Exponential Sums and Differential Equations. (AM-124), Volume 124 9781400882434

Table of contents :
Contents
Introduction
Chapter 1- Results from Representation Theory
1.0-1.6 Statements of the main results
1.7 The proofs
1.8 Appendix: direct sums and tensor products
Chapter 2- D.E.'s and D-modules
2.1 The basic set-up
2.2 Torsors and lifting problems
2.3 Relation to transcendence
2.4 Behavior of Ggal under specialization
2.5 Specialization of morphisms
2.6 Direct sums and tensor products
2.7 A basic trichotomy
2.8 The main D.E. theorem
2.9 Generalities on D-modules on curves
2.10 Some equations on A^1, with a transition to Gm
2.10.16 Location of the singularities of a Fourier Transform
2.11 Systematic study of equations on Gm
Chapter 3- The generalized hypergeometric equation
3.1 Basic definitions
3.2-3.3 Basic results about irreducibility and contiguity
3.4 Duality
3.5 The case n= m; Lie irreducibility, rigidity, Belyi and Kummer induction
3.6 The case n ≠ m
3.7 Intrinsic characterization; rigidity for n ≠ m
3.8 Direct sums, tensor products, and Kummer inductions
Chapter 4 Detailed analysis of the exceptional cases
4.0 Eliminating a few cases
4.1 The G2 case
4.2 The Spin(7), PSL(3) and SL(2)xSL(2)xSL(2) cases
4.3 The PSL(3) case: detailed analysis
4.4 The Spin(7) case: detailed analysis
4.5 The SL(2)xSL(2)xSL(2) case
4.6 The SL(3)xSL(3) case
Chapter 5- Convolution of D-modules
5.1 Generalities
5.2 Convolution on Gm and Fourier Transform on A^1
5.3 Convolution of hypergeometrics on Gm
5.4 Motivic interpretation of hypergeometrics of type (n, n)
5.5 Application to Grothendieck's p-curvature conjecture
Chapter 6- Fourier transforms of Kummer pullbacks of hypergeometrics
6.1 Some D.E.'s on A^1 as Kummer pullbacks of hypergeometrics
6.2 Fourier transforms of Kummer pullbacks of hypergeometrics: a remarkable stability
6.3 convolution of hypergeometrics with nondisjoint exponents, via a modified sort of hypergeometric
6.4 Applications to Fourier transforms of Kummer pullbacks of hypergeometrics
Chapter 7- The l-adic theory
7.1 Exceptional sets of primes
7.2 l-adic analogue of the main D.E. theorem 2.8.1
7.3 Construction of irreducible sheaves via Fourier transform
7.4 Local monodromy of Fourier transforms d'apres Laumon
7.5 "Numerical" explicitation of Lauman's results
7.6 Pseudoreflection examples and applications
7.7 A highest slope application
7.8 Fourier transform-stable classes of sheaves
7.9 Fourier transforms of tame pseudoreflection sheaves
7.10 Examples
7.11 Sato-Tate laws for one-variable exponential sums
7.12 Special linear examples
7.13 Symplectic examples
7.14 Orthogonal examples
Chapter 8- l-adic hypergeometrics
8.1 Rapid review of perversity, Fourier transform, and convolution
8.2 Definition of hypergeometric complexes and hypergeometric sums over finite fields
8.3 Variant: hypergeometric complexes over algebraically closed fields
8.4 Basic properties of hypergeometric complexes; definition and basic properties of hypergeometric sheaves
8.5 Intrinsic characterization of hypergeometrics
8.6 Local rigidity
8.7 Multiplicative translation and change of ψ
8.8 Global and local duality recognition
8.9 Kummer induction formulas and recognition criteria
8.10 Belyi induction formulas and recognition criteria
8.11 Calculation of Ggeom for irreducible hypergeometrics
8.11.7 Direct sums and tensor products
8.12 Arithmetic determinant formula
8.13 Sato-Tate law for hypergeometric sums; nonexceptional cases
8.14 Criteria for finite monodromy
8.15 Irreducible hypergeomtrics with finite Ggeom
8.16 Explicitation via Stickelberger
8.17 Finite monodromy for type (n, n), intertwining, and specialization
8.18 Appendix: semicontinuity and specialization for Ggeom, d'apres R. Pink
Chapter 9 G2 examples, Fourier transforms and hypergeometrics
9.1 Another G2 example
9.2 Relation of simple Fourier transforms to hypergeometrics
9.3 Fourier transforms of Kummer pullbacks of hypergeometrics
9.4 Reduction to the tame case
Chapter 10- l-adic exceptional cases
10.0 Introduction
10.1 The G2 and Spin(7) cases
10.2 The PSL(3), SL(2)xSL(2)xSL(2) and SL(3)xSL(3) cases, via tensor induction
10.3 Short review of tensor induction
10.4 A basic example; tensor induction of polynomials
10.5 The geometric incarnation
10.6 Tensor induction on Gm
10.7 Return to the PSL(3), SL(2)xSL(2)xSL(2) and SL(3)xSL(3) cases
10.8 The SL(2)xSL(2)xSL(2) case
10.9 The SL(3)xSL(3) case
Chapter 11- Reductive tannakian categories
11.1 Homogeneous space recovery of a reductive group
11.2 First analysis of finiteness properties
11.3 Transition away from tannakian categories
11.4 Mock tannakian categories
11.5 Statement of the reductive specialization theorem
11.6 Proof of the reductive specialization theorem
11.7 A minor variant on the the reductive specialization theorem
Chapter 12- Fourier universality
12.1 The situation over C
12.2 Additive convolution, exotic tensor product, and Fourier transform on A^1 over C
12.3 The tannakian category DA,B
12.4 The tannakian category DA,RS
12.5 A minor variant: ! convolution of D-modules
12.6 Brief review of Riemann- Hilbert; transition from D-modulesto D^bc
12.7 Transition to Ql coefficients
12.8 Recapitulation of the situation over C
12.9 The situation in characteristic p
12.10 The tannakian category Perv A!,B(A^1, Ql)
12.11 The tannakian category PervA!,tame(A^1, Ql)
Chapter 13- Stratifications and convolution
13.1 Generalities on stratifications and convolution
13.2 Interlude: review of elementary stratification facts about normal crossings
13.3 The special case of A^1
13.4 Location of the singularities of a convolution
13.5 The bookkeeping of iterated convolution
13.6 Various fibre-wise categories
Chapter 14- The fundamental comparison theorems
14.1 The basic setting
14.2-14.10 Reductive comparison theorem, semisimple comparison theorem, and sharpened reductive comparison theorem
14.11 interlude: a sufficient condition for the reductivity of Ggal
14.12 Application to hypergeometric sheaves
14.13 Application to Fourier transform of cohomology along the fibres
14.14 Examples
References

Citation preview

Annals of Mathematics Studies Number 124

Exponential Sums and Differential Equations by

Nicholas M. Katz

PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY

1990

Copyright

©

1990 by Princeton University Press

ALL RIGHTS RESERVED

The Annals of Mathematics Studies are edited by Luis A. Caffarelli, John N. Mather, John Milnor and Elias M. Stein

Princeton University Press books are printed on acid-free paper, and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources

Printed in the United States of America by Princeton University Press, 41 William Street Princeton, New Jersey

Library of Congress Cataloging-in-Publication Data Katz, Nicholas M., 1943Exponential sums and differential equations / by Nicholas M. Katz p.

em. -

(Annals of mathematics studies ; no. 124)

Includes bibliographical references. ISBN 0-691-08598-6 (cloth) ISBN 0-691-08599-4 (pbk.) 1. Exponential sums. 2. Differential equations.

