4 whose moduli point is on of the MumfordShimu.I·a curves is called ''a curve of which the Jacobian is of Mumford type". Hecke correspondences can be applied to obtain co11ntably many of such mutually diffe1..ent curves. The union of all MumfordShimura curves will be denoted by M 84 c Ai ® C. The image of tl1e Torelli morphism M 9 ► A 9 is called the open Torelli locus. The Zariski closure
r:
Tg
:==
o)Zar (½
C
A 9 ,1
is called the closed Torelli locus. Question 23.2. Is any of the MumfordShimura curve.~ conta,ined in the To1·elli
locus
4 c ~?
Note that none of the MumfordShimura curves is contained in 8(7:i) :==
4 \ ~o.
More generally, one can ask for Shimura varieties in A 9 conta.ined in Tg, meeting the open Torelli locus Yg0 c Tg; for a discussion, and some references see [66], i11 particular Expectation (4.2) and further discussions.
It seems hard to give examples of curves where the moduli point is on a MumfordShimura curve. In [98], on page 716 we find a discussion of sucl1 curves: "Il sera it specialement amusant de trouver une cou1·be de genre 4 dont la jacobienne soit du type Mumford''; in comn1ent S12 on page 898 Serre remar·ks that it seems we still do not know a single example.
Question 23.3. Is the intersection of the union M S4 of all MumfordShirriura curves with the open Torelli locus ½o c A 9 finite or infinite?
On page 714 of [98) we see that the suggestion that this set of points is finite. Note that every MumfordShimura curve has a nonempty intersection with the closed Torelli locus. Hence for (MS4 n 4) C A 9 (C) we have
It is easy to give infinitely many examples of reducible
g == 4 curves whose gen
eralized Jacobian is an abelian variety (curves of "compact type'') such that the
Appendix 3
Questions in Arithmetic Algebraic Geometry
317
moduli point of tllis principally polarized abelian variety is on a MumfordShimura curve:
with
I thank Rutger Noot
a11d
Ben Moonen for helpful discussions on this topic.
For a furtl1er discussion, see Cl1apter 4 of this volume, and see [82].
24. Do we know special subvarieties of positive dimension generically contained in the open Torelli locus?
(Frans Oort) We consider the open and the closed Torelli locus
~ C Ty C Ag:= Ag,l @C.
We say tl1at a variety X C A 9 is generically contained in ~ if X c Ty
a11d
X ¢:. 8(Tg) := Tg \
Tyo.
As Tg = A 9 fo1~ g ~ 3 in those cases every subvariety is generically contained in For 4 ~ g $ 1 we know examples of special subvarieties of positive clime11sio11 generically contai11ed in 0 ; see [66], 5.15 of that paper for references and for a survey of known cases. However for g > 7 no such exam1)les are known. Many partial results have lJeen obtained· see (66] for references.
½o.
½
Expectation 24.1. For large g tliere cloes not exist a special subvariety of positive di1nension ge11.erically contained ·in the open Torelli locus
½o.
See [77}, Section 5; see [1 13], Question 7. See Chapter 4 of tltls volume. Question 24.2. De we kn.ow any g > 0 and any special subvariety Z c A 9 g·enerically co1itai1ied in the open To·relli loc·us suc}i that the fiber Xr; over the geometric ge·n eric point r; has th.e property End(X77) = Z?
25. Defor1nations of curves keeping the Newton Polygon constant. (Frans Oort) 25.1. In orcler to have a positive answer to 21.1 in some cases, we study Newton Polygo11s witl1 slopes appearing for smaller values of g. Suppose (' and ~11 are sy1nmetric Newton Polygons~ we write = (,' u {'' for t11e NP ol)tained lJ)r taking the join of all slopes appearing in ~, and in ~'' with their 1nultiplicities and ordering them in nondecreasing order.
e
318
Editor: Frans Oort
Conjecture 25.2. Suppose~' does appear on Mg (€' ) ® IFp an,d ( 11 does appear on M 9 c~1') ® 1Fp Then(?)~==~, U('' does a,ppear on Mg({) ® 1Fp. See (115], 8.5.7.