II. Series. QA246.7.K38 1990 512'.73-dc20

90-34934

I. Title.

Contents Introduction Chapter 1- Results from Representation Theory 1.0-1.6 Statements of the main results 1.7 The proofs 1.8 Appendix: direct sums and tensor products

Chapter2- D.E."s and JJ-modules 2.1 The basic set-up 2.2 Torsors and lifting problems 2.3 Relation to transcendence 2.4 Behavior of Ggal under specialization 2.5 2 .6 2.7 2.8 2.9

9

31

Specialization of morphisms Direct sums and tensor products A basic trichotomy The main D.E. theorem Generalities on JJ-modules on curves

2.10 Some equations on

t..i,

with a transition to IGm

2.10.16 Location of the singularities of a Fourier Transform 2.11 Systematic study of equations on IGm 92 Chapter 3- The generalized hypergeometric equation 3.1 Basic definitions 3.2-3.3 Basic results about irreducibility and contiguity 3.4 Duality 3.5 The case n= m; Lie irreducibility, rigidity, Belyi and Kummer induction 3.6 The case n z m 3.7 Intrinsic characterization; rigidity for n z m 3.8 Direct sums, tensor products, and Kummer inductions

Chapter 4 Detailed analysis of the exceptional cases 4.0 Eliminating a few cases 4 .1 The G2 case 4.2 4.3 4.4 4.5 4.6

The The The The The

Spin(7), PSL(3) and SL(2)xSL(2)xSL(2) cases PSL(3) case: detailed analysis Spin(7) case: detailed analysis SL(2) x SL(Z) x SL(Z) case SL(3)xSL(3) case

v

122

vi

Contents

Chapter 5- Convolution of .D-modules 5.1 Generalities

161

5.2 Convolution on 113m and Fourier Transform on A1 5.3 Convolution of hypergeometrics on IGm 5.4 Motivic interpretation of hypergeometrics of type (n, n) 5.5 Application to Grothendieck's p-curvature conjecture Chapter 6- Fourier transforms of Kummer pullbacks of h ypergeometrics

178

6.1 Some D.E.'s on A1 as Kummer pullbacks of hypergeometrics 5.2 Fourier transforms of Kummer pullbacks of hypergeometrics: a remarkable stability 5.3 convolution of hypergeometrics with nondisjoint exponents, via a modified sort of hypergeometric 5.4 Applications to Fourier transforms of Kummer pullbacks of h ypergeometrics Chapter 7- The t-adic theory 7.1 Exceptional sets of primes 7.2 ~-adic analogue of the main D.E. theorem 2.8.1 7.3 Construction of irreducible sheaves via Fourier transform 7.4 Local monodromy of Fourier transforms d'apres Laumon 7.5 "Numerical" explicitation of Lauman's results 7.5 Pseudoreflection examples and applications 7.7 A highest slope application 7.8 Fourier transform-stable classes of sheaves 7.9 Fourier transforms of tame pseudoreflection sheaves 7.10 Examples 7.11 Sato-Tate laws for one-variable exponential sums 7.12 Special linear examples 7.13 Symplectic examples 7 .14 Orthogonal ex am ples

193

Chapter 8- ~-adic hypergeometrics 251 8.1 Rapid review of perversity, Fourier transform, and convolution 8.2 Definition of hypergeometric complexes and hypergeometric sums over finite fields 8.3 Variant: hypergeometric complexes over algebraically closed fields.

vii

Contents 8.4 Basic properties of hypergeometric complexes; definition and basic properties of hypergeometric sheaves 8.5 Intrinsic characterization of hypergeometrics 8.6 Local rigidity 8.7 Multiplicative translation and change of 72d2. This given Theorem 1.4 (whose h has d= 4) as soon as dimV > 72x16. Theorem 1.5 (Kazhdan-Margulis, Gabber, Beukers-Heckman [B-H]) Let ~ be a semisimple Lie-subalgebra of End(V) which acts irreducibly on V. Suppose that ~ is normalized by a pseudo-reflection ¥ in GL(V). Then ~ is either .S.t(V) or .S~(V) or (for dimV even) ,S"ij)(V). Moreover, if det¥ ;t. ± 1, then ~ = .S.t(V); if det¥ = +1, then~ = .S.t(V) or (for dimV even) ,S"ij)(V); if det¥ = -1, then~= .S.t(V) or .S~(V). Theorem 1.6 (Gabber) Let~ be a semisimple Lie-subalgebra of End(V) which acts irreducibly on V. Suppose that dimV is a prime p. Then~ is either .SJ:(Z) in SymP-l(std), or .SJ:(V) or ,S~(V) or, if n=7, possibly Lie(Gz) in the seven-dimensional irreducible representation of Gz.

12

Chapter 1

1.7 The proofs Notice that in Theorems 1.1, 1.2, 1.3, 1.4, 1.5, 1.6 the cases listed can be easily checked to have the property in question; the problem is to show that these are the only cases. In doing this we will make explicit use of classification, via the Bourbaki tables. Notice also that the " ..i(h) of h-weights. This is a nonincreasing sequence of integers, whose successive drops are the integers «i(h), where 0, and pick a chain which 1

runs from HOC'. up to the highest dual root Hhighest; we get 1 + (:>.., ~pas"' HOC'.) =1 1

+ (>., Hhighest) + ~"' pos, HOG not highest (:h, HOC'.)

1 +

(:h, Hhighest)

+ ~chain omitting Hhighest (:h, Hin chain)

21 + (>.+p, Hhighest)- (p, Hhighest)+ ~chainomittingHhighest (:h, Hinchain). Now we will establish that in fact 1 + ~chainomittingHhighest (>., Hinchain)

1

(p, Hhighest).

Writing Hhighest = ~simple~ n~H~ as a sum of simple dual roots, and recalling that p is the sum of the fundamental weights, we see that

24

Chapter 1

(p, Hhighest) = Ln~

= the

"length" of Hhighest'

so what is to be proved is 1 + Lchain omitting Hhighest (:X, Hin chain)

l

length of Hhighest·

This inequality is obvious, because the length of any chain from the simple root H G -> H -> 1, the cohomology sequence H1 (Xet- f) -> H 1 (Xet• G) -> H 1 (Xet• H) -> H2 (Xet• f)

shows that the obstruction to lifting (the isomorphism class of) a given H-torsor P is in H2(Xet• f), and the indeterminacy in lifting it is in H 1 (Xet• f). QED

Corollary 2.2 .2 .1 If K is algebraically closed and if X/K is an open curve, then liftings exist. ( 2 .2 .3)

Suppose now that K is C. The ex act tensor functor V

~--+

van

from D.E.(X/C) to D.E.(xan) ~ Rep( rr 1 top(xan, x)) defines a homomorphism " : rr 1 top -> rr 1 cliff from the topological rr 1 of the complex manifold xan to rr 1 diff(x, x). With respect to this homomorphism " : rr 1 top -> rr 1 cliff, we have the following variant of the lifting problem: Let G

36

Chapter 2

with cp = pep. Remark Z.Z.4.Z Here is a slightly more cohomological formulation. Given a field K of characteristic zero, and a smooth X/K, we have the "crystalline-etale site" of X/K, noted (X/K)crys-et· Its objects are the pairs (U/X, 1.: U--+T) consisting of an etale X-scheme U/X and a closed K-immersion 1. of U into a K-scheme T such that U is defined in T by a nilpotent ideal of ~T· The morphisms are the obvious commutative diagrams, and a family of objects (U/X, 1.: Ui--+Ti) over (U/X, 1.: U--+T) is a covering if and only if the Ti --+ T are an etale covering of T. Given any smooth K-groupscheme G, we get a sheaf, still noted G, on (X/K)crys-et by the rule (U/X, 1.: U--+T) ~ G(T). It is essentially tautological that a right G-torsor on (X/K)crys-et is the same as a right G-torsor on Xet endowed with a right G-equivariant integrable connection. Now suppose that X(K) is nonempty. In view of the above discussion, we see that the set H1((X/K)crys-et• G) of isomorphism classes of right G-torsors on (X/K)crys-et• is none other than the quotient set HomK-gpschhr 1 diff(X, x), G) modulo the conjugation action of G (this action because we have not specified the trivialization over x). In other words, we have H1((X/K)crys-et• G) ~ HomK-gpsch(1T 1 diff(x, x), G)/G, so that 1T 1 diff(X, x) is a kind of "fundamental group" of (X/K)crys-et· The point is that if we have p: G --+ Has above with finite etale central kernel r, then we can use the standard cohomological setup on (X/K)crys-et to investigate our lifting problem. Consider the exact sequence of sheaves on (X/K)crys-et 1 --+ f --+ G --+ H --+ 1, which gives rise to an exact sequence of cohomology Hl((X/K)crys-et• f) --+ H1((X/K)crys-et• G) --+ --+ H1((X/K)crys-et• H) --+ HZ((X/K)crys-et• f). By its very construction the site (X/K)crys-et maps to (both the usual crystalline site (X/K)crys• and to) the etale site Xet· The CechAlexander calculation (cf. [Gro-CDR, 5.5], [Bert, V, 1.2]) of the cohomology shows that for any etale (resp. and commutative) Kgroupscheme r ' the canonical maps Hi((X/K)crys-et• f) --+ Hi(Xet• f)

D.E.'s and

37

~-modules

are isomorphisms for i=0,1 (resp. for all i). On the other hand for the smooth groups G and H, the canonical maps H1((X/K)crys-et• G (resp. H)) ~ H1 (Xet• G (resp. H)) correspond to the map "forget the connection". Thus we find once again that the obstruction and indeterminacy in our lifting problem lie in H2(Xet• f) and H1(Xet• f) respectively, and are the same as the obstruction and the indeterminacy in the lifting problem for the underlying "naked" torsors without connection.

2 .3 Relation to Transcendence We now turn to a brief discussion of the relation between the differential galois group of a D.E. and the transcendence properties of its power series solutions. This material is "well-known", indeed it was a large part of the basic motivation for the classical differential galois theory, but it does not seem to be written down anywhere in the Tannakian context (however cf. [De-CT, 9] for a Tannakian proof of the general existence of Picard-Vessiot extensions). Proposition 2.3.1 Let V be a D.E. on X/K, G := Ggal(V, x) its differential galois group, v e: Vx, and

v e:

V®("x.x>" be the corresponding

horizontal section. Suppose that K is algebraically closed. The the transcendence degree over K(X) of the coefficients (with repect to any K( X)- basis of V ® ~ XK( X)) of is the K- dimension of (the closure in V x

v

of) the G-orbit Gv.

proof Inside vv, the annihilator of

v is a

horizontal submodule W, so it

corresponds to a G-stable subspace Wx of (Vx)v which lies in the annihilator of v. Being G-stable, Wx must annihilate the entire G-orbit of v, so W x is contained in the annihilator Sx of Gv; this Sx is a G-stable subspace of (Vx)v, so it corresponds to a sub-D.E. S of vv. Since Sis horizontal, and Sx annihilates v, S annihilates

v. Thus S

C

W, and as

Wx c Sx we haveS= W. Therefore the K-dimension of the space Sx of K-linear forms on Vx which annihilate Gv is the same as the of W, i.e., the same as the K(X)-dimension of the space K(X)-linear forms on V®"xK(X) which annihilate

~x-rank

W®~XK(X)

v.

Applying this equality of dimensions to the situation

of

38

Chapter 2

E13 j s. nSymmj( v)

E:

E13 j s. nSymmj(V)x, for all n, we obtain equality of the

corresponding Hilbert polynomials, whence the asserted equality of dimensions. QED

Corollary 2.3.1.1 Suppose K algebraically closed. The transcendence degree over K(X) of the (rank(V))Z matrix coefficients of any fundamental solution matrix at x is the dimension of Ggal· proof Simply apply the above result to the internal hom D.E. W := Hom(Vx®KC}'X, V), whose fibre Wx is End(Vx) with G acting by right translation. Fundamental solution matrices at x are precisely those horizontal w's in W®(C}'x,x)"' whose value w at

X

lies in GL(Vx)·

Since G acts freely on GL(Vx), the orbit dimension is dimG. QED

Remark 2.3.2 Here is a slightly more precise version of the above numerical result. Consider the right G-torsor P on X corresponding to the homomorphism rr 1 diff(X, x) ~ G which "classifies" the D.E. V. An a priori description of it is this. Consider the subcategory of D.E.(X/K). On it we have two obvious C}'x-valued fiber functors, namely Wx: W

H

wid : W

H

Wx®KC}'X, W as C}'x-module.

It is essentially tautological that P is the right G-torsor

Isom® ,(wx, wid)· The horizontal sections of P over (C}'x,x)"' are precisely the set (actually a right G(K) torsor) S of those fundamental solution matrices w which have the following property: for every "construction of linear algebra" Constr(V), and every sub- D.E. W of Constr(V), the induced horizontal section Constr(w) of Isom(Constr(V)x®KC}'X, Constr(V)) maps Wx®KC}'X toW. We claim that if G(K) is Zariski dense in G (e.g., if K is algebraically closed), then this setS is Zariski dense in P. Indeed, this is more general nonsense: Suppose that G(K) is Zariski dense in G. Then for any right G-torsor P on X with right G-equivariant integrable connection, the set (actually a right G(K) torsor) S of its horizontal sections over (C}'x,x)"' is Zariski dense in P. [proof: the annihilator ideal I of Sin A is horizontal, the union of the D.E.'s In An, so determined by its fibre at x. But at x it annihilates all the K-rational points G(K) of G :::: Px, and as these are Zariski dense I

= 0.]

D.E.'s and :tl-modules

39

Corollary 2.3 .2 .1 Suppose K is algebraically closed. The algebraic group G acts transitively on V x - {0} if and only if for every nonzero v e: V x•

v

the corresponding horizontal section has its n=rank(V) coefficients algebraically independent over K(X). proof If G acts transitively, then for each nonzero v, the orbit Gv is ndimensional. Conversely, let v.:O. If Gv is n-dimensional, then Gv must be all of Vx - {0}, simply because Gv is constructible. QED

2.4 Behavior of Ggal under Specialization Let K be a field of characteristic zero, t an indeterminate, R the ring K[[t]l, X/R a smooth separated R-scheme of finite type with geometrically connected fibres, and xe:X(R). Suppose we are given a locally free "x-module V of finite rank n together with an integrable connection o : V -+ V®"xo1X/R relative to the baseR. Let us denote by 'f): R -+ K((t)) the inclusion (generic point of Spec(R)), and by s: R -+ K the specialization map t~--+0 (special point of Spec(R)). Via these extensions of scalars, we obtain D.E.'s V(Y)) on XY)/K((t)) and V(s) on Xs/K, and rational points x'f)

E:

XY)(K((t))) and Xs e: Xs(K). Thus we can

speak of the differential galois groups Ggal(V(Y)), x'f))

C

GL(V('f1)x'f)) and

Ggal(V(s), xs) c GL(V(s)xs)

Specialization Theorem 2.4.1 (Ofer Gabber) Let G/R be the closed Rflat subgroupscheme of GL(V x> ~ GL(n)/R obtained as the schematic closure of Ggal(V(Y)), x'f)), and let Gs denote its special fibre. Then there is natural inclusion Ggal(V(s), xs)

C

Gs (inside GL(V(s)Xs) ~ GL(n)/K).

proof Denote by A the coordinate ring of GL(Vx), by Ad c AtheRsubmodule of those functions which are the restrictions of polynomials of degree s. d in the nZ matrix coefficients Xi,j and the function 1/det(Xi,}• and by IY) c AY) the ideal defining Ggal(V(Y)), x'f)) in GL(V(Y))). Then the schematic closure G of Ggal(V(Y)), x'f)) in GL(V x> is defined by the ideal IY)nA. Since A is noetherian, this ideal is generated by IY)nAd for sufficiently large d. For each d, there is a natural "construction of linear algebra" Ad := Constrd(V) whose pullback (Ad >x by the section x is Ad. Since the intersection IY)n(Ad)Y) is tautologically Ggal(V(Y)), x'f))-stable, it makes

40

Chapter 2

sense to speak of the horizontal submodule Md of Ad(TJ) whose xT]-fibre is r11 n(Ad) 11 . Let :md denote the intersection MdnAd inside Ad(fl). Then :md is visibly horizontal, R-flat, and C)"x-coherent. Hence it is C)"x-locally free, as results from Lemma 2.4.2 Let X/R as above. Let (:JTl, o) be an C)"x-coherent module

:m together with an integrable connection D: :m

~ :JTl®C)-X0 1 x/R

relative to the base R. Then (1) is C)"x-locally free if (and only if) it is R-flat.

:m

(2) If there exists a section xE:X(R), then only if) it :mx := x *

:m is C)"x-locally free if (and

:m is R-flat.

proof of Lemma The "only if" is trivial, since X/R is flat. Because C)"x-coherent and is endowed with an integrable connection, it is

:m

is

:m

automatically locally free on x 11 . Suppose that is R-flat. To prove (1), it suffices to show that is locally free over the local ring "x,p of X at

:m

every closed point p of the special fibre. Since finite extensions of K are harmless, we may assume that pis K-rational. By faithful flatness of the completion, it suffices to treat the formal case, where X is the spec of R[[x1, ... , xnll. But for any noetherian 0-algebra R, the functor

:m

"horizontal sections", H :JTl 0 defines an equivalence of categories {coherent R[[x 1• ..., Xnll-modules with integrable connection over R} ::::: ::::: {coherent R-modules}, whose inverse functor is M H M[[x1, ..., xnll with the trivial connection 1®d. From this explicit description of the inverse, it is obvious that for R local we have

:m is R-flat

#

:m 0 is R-flat

#

:m 0 is R-free

#

:m is R[[x1,

..., xnll-free.

Supose that there exists a section xE:X(R), and denote by p the point Xs· Working at pas above, we see from the explicit description

:m

that :JTl 0 is R-isomorphic to x*:JTl. So if x*:JTl is R-flat, then is R-flat in a neighborhood of pin X. But the locus of non R-flatness of 1f1. is the support of Ker( Left(t): ~ 1f1.); but this is a coherent C)-X -module

:m

s

with integrable connection, i.e., a D.E. on X/K, so it is "x -locally free; s

as it vanishes near p, it must be zero. QED By the definition of md c Ad, the quotient Ad/md is R-flat.