If this conjecture l1olds, then Newton Polygons with only small slopes do appear on M 9 . 25.3. Corollary of the conjecture. For every p and every g the sitpersingular Newton Polygon a 9 does appear on M 9 (e) ® F1)•
We make 25.2 more precise.
C''
be
and these components are connected via a transversal crossing at the points P' P == P'' on Co= C' Up C''. We write
==
Conjecture 25.4. Let k => JFP be an a,lgebraically closed field4 Let C', arid algebraic curves ouer k ( irreducible, com,plete, nonsingular). We 'UJrite
Co= C' Up C 11 ,
where P' E C'(k), P''
E
C''(k)
e==N(Go) ==N(C') uN(C''). We conjecture: there exist nonempty Zariski closed sets
S(C', C'') c C',
S(C'', C') c C''
such that a deformation of Co to a nonsingular· generic fiber Cr, exist.s
if
with N(Co) = ( == N(Cr,) and only if P' E S(C', C'') and P'' E S( C 11 , C').
In other words: deformation to a nonsingular curve with the same NP exists in case the points where the curves are attached are chosen i.J1 tl1ese nonempty closed sets.
In some special cases we l{now this is true. We have examples where S( C', D 1 ) i= S( C', D2) a1~e different subsets of C' for different D 1 and D 2 ; i.e. this subset S (C', C'') C C' depends on C' and on C'' in general. We expect that tl1e construction and properties of these subsets S(C', C'') depend on the choice of p.
26. Lifting up to isogeny in Shimura varieties.
(Frans Oort) 26.1. Question. Suppose given a Shimura variety S of abeliar, type in mixed
characteristic, and A an abelian 'Variety plus extra structure over K == 1Fq giving a point x ES("") Does there exist am, isogeny defined over Ki giving y E S(K) such that this with the given Shimura data lifts to clia racter·istic zero? 1
Appenclix 3
Questions
i11
Arit}11netic Algebraic Geometry
319
26.2. Some comments . 26.2.1. In tl1e HondaTate theoI)r we see that the isogeny class any abelian va,r iety over FP ca.11 be C!\ 1lifted; see [50], (104]. 1
26.2.2. Tl1ere do exist abelia.n varieties over ]FP tl1at ca1111ot be CMlifted: in general an isoge11y is 11ecessary; see (76]. Also see Chapter 3 of [19].
26. 2.3. In tl1e IIouda. Tate tl1eo1·y a11alytic n1ethods are used in order to produce such a Cl\lIlifti11g. There exists also a pu1·ely algebraic proof of this fact [18]. 26.2.4. In [19] we show tl1at an abelian variety Ao defined over
=
lF q admits a11 isogeny Ao Bo defin,ed over K such that Bo admits a CMlifting. Note that such a. lifting is to a rn ixed cha1~acteristic integral domain that i11 general carn1ot be chose11 norn1al,. see tl1e residual reflex conditio11 in [19]. K
t"'J
26.2.5. One can wonder whether this theory can be formulated and p1·oved for arbitrary Sl1i1nura va1·ieties. See [111], [107]: [58], whe1·e we see that a Shimura lift does exist for x E S (IFP) after an appropriate isogeny x y over JFP. ('.J
26.2.6. The c111estion 26.1 asks whetl1er the results mentioned in 26.2.4 and 26.2.5 can be combined into a theory that does not need an extension of the given finite field. We see in [26] how to circumvent this problem in questions st11died in ,,01nmelin 's PhDthesis, but we also see that the theory ca11 be simplified i11 case w wot1ld have a positive answer to 26.1.