D.E.'s and

41

'!:~-modules

Therefore it too is G"x-locally free; in other words, :md is locally a direct factor of Ad. Pulling back by the section x, we find that (Jnd)x is a direct factor of (Ad)x. = Ad. As the generic fibre of (Jnd)x is IT) n (Ad)TJ, we conclude that (Jnd)x is IT) n Ad Now consider the special fibre Gs of G. It is defined by the ideal (ITJ n A)s of As. Since ITJ n A is the union of the IT) n Ad, and each ITJ n Ad is a direct factor of Ad, we see that (IT)nA)s is the union of the (IT)nAd)s. But (IT) n Ad)s is the Xs -fibre of (Jnd)(s). Since :md is a locally direct factor of Ad, (Jnd)(s) is a sub-D.E. of (Ad)(s), and therefore its xs-fibre (IT) n Ad)s is a Ggal(V(s), xs)-stable subspace of (Ad)xs = (Ad)s· Therefore the entire ideal (ITJnA)s of As is Ggal(V(s), Xs)-stable. As this ideal kills the identity element in GL(n), it kills all of Ggal(V(s), xs)· This means precisely that Ggal(V(s), xs) c Gs· QED Corollary 2.4 .3 .1 Hypotheses and notations as in the specialization theorem 2.4 .1 above, we have the inequality of dimensions dimK(Ggal(V(s), xs)) i dimK((t))(Ggal(V( T)), xTJ)). By successive specialization, we find Corollary 2 .4 .3 .2 Let K be a field of characteristic zero, R a smooth geometrically connected affine K-algebra or a power series ring in finitely many variables over K, X/R a smooth separated R-scheme of finite type with geometrically connected fibres, and xE:X(R). Suppose given a locally free G"x-module V of finite rank n with an integrable connection o : V ~ V®G"xo1X/R relative toR. LetT) be the generic point of Spec(R), and s any closed point of Spec(R). Then we have the inequality of dimensions dimK(s)(Ggal(V(s), xs)) i dimK(T)) in

iJ.b(C). In particular, 'J' contains Diag(1,-1,0, ... ,0), and so ", and so we find fE:xn(~x,o;)"). To split 1T, it suffices to show that 1T maps C onto B. (For 1TIC : C -+ B is automatically injective, as C () (~x,o;)"' = 0.) Given an element

~

of B, choose for each n

which lifts x-n~. For each n

l

~

0 an element o;nE: A

0, let fn := o;n- xo;n+1· Then 1T(fn)=O,so

fnd~x,o;>". The series Ln~O xnfn converges in (~x,o;>"', say to F. Now define new liftings '6nE:A of the x-n~ by '6n := o;n- F. With this choice, the differences cn := '6n- x'6n+1 are cn = fn- (1-x)F, so Ln~O xncn = 0. For each n~O. define CnE:(~x,o;>"" to be Cn := 2:i~O xici+n· Then v

•O

and so '6 0

E:

-

xn+1v•n+ 1 --

'6o =

L·1= 0 , ...,n xic.1 --

-L·Hn+ 1 xic.1 --

-xn+1c n+1•

xn+1('6n+l -Cn+l) lies in xn+1A for every n ~ 0, and hence

C is a lifting of

~

to C. QED

This Proposition leads immediately to the following .D-module complement to Deligne's Euler-Poincare formula !De-ED,Il, 6.211, which was suggested to me by Ofer Gabber. Recall that for a holonomic .Dmodule m. on X, we define X(X, Jn) := X(H*DR(X, Jn)) = 2:(-1)idimcHiDR(X, Jn). Corollary 2.9 .8 .1 Let j : U -+ X be the inclusion of a nonempty open set. Let m be a holonomic .D-module on U. For each o;E:X-U, denote by Solno; the finite-dimensional C-vector space of formal meromorphic solutions of mat o;. Then x-+e"'x. (2.10.2) Thus Jn>-+FT(Jn) is an exact autoequivalence of the category of left (resp. left holonomic) .D-modules. If we iterate FT, we find FT(FT(Jn)) ~ [-1]*(Jn). A key point for later applications is the apparently trivial consequence that a holonomic left .D-module is irreducible if and only if FT(Jn) is irreducible. Here is a simple illustration :

m

Theorem 2.10.3 Let P:=Pn(x) = L Pixi and Q:=Qm(x) = L qixi be nonzero polynomials in IC[x), of degrees n and m respectively, and suppose that (1) if ex is a simple root of Q, then P(«)/Q'(«) is not in Z. (2) if« is a multiple root of Q, then P(cx);tO. Denote by L the operator L := P(a) + xQ(a) , :=.D/ .DL. Then ( 1)

m is

m

an irreducible .D-module on A l

m

(2) If n > m, is a Lie-irreducible object of D.E.(A 1/te), whose largest slope at oo is (n+1-m)/(n-m), with multiplicity n-m. (3a) If n i m, then for « := -pm/qm,

m I A1

- {«} is an irreducible

object of D.E.(A 1 - {«}/IC). Its local monodromy at « is a pseudoreflection of determinant exp(-2rri~), where ~ is given by 2 ~ := ./(x - oc.i) + g'(x) with (P,Q)=1. If g=O, then n=m-1; otherwise m-1-n = ord 00 (g). We will see below that already the sequence of functions x-1/Zexp(-xn/n) leads to some surprises. Theorem 2.10.4 Let P:=Pn(x) = L PiXi and Q:=Qm(x)

=L

qixi be

nonzero polynomials in C[x]. of degrees n and m respectively, and suppose that (1) if oc. is a simple root of Q, then P(oc.)/Q'(oc.) is not in Z. (2) if oc. is a multiple root of Q, then P(oc.);tO. Suppose n > m. The di:(ferential galois group G of P(a) + xQ(a) on A 1 is connected and reductive. If Pn- 1 = qn_ 1 = 0, then G = (30,der; otherwise G = 1Bm(30,der The possibilities for GO,der are given by:

74

Chapter 2

(1) If n-m is odd, GO,der is SL(n); if n-m= 1, then G is GL(n). (2) If n-m is even, then either GO der is SL(n) or SO(n) or (if n is even) SP(n), or n-m=5, n=7,8 or 9, and GO,der is one of n=7: the image of G2 in its 7-dim'l irreducible representation n=8: the image of Spin(7) in the 8-dim'l spin representation the image of SL(3) in the adjoint representation the image of SL(2)xSL(2)xSL(2) in std®std®std the image of SL(2)xSp(4) in std®std the image of SL(2)xSL(4) in std®std n= 9: the image of SL(3) x SL(3) in std® std. proof In view of 2.10.3(2) above, this is just the Main D.E. Theorem 2.8.1 on A1, with alb = (n+1-m)/(n-m), together with the remark that on A 1 one has detG = {1} or tUm, and detG = {1} if and only if the coefficient of an-1 vanishes. QED To give a concrete illustration of this theory, let us compute G for the operator an- xa - 112, whose FT defines x-1/Zexp(-(-x)n/n). Theorem 2.10.5The differential galois group G of an- xa - 1/2 on A1 is GL(2) for n= 2, SL(n) for n even 1 4, SO(n) for n~7 odd 1 3, G2 for n=7. proof This is an instance of the above theorem with P(x)= xn - 1/2, Q(x) = x, m= 1. For n even, n-m = n-1 is odd, and so G is SL(n) or GL(n); looking at the an-1 term, we see that G is inside SL iff n > 2. If n is odd, this operator is self-adjoint (up to a sign), and as n is odd the resulting autoduality is necessarily symmetric. Therefore G is inside SO(n) for n13 odd; in view of the limited possibilities for G, it must be SO(n) except for n=7, where the (only) other possibility is G2. That it is G2 in this case results from the following Theorem 2.10.6 For any polynomial fin IC[x] of degree k prime to

Gz

6, the differential galois group G of a7- fa- (1/2)f' on A1 is G2 . proof We first prove that the D.E.on A 1

3Tt := .D/.DL, L := a7- fa- (1/2)f is irreducible. Its oo -slopes are 1 + (k/6) with multiplicity six and one slope 0. Since (k, 6)=1 by hypothesis, the ! 00 -representation is the direct

D.E.'s and '!l-modules

75

sum of an irreducible of dimension 6 and a tame character. So if m is reducible on A1, its Jordan-Holder constituents must be an irreducible D.E. non A1 of rank six and a rank one D.E. I.. on A1 which is regular singular at oo, and therefore isomorphic to the trivial .b-module ~. So either ~ is a quotient of m, or it is a subobject. Since m is self-adjoint, and n and~ are nonisomorphic irreducibles, m ~ n EB ~.Therefore ~ is a quotient of m. This means that the equation Lcp=O has nonzero solutions in C[x). But L acts injectively on C[x); indeed if f = Axk + ... , then L maps xd + lower terms to (-d- (1/2)k)Axd+k-1 + lower terms, and ask is odd (being prime to 6), (-d- (1/2)k) is nonzero for all dE:Z. Therefore m is irreducible. Once m is irreducible, it is Lie-irreducible (we are on A1). Its ooslopes qualify it for the Main D.E. Theorem. Because it is self-adjoint, the only possibilities are Gz or S0(7). It thus suffices to rule out S0(7). Because /\ 3 (std7) is irreducible for S0(7), it suffices to show that m has a nonzero horizontal section in /\3m. We can view mas the free~­ module with basis eo, ..., e6, where a acts by aei = ei+1 for i=0,1, ... ,5

ae6 = (1/2)feo + fe1.

Then one readily verifies that in /\3m the element eo-"e4-"e5 + ezAe3-"e4 + 2e1AezAe6 e1 Ae3Ae5

_ eoAe3Ae5

is killed by a . QED

Theorem 2.10.7 Let P:=Pn(x) = ~ PiXi and Q:=Qm(x) = ~ qixi be nonzero polynomials in C[xl, of degrees nand m respectively, and suppose that (1) if 0 Oo = Q, deg(Qi) s. N- M fori> 0, with equality if deg(Ai)=N.

proof If such a factorization exists, then equating like powers of x and using the commutation relation Pi(D)xj = xjPi(D+j), we find An(t) =

~i+j=n Pi(t+j)QjO and JTl. is reducible, it has either a nonzero subobject or a nonzero quotient n in D.E.(IBm/C) all of whose slopes at both zero and oo are 0.

89

D.E.'s and :t'-modules Replacing m by its adjoint if necessary, we may assume that it has such a quotient. But such an n is itself a successive extension of .D-

modules of the form x«C[x,x-1], so M would admit some x«c[x,x-1] as a quotient. Concretely, this means that L kills some nonzero element of x«c[x,x-1], say x«f(x) where f = L:i=a, ...,b :Aixi is a Laurent polynomial of bidegree (a,b). Looking at the highest order term in L(x«f)=O, we see that Pd(oc:+b)=O; looking at lowest order terms we see that Po(oc:+a)=D, contradicting (c). Thus m is irreducible. If m is not Lie-irreducible, it is Kummer induced of some degree D > 1. Looking at the slope decompositions of the Io- and I 00 representations, we see that every slope=:A component of each decomposition is itself Kummer induced of degree D. Therefore D must divide the multiplicities with which any of the slopes occurs, whence D divides gcd(n,m). Looking at the semisimplification of the slope=O part at zero (resp. oo), we see that the roots mod l of Po (resp. P d) are Kummer induced of degree D. This concludes the proof of (1). We now turn to the proof of (2). It is clear that j*m is a D.E. on U, which is regular singular at 0, oc:, and oo. Let us admit temporarily that m~j!*j*m.

Then from Pochammer's Lemma 2.9.7 we see that local

monodromy around oc: is a pseudoreflection. So if j*m is not irreducible, it has either a nonzero subobject or a nonzero quotient n in D.E.(U/C) which is regular singular at O,oc:,oo and whose local monodromy at oc: 1s

trivial. Such ann is a successive extension of j*(x«c[x,x-1])·s, and hence, at the expense of replacing m by its adjoint, we may assume that j*m has a quotient j*(x«c[x,x-1]). This means that m~j,*j*m has a nonzero map to j!*j*(x«C[x,x-1]) =x«c[x,x-1], which means precisely that L kills a nonzero element of x«C[x,x-1]. Exactly as above, this contradicts (c), and so establishes the irreducibility of j*m. Again using m~j,*j*m, we find that m itself is an irreducible .D-module on

6m. Thus it remains only to establish that

m~j,*j*m

in case (2). This

means showing that both L and L * act injectively on the delta-module Soc:. As the hypotheses are self-adjoint, it suffices to prove this for L. At the expense of a multiplicative translation, we may assume that oc: = 1. We write

90

Chapter 2 L= P(D) - xQ(D) with both P, Q of degree n

~

1.

Passing to the formal parameter tat oc:=1 such that x= et, our operator becomes L=P(d/dt) - etQ(d/dt), and we must show that this operates injectively on So = IC((t))/IC[[t]). This is the Fourier Transform of the equivalent problem of showing that, denoting by Sub = exp(-d/dt) the endomorphism of IC[t] given by t~-->t-1, the operator P(t)- SuboQ(t) is injective on IC[t). But if f(t) is a nonzero polynomial which satisfies P(t)f(t) = Q(t-1)f(t-1), i.e., f(t)/f(t-1) = Q(t-1)/P(t), then for each k ~ 2 we have f( t)/f( t- k) = [Q( t-1)Q( t-2) ... Q( t- k)l/[P(t)P( t-1) ... P(t-(k-1))). By hypothesis (c), the right hand fraction is in lowest terms, which implies that f(t-k) has at least nk zeroes. Therefore no such nonzero f exists. QED Applying the Main D.E. Theorem 2.8.1 in the case n..em, (we will take up the case n=m further on, in 3.5, 3.5.8) we obtain Theorem 2.11.10 Let (d,n,m) be a triple of nonnegative integers with d ~ 1, n..e m and gcd(d,n-m) = 1. Suppose the operator L := 2:xiPi(D) satisfies the following four conditions: (a) Pi =0 except for i::O, ...,d; Po and Pd are nonzero. (b) degPo = n, degPd = m, and min(n,m) ~ degPi for DO has irregularity one, this contradicts [Ka-DGG, 2.3.8]. QED Remark 3.3 .1 If we knew that the isomorphism class of the semisimplification of ·X as .D-module on IBm depended only on the data (or.i mod l, r>j mod l, A), then at the last step we could replace "isomorphism" by "isomorphism of semisimplifications". This would still be adequate to show IJ.=1, using [Ka-DGG, 2.3.81. Duality Recognition Theorem 3.4 Suppose that X:=XA(or.i's; r>j's) is an irreducible hypergeometric .D-module on IBm of type (n,m). In order that there exist an isomorphism of X with its adjoint, it is necessary and sufficient that the following three conditions hold: ( 1) n-m is even. (2) there exists a permutation iHi' of [1, ... ,n] such that j' + r>j E: l. Moreover, if these conditions are satisfied, then the resulting autodua!ity of X (on the dense open set where X is a D.E.) is alternating if and only if max(n,m) is even, and¥:= Lr>j -Lor.i E:l; otherwise (i.e., if max(n,m) is odd or if ¥ E: 1/2 + l) it is symmetric. proof Indeed, the adjoint is X,.j's), so the first assertion is obvious from 3.2 and 3.3. Because X is irreducible!, it has at most one autoduality (up to a ex factor) , which is either alternating or symmetric. If the generic rank max(n,m) of X is odd, the autoduality has no choice but to be symmetric. The only problem comes when max(n,m) is even. At the expense of an inversion, we may assume n ~ m. If n=m, then local monodromy at A is a pseudoreflection of determinant exp(21Ti¥). If ~E:Z, this is a unipotent pseudoreflection, and

Generalized Hypergeometric Equation

97

so cannot be symmetric. [For denoting by N the log of this unipotent pseudoreflection, + =O, so if =, we find = 0. Since N has one-dimensional image, say ICe, we have =O if Nx:tO, and then for all x, whence e=O, contradiction.] Conversely, if ¥ E: 112 + Z, then we have a true reflection, which lies in no symplectic group (in Sp(2d), the eigenvalues of any element can be grouped into d pairs of inverses). If n > m, and both nand n-m are even, then detX is a rank one D.E. on ~m which has slope zero at both 0 and oo (because all the ooslopes of X are 1/(n-m) < 1). So detX must be of the form x&IC[x,x-1) for some&. Looking at the slope decompositions of X at both 0 and oo, and at the formal Jordan decompositions of the slope zero parts, we see that det(X®IC((x))) :::: xO is irreducible (because it has rank n-m and all slopes 1/(n-m)), it has at most one autoduality (up to a ex factor), say (x,y), and that autoduality has a sign (i.e., is either symmetric or alternating) which must be the same sign as that of our global one . So our sign rule follows from the following Lemma :3.4.1 Let d:?. 2. Let W be a D.E. on C((1/x)) of rank d all of whose slopes are 1/ d. (1) The isomorphism class of W is determined up to a multiplicative translate by the isomorphism class of det(W). (2)W is self dual if and only if dis even and det(W)®2 is trivial, and the duality is alternating if and only if det(W) is trivial. proof Any D.E. Won IC((1/x)) of rank d all of whose slopes are 1/d is a multiplicative translate of one of the form ([d)*!.)® x>. where !. is the rank one D .E. for eX, and >.