27. The prank O strata of the Torelli locus.
(Rachel Pries) For a prime number p and natural n,1mber g, conside1· tl1e n1oduli space A 9 : A9 ® lFP of principally polarized abelian varieties of dimension .9 in characteristic p a.nd consider the closed Torelli locus Tg C A 9 . Let At= 0 be the subsche1ne that parameterizes abelian varieties of dimension g with prank f = 0. For 9 ~ 3, it is 0 is irreducible, tl1at its generic point has anumber 1, and that known that 0 its Nev\rton polygon has slopes 1 / g and (.q  1) / g (16]. Let T 0 = T n .
At=
/=
At=
Qt1estion 27.1. For p prime and g 2: 3, is T/= 0 irreducible? For the gerieric point of each, of its componen,ts_. is it true th,at the anumber is l and the Newton, polygon has slopes 1/ g and (g  l) / g ? The a.11swer is yes when g
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o,nthe inverse
[14] Ph. Casso11Nogues, T. Chinburg, B. Morin a11.d M. J. Taylor  The classifying topos of a gro'ttp scheme and invariants of symmetric bundles, Proc. Land. Math. Soc. (3) 109 (2014), no. 5, 1093~1136.
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Below we record s01ne earlier collectio11s of open proble1ns and questio11s. (113] F. Oort Some question.sin algebraic ,qeonietry. June 1995, at, tl1e occa~io11 o f tl1e co1iference "Arith1netic arid geon1etry of abelian ,,.arieties·,. See Appendix 1 in this vol t1me. http://www.staff.science.uu.nl/oort0109/ [114] S. Edixhoven, B. l\lloonen and F. Oo1·t (Editors)  Open problems i·n algebraic geometry. Jtme 2000, at the occasion of the workshop '
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37. Proceedings of The 6th International Congress of Chinese Mathematicians (Vol. 11) (2016) (Editors: ChangShou Lin, Lo Yang, SbiogTung Yau, Jing Yu)
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35. The Legacy of Bernl1ard Riemann After 011e Hu11dred and Fifty Years (2016) (Editors: Lizhen Ji, Frans Oort, Sl1ingTu11g Yau)
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34. Some rfopics in Ilarmonic Analysis a11d Applicatio11s (2015) (Editors: J11nfeng Li, Xiaochur1 Li, Guozhen Lu)
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33. Introduction to Moder11 Mathematics (2015) (Editors: ShiuYuen Cheng, Lizhen Ji, YatSun Poo11, Jie Xiao, Lo Yang, ShingTung Yau)
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ADVANCED LECTURES IN MATHEMATICS 28. Selected Expository Works of Sl1i11gTung Yau with Comr11entary Vol. I (2014) (E
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19. Ar ithmetic Geon1etry and A11tomorphic For111s (201 1) {Editors: Ja1nes Cogdell, Je11s F1111ke Micl1ael RapoJ)ort, rrongl1ai Yang) 18. Gcon1etry a11d A11alysis Vol. II (2010) (Editor: Lizhen Ji)
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16. Transformation Groups a11d rvtoduli Spaces of Curves (2010) (Editors: Lizben Ji. Shl11gTung Ya11)
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6. Geon1etry, Analysis and Topology of Discrete Groups (2008) (Editors: Lizhen Ji, Kefeng Liu, Lo Yang, ShingTung Yau)
5. Proceedings of The 4th International Congress of Chinese Mathematicians Vol. I j II (2007) (Editors: Lizhen Ji, Kefeng Liu, Lo Yang, ShingTung Yau)
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ALM The launch of this Advanced Lectures in Mathematics series is aimed at keeping mathematicians informed of the latest developments in mathematics, as well as to aid in the learning of new mathematical topics by students all over the world. Each volume consists of either an expository monograph or a collection of signifi~ cant introductions to important topics. This series emphasizes the history and sources of motivation for the topics under discussion, and also gives an overview of the current status of research in each particular field. These volumes are the first source to which people will t urn in order to learn new subjects and to discover the latest results of many cuttingedge fields in mathematics. 1
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