E: IC

. Notice that the isomorphism class of

98

Chapter 3

([d)*J:)®x). ~ [d)*(J:®xd).) depends only on d:>. mod Z. Now det(([d)*J:)®xA) ~ det([d)*J:)®xd). visibly determines d). mod Z. Because [d)*J: has all slopes< 1, its determinant is of the form xi-IC((x)), so the isomorphism class of det([d)*J:) is translation invariant. Therefore the isomorphism class of det(W) is translation invariant, and hence det(W) determines d). mod Z. Therefore det(W) determines the isomorphism class of W itself, up to a multiplicative translation. This proves (1). We next observe that if d is odd, then W cannot be self dual, because W®W has all slopes 1/d for odd d. [To see this, use the fact that [d]*W as representation of the upper-numbering subgroup (! 00 )(0+) is, for some aE: C x, [d]*W I (Ioo )(0+) ~ E9 \:E: 1-ld "X a~:• where "Xa~: is the character of (! 00 )(0+) given by the rank one D.E. for ea!:'X.] If W is self dual, so is det(W), whence det(W) ® 2 is trivial. Suppose now that det(W)®2 is trivial. We argue globally as follows. Consider a Kloosterman equation of even rank d,i.e., a hypergeometric equation of type (d,O), say X).(a1, ..., ad; ¢).Since d12, its determinant is xaC[x,x-1] for a=Lai, so over C((llx)) we obtain a Was above with det(W) xaC((1/x)). As). varies over

~

ex, but the ai remain fixed, we obtain all

translates of this W. So it suffices to analyse the sign of the autoduality for the two particular Kloosterman equations of even rank d X).(O, 0, 1/(d-1), ..., (d-2)/(d-1); ¢), X).(1/2,

o,

1/(d-1), ... , (d-2)/(d-1); ¢).

Consider for each the d-1'st power of local monodromy at zero. For the first, it is a unipotent pseudoreflection (not in any O(d)), and for the second it is a true reflection (not in any Sp(d)). QED

Corollary 3.4 .1.1 Hypotheses and notations as in the above lemma, denote by J: the rank one D.E. for eX. Then (1) det([d)*J:) ~ xIC[x, x-11, simply because its slope is 0 at zero and i 1/ d < 1 at oo, so also 0 at oo. To evaluate&, we look at local monodromy at zero; this shows & = (d-1)/2. Now looking over C((1/x)) we get the asserted formula for det([d)*L).

'*

That (1) (2) was proven as part (1) of the previous Lemma. To prove (3), notice that ford odd, the dual of X1(0, 1/d, 2/d, ... , (d-1)/d; ¢)is its multiplicative translate by -1. QED

(3.5) We now study the Lie-irreducibility of the hypergeometric equation in the case n=m. Our analysis is a geometric version of the more group-theoretic one of [B-H, 5.81. Given a pair (a, b) of strictly positive integers, and A in IC x, consider the "Belyi polynomial" Bela,b,A(x) := AIJa,bxa(1-x)b, where 1-la,b := (a+b)a+b/aabb. We call it the Belyi polynomial because of the brilliant use Belyi makes of it in [Bel, Part 4]. It is a morphism of degree n:=a+b from IP1 to IP1, which induces a finite etale covering IP1- {0, 1, a/(a+b), oo} ~ 1?1- {0, A, oo} whose ramified fibres are over 0: exactly two points; one with mult. a and one with mult. b, over oo: exactly one point, over A: exactly n-1 points. We call this covering the Belyi covering of type (a,b). The covering defined by 1/Bela,b,,_-i(x) we call the inverse Belyi covering of type (a, b). Lemma 3.5.1 Over IC, any finite etale connected covering x~ IP1- {0, A, oo} of degree n such that the induced map of complete nonsingular models -rr:X~ 1?1 has exactly n-1 points in the fibre over A is isomorphic to either a Belyi covering or an inverse Belyi covering of type (a,b) for some partition of n =a+b as the sum of two strictly positive integers. proof Since X- ,.-1(0, A, oo) is finite etale over 1?1- {0, A, oo} of degree n, and we are in characteristic zero, the Euler characteristics multiply:

100

Chapter 3 2- 2g(X)- Card(1T-1(0)) -Card(1T-1(:>.))- Card(1T-1(oo))

= -n.

By assumption, Card(1T-1(:>.)) = n-1, so we find 2 - 2g(X) - Card(1T-1(0))- Card(1T-1(oo))

= -1, which we rewrite

2g( X) + Card( 1T-1(0)) + Card( 1T-1( oo )) = 3. Since each of Card(1T-1(0)) and Card(1T-1(oo)) is~ 1, and g(X) ~ 0, we see that g(X) = 0, and Card(1T-1(0)) + Card(1T-1(oo)) = 3. After a multiplicative inversion on the base, we may assume that Card(1T-1(0)) = 2,

Card(1T-1(oo)) = 1.

Since X is a noncanonical IP 1, we may decree that oo is the unique point over oo, and that the two points over 0 are 0 and 1. That oo is the unique point over oo means that our covering is given by a polynomial of degree n, and looking at the fibre over 0 shows that its only zeroes are 0 and 1. So our covering is given by .} of the d-fold Kummer covering of Gm by itself) or d=n and the covering is isomorphic to either a Belyi covering or an inverse Belyi covering of type (a, b) for some partition of n =a+ b as the sum of two strictly positive integers. Moreover, in the case of a Belyi or inverse Belyi covering, local monodromy of X around :>. is a true reflection. proof Suppose that the fibre of 1T over :>. consists of points Pi with multiplicities ei. Then X®C((x-:>.)) is the direct sum of the ecfold Kummer inductions of the V'®IC((x-pi)): X® IC( (x-:>.))

~

EB i [ei] * ('})' ® IC( (x-pi))).

Because X® C((x-:>.)) is of slope zero, with local monodromy a pseudoreflection, we see all the terms [ei]*(V'®C((x-pi))) are of slope zero, exactly one of them (say i= 1) has local monodromy a pseudoreflection, and all the others have trivial local monodromy. Now in order for [e1]*(V'®IC((x-p1))) to have local monodromy a pseudoreflection, either e1 = 1 and V' ® IC((x-p1)) itself has local

101

Generalized Hypergeometric Equation monodromy a pseudoreflecton, or e1 = 2 and V' is of rank one with

trivial local monodromy around P1· In this second case, notice that the pseudoreflection is a true reflection. If [ei]*(V'®C((x-pi))) with i 1 2 is of slope zero and has trivial local monodromy, then ei = 1, and V' is of slope zero at Pi and has trivial local monodromy around Pi· If all the ei = 1, then our covering is unramified over :>., so it is the restriction to Gm - (:>.} of a d-fold connected finite etale covering of Gm, necessarily the d-fold Kummer covering of Gm by itself. If e1 = 2 and all ei 2 2 = 1, then V' has rank one, whence d=n, and there are n-1 points in the fibre over:>., so our covering is either Belyi or inverse Belyi. QED Belyi Recognition Lemma 3.5.3 Suppose that X:=X:>.(.}.We must show that V' is isomorphic to X on

Chapter 3

102

6m - {;>,}. Because both 1J and X have regular singular points, it suffices to show that on (6m - {:h})an they give rise to isomorphic local systems. This is given by the following rigidity theorem 3.5.4. QED The rigidity of the hypergeometric equation in the case n= m is given by the following theorem. In the case n= 2 it goes back to Riemann (his ""o:>-scheme"; the point is that for n=2, we may always twist by some (x-:>..)'6 to make local monodromy around :h a pseudoreflection). Levelt gave a simple group-theoretic proof [Lev-HF] in the general case (cf [B-H, 3.5]). The proof we give below is due to Ofer Gabber.

Rigidity Theorem 3.5.4 Let 1 and .})an of the same rank n ~ 1, and suppose that

(a) the local monodromies at ), of both r and .. is of the form A®B with A in GL(d) and B in GL(d'). But if both d and d' are 1 2, no pseudoreflection can be of this form. Therefore d= 1, as required. Then detX ~ w®n® detK, with detK of finite order, so W is of finite order if and only if detX is of finite order; if it is, then X ~ W® K itself has Ggal finite. QED Theorem 3.5.8 ([B-Hl, 5.5]) Suppose that X:=X:>.. («i's; ~ j's) 1s an irreducible hypergeometric .D-module on IBm of type (n,n) which is neither Kummer induced nor Belyi induced nor inverse Belyi induced. Denote by G the differential galois group of X I IBm-{:>..}, by ¥:=L~rL«i· ( 1) The group G is reductive. If ¥, L«i, and

L ~ j are all in Q (i.e., if detX

is of finite order), then GO =GO,der. Otherwise, GO = IBmGO,der. (2) The group GO,der is either {1}, SL(n), SO(n), or (if n is even) Sp(n). (3) if exp(21Ti¥)

:t

± 1, GO,der::: {1} or SL(n).

(4) if exp(21Ti¥) = -1, GO,der = {1} or SL(n) or SO(n). (5) if exp(21Ti¥) = +1, GO,der ::: SL(n) or (for n even) Sp(n). (5) if ¥ is irrational, G= GL(n). proof The local monodrom y around :>.. is a pseudoreflection of determinant exp(21Ti¥). So if X I IBm - {:>..} is Lie irreducible, the theorem is an immediate consequence of the Pseudoreflection Theorem 1.5. In view of the preceding Lemma, the only other case is when X I IBm - {:>..}

Generalized Hypergeometric Equation

107

is the tensor product W®K of a D.E. W of rank one with an irreducible D.E. K of rank n whose Ggal is finite. In this case GO is either {1} or 6m, depending on whether or not W, or equivalently detX I 6m - {A}, is of finite order. So (1) through (4) hold (trivially) in this case. If 'g" is either in Z or is irrational, then we cannot be in this case, for then local monodromy around A is a either a unipotent pseudoreflection or is Diag(exp(2TTi'(), 1, 1, ... , 1), no power of which is scalar. QED We can be more precise about the distinguishing the the various Lie-irreducible cases. (We will discuss later, in 5.4-5.5 and then again in 8.17, how to detect the case when GO,der is {1}.) Corollary 3.5 .8 .1 Notations and hypotheses as above, suppose further that GO,der ~ {1}. Then GO,der is SO(n) (respectively Sp(n)) if and only if there exists SE:IC such that X®xS :=XA(«i + S's; ~j + S's) is self dual and its autoduality pairing is symmetric (resp. alternating). proof Notice that GO,der is the same for any twist X®xS as for X. So if some twist X® x S is self dual, then GO, der is contained in SO( n) or Sp(n), depending on the "sign" of the autoduality. In view of the paucity of choices for GO,der, GO,der must be SO(n) or Sp(n). Conversely, suppose that GO,der is SO(n) (son ~ 3 since S0(2) is not semisimple) or Sp(n). We must distinguish several cases.

If GO,der is Sp(n), then G must be contained in the normalizer in GL(n) of Sp(n), which is 6mSp(n). Since the only scalars in Sp(n) are ± 1, we cen construct a character 'X. of G by writing an element of G as tA (tin 6m, A in Sp(n)) and defining ')(.(g)=')(.(tA) := t2. This character of G corresponds to a rank one D.E. which is in the tensor subcategory . Now X has regular singularities at O,A, and oo, and its local monodromy around A is a unipotent pseudoreflection (otherwise we can't have GO,der = Sp(n)). Therefore every object in is regular singular at 0, A, and oo and has unipotent local monodromy at A. Therefore the rank one D.E. on 6m - {A} corresponding to 'X. is regular singular at 0, A, and oo and has trivial local monodromy at

A,

so it must be the D.E. for xS

for someS. Then X®x-S/2 has its differential galois group inside ±Sp(n) = Sp(n), as required. If GO,der is SO(n), n ~ 3, then G must be contained in the

108

Chapter 3

normalizer in GL(n) of SO(n), which is 6mO(n). As O(n) contains no scalars except ±1, so we get a character X of 6mO(n) by x(tA):=t2. The corresponding rank one D.E. is in . But X has regular singularities at O,A, and oo, and its local monodromy around A is a true reflection (otherwise we can't have GO,der = SO(n)), so any object in is regular singular at 0, A, and oo. Let us denote by TA the local monodromy of X around A. If we can show that ')(.(T:h) = 1, then just as above X corresponds to the D.E. for some xS, and X®x-S/2 has its differential galois group inside ±O(N) = O(N), as required. To show that x(TA) = 1, suppose not; then ')(.(TA)

= ~2,

where ~2

;t

1. This means that

TA =~-1A with A in O(n). But in a suitable basis e 1 , ..., en of the representation space, the reflection TA is Diag(-1, 1, ..., 1). So we find that the matrix A:=Diag(-~, ~ •... , ~) E: O(n) for some nondegenerate quadratic form < , >.To see that this is impossible for n 2 3, we argue as follows. Denote by V the line Ce1, and by W the C-span of e 2 , ... , en. Writing vectors in the form v+w, with ve:V and we:W, we have A(v+w) = -~v + ~w. As Ae:O(N), we have = = . Expanding out, we find + 2 +

= ~2 - 2~2 + ~2.

Taking v=O (resp. w=O), we see that ~2 ;t 1 forces = 0 =. Therefore V and Ware totally isotropic, and so Wn.L(V) = 0. But both W and .L (V) are codimension one subspaces, so W n .L (V) = 0 is impossible if n 2 3. QED In the case n;t m, we have

Theorem 3.6 Suppose that X:=XA(«i's; llj's) is an irreducible hypergeometric .D-module on 6m of type (n,m), n

;t

m, which is not

Kummer induced. Let N:=max(n,m) be the rank of X, and G its differential galois group. Then (1) G is reductive. If detX is of finite order (i.e., if ln-ml > 1 and if ~ m (resp. ~ 11 j e: iQ when m > n)) , then GO = GO,der;

otherwise GO = QlmGO,der. (2) If ln-ml is odd, GO,der is SL(N). If ln-ml = 1 then G is GL(N). (3) If ln-ml is even, then GO,der is SL(N) or SO(N) or (if N is even) SP(N), or ln-ml=6, N=7,8 or 9, and GO,der is one of

Generalized Hypergeometric Equation

109

N=7: the image of G2 in its 7-dim'l irreducible representation N=8: the image of Spin(7) in the 8-dim'l spin representation the the the the N=9: the

image image image image image

of of of of of

SL(3) in the adjoint representation SL(2)xSL(2)xSL(2) in std®std®std SL(2)xSp(4) in std®std SL(2)xSL(4) in std®std SL(3)xSL(3) in std®std.

proof This theorem, "mise pour memoire", is just the special case d= 1 of 2.11.10. QED The discrimination among the various possible cases is aided by Corollary 3.6.1 Notations and hypotheses as above, GO,der is contained in SO(N) (resp. in Sp(N)) if and only if there exists Se:IC such that X®xS := X:\(o:.i + S's; ~j + S's) is self dual and its autoduality pairing is symmetric (resp. alternating). Moreover, if N is odd, then GO,der is contained in SO(N) if and only if there exists Se: IC such that X®xS := X:\(o:.i + S's; ~j + S's) has its Ggal c SO(N). proof The proof is entirely analogous to that of 3.5.8.1. If some twist X®xS of X is self dual, then GO,der is certainly contained in SO(N) or in Sp(N), depending on the sign of the autoduality. If GO,der c O(N) (resp. Sp(N)) , then G c IGmO(N) (resp. IGmSp(N)).

Indeed, if r is any irreducible subgroup of GL(N) which respects a nonzero bilinear form < , >, then its normalizer in GL(N) lies in in the corresponding similitude group. [For if Ae:GL(N) normalizes r then the form (x,y) := is also f-invariant, so a scalar multiple of < , >.l Now consider the character 'X. of G defined by 'X. (g)= 'X. ( tA): = t 2. The corresponding rank one D.E. on IGm is in . Now ln-ml is nonzero (by assumption) and even (otherwise GO,der is SL(N)), hence ln-ml 1 2. Therefore all slopes of X at 0 or oo are i 1/ln-ml < 1, and hence every object of has all its slopes < 1 at 0 or oo. So any rank one object in has slope zero at 0 and oo, so is the D.E. for xS for some S. Taking the S corresponding to ')(.., X®x-S/2 has its differential galois group in ±O(N) = O(N) (resp. in ±Sp(N) = Sp(N)). If N is odd, then O(N) is the product{± 1}xSO(N), so if Ggal CO(N)

110

Chapter 3

but Ggal (/. SO(N), its projection onto the { ± 1} factor is a character of order two corresponding to a rank one object of , necessarily the D.E. for x112. QED

Lemma 3.6.2 Let X:=X:>..(oc(s; ~j's) be a hypergeometric .D-module on Gm of type (n,m), n > m. Then Ggal c SL(n) if and only if n-m Loci

E:

2

2 and

Z.

proof If n-m= 1, then detX has slope 1 at oo, so nontrivial. If n-m

1

2,

then det(X) is necessarily the D.E. for xS, someS; looking at zero we see that S = Loci mod Z. QED

3.7 Intrinsic characterization of hypergeometric equations We now turn to the intrinsic characterization of irreducible hypergeometric .D-modules on Gm among all irreducible .D-modules on Gm. Theorem 3.7 .1 Let :m be an irreducible, non punctual holonomic .Dmodule on Gm. Then :m is hypergeometric if and only if its Euler characteristic ')((Gm, 1n):= ')((H*DR(Gm, 1n)) is -1.

proof It is obvious from the elementary Euler Poincare formula on Gm (2.9.13) that X(Gm, X)= -1 for any hypergeometric X on Gm, irreducible or not. Suppose now that :m is an irreducible, nonpunctual .D-module on Gm with ')((Gm, 1n)= -1. We will make essential use of

Lemma 3.7.2 (compare 3.5.5) If :m is an irreducible, nonpunctual .Dmodule on Gm with ')((Gm, 1n)= -1, then for any twist 1n®xs, dimcSolno(1n®x 8 ) + dimcSoln 00 (1n®xS)

5.

1.

proof The twist 1n® xS is also irreducible on· Gm, and its 'X is the same as that of :m (this is obvious from the Euler-Poincare formula), so it suffices to prove dimcSoln 0 (:m) + dimcSoln 00 (1n) 5. 1. Let k: Gm -+ IP1 denote the inclusion. Then by 2.9.8 we have a short ex act sequence on IP 1 0-+ k!*:m-+ k*:m-+ So®Homc(Soln 0 , IC) E9 S 00 ®Homc(Soln 00 , IC)-+ 0 Let us admit temporarily that

Generalized Hypergeometric Equation

Ill

( .. ) dimcH1DR(IP1, k*:Jn) = 1, H2DR(IP1, k 1.. m) = 0. Then as H1DR(IP1,

s0 ):::::

IC ::::: H1DR(IP1, S 00 ), the long exact cohomology

sequence gives us a surjection H1DR(IP1, k*:Jn)--» Homc(Soln 0 , IC) EfJ Homc(Soln 00 , C), whence the required inequality on dimensions. To prove( .. ), we argue as follows. We have HiDR(IP1, k .. :Jn) ::::: HiDR(Gm, :Jn), and only HO and Hi are possibly nonzero. But HO = Hom.J)(~ ,:Jn) vanishes, because if not then m, being irreducible, would be isomorphic to~. which is nonsense because x.(\X1, ... , \X7; ~1) is irreducible with Ggal

C

S0(7) if and

only if there exist x,y,z E:C such that after renumbering we have (\Xi, ..., \X7)

=(0, x, -x, y, -y, z, -z) mod z7, and ~1 = 1/2 mod Z, and

none of x,y,or z is = 1/2 mod Z. proof First of all, such an equation X:>. (0, x, -x, y, -y, z, -z; 112) is

irreducible, self dual and of determinant one. If X:>.(\X1, ... , \X7; ~1) is self dual, then ~1 must be 0 or 1/2 mod Z. If Ggallies in S0(7), then its local monodromy at zero must lie in S0(7). But the eigenvalues of any element of S0(7) are of the form (1, a, a-1, b, b-1, c, c-1) for some a,b,c in C x. As the eigenvalues of local monodromy at zero are the exp(2'1Ti\X }'s, we get the existence of x,y,z E:C such that after renumbering we have (\X1, ... , \X7)

=(0, x, -x, y, -y, z, -z) mod z 7 .

Since X is irreducible, ~1 cannot be 0 mod Z, so ~1 Irreducibility now insures that none of x,y,z can be

= 1/2 mod Z. = 1/2 mod Z.

QED

Lemma 4.1.3 If X:>.(\X1, ..., \X7; ~1) is irreducible with Ggal c G2 c S0(7), then there exist x,y E:C such that after renumbering we have

= (0, x, -x, y, -y, x+y, -x-y) mod z7, and ~1 = 1/2 mod Z, and none of x,y,or x+y is = 1/2 mod Z. (\X1, ..., \X7)

proof The eigenvalues of any element of G2 in its seven-dimensional representation are of the form (1, a, a-1, b, b-1, ab, (ab)-1) for some a,b in C x. Proceed as above. QED In view of the above lemmas, the problem of recognizing which

Chapter4

124

hypergeometrics have GO,der = G2 is sompletely solved by

Theorem 4.1.4 Let x,y E:IC such none of x,y,or x+y is any A E:

ex, X := XA(O,

X,

= 1/2 mod Z. For

-x, y, -y, x+y, -x-y; 1/2) has Ggal = G2.

In view of the preceding Lemmas, this implies

G2 Recognition Theorem 4.1.5 Let x,y E:IC such none of x,y,or x+y is

= 1/2 mod Z. Then for any A E: IC x, X := XA (0, x, -x, y, -y, x+y, -x-y; 1/2) has Ggal = G2 . These are all the hypergeometric of type (7, 1) with Ggal = G2. The hypergeometrics of type (7,1) with GO,der = G2 are precisely the xS twists of these.

proof of 4 .1.4 .The only two possibilities for Ggal are S0(7) or its subgroup G2. These two cases may be distinguished by the fact that for S0(7), /\3(std 7 ) is irreducible, while G2 has a non-zero (in fact onedimensional) space of invariants in /\3(std7)· Thus we must show that for X as above, /\3(X) has HODR(6m, /\3(X)) nonzero. Here is a proof suggested by Ofer Gabber, analogous to the proof of 3 .7 .3. Denote by j: 6m -+ IP1 the inclusion. Since /\3(X) is self dual (because X is), its middle extension j! * 1\ 3(X) is also self dual. By global duality, the two cohomology groups HiDR(IP1, j!*/\3(X)) for i=O and i=2 are dual to each other. By 2.9.1.3, HODR( IP 1' j! * 1\ 3(X)) = HomJ)(~ IP 1, j! * 1\ 3(X)) = = Hom.n(j!*~6m' j!*/\3(X)) = Hom.J)(~6m' /\3(X)) = HODR(6m, /\3(X)). Therefore the nonvanishing of HODR(6m, /\3(X)) will result from the estimate ')(.(IP1, j!*/\3(J{))

2

2 > 0.

By the Euler-Poicare formula, we have ')(.(IP1, j 1*/\3(X)) = -Irr0 - Irr 00 + dimcSolno + dimcSoln 00 We will show that (1) dimcSolno 1 5.

.

Detailed Analysis of Exceptional Cases

125

(2) Irro = 0. (3) dimcSoln 00 = 2. (4) Irr 00 = 5. In order to prove (1), let us denote by T the local monodromy of X around zero, and by P(T) its characteristic polynomial. We know that as C[T]-module, X is C[T]/(P(T)). In terms of the quantities a:= exp(21Tix), b:=exp(21Tiy), the roots of P(T) are (1, a, 1/a, b, 1/b, ab, 1/ab). Since X is regular singular at zero, we have dimcSolno(/\3(X)) = dim Ker(T-1 acting on /\3(C[T]/(P(T))). We claim that this dimension is ~ 5. To see this, we will resort to a specialization argument to reduce to the case in which P has all distinct roots. We first treat the case where P has all distinct roots. Then T is diagonalizable, say T ~ Diag(a1, ..., a7l, hence /\3(T) is diagonalizable with eigenvalues exactly all triple products aiajak with i < j < k. If we number the ai so that they are (1, a, 1/a, b, 1/b, ab, 1/ab), then the five triple products indexed by (1,2,3), (1,4,5), (1,6,7), (2,4,7), (3,5,6) are all 1, so dimKer(/\3(T) - 1) ~ 5, as required. In the general case, we argue as follows. Let us define, for indeterminates A, B , the polynomial P A,B(T) := (T -l)(T- A)(T- 1/ A)(T- B)(T- 1/B)(T- AB)(T- 1/ AB). Then over the ring R := C[A, Bl!1/ ABL we can form the R[T]-module M := R[T]/(P A,B(T)), which is free of rank seven over R. The general case results immediately from the following elementary lemma, applied to S:=Spec(R), :Jn.:= /\3(M), 'J:= /\3(T)- 1.

Specialization Lemma 4.1.6 LetS be a scheme, :m. an C)"s-module which is locally free of finite rank n, and 'J

E:

EndC)- 5 (:Jn.). For each point

s in S, consider the induced endomorphism 'J s of then-dimensional K(s)-vector space :m.s· For any integer i

1

1, the set

{s in S where dimKer('J s ) ~ i} is Zariski closed in S. proof Since dimKer('T s ) + dimim('T s ) = n, this is also the set where dimim('J s)

i

n-i. But dimim('J s)

i

n-i

#

/\1+n-i('J s> = 0. Thus our set

Chapter 4

126

is the locus of vanishing of all minors of

'T

of a given size. QED

This concludes the proof of (1). Since X is regular singular at zero, (2) is obvious. We now turn to the proofs of (3) and (4), both of which are tedious but straightforward. Let us denote by W the sixdimensional wild part of X®C((1/x)). Since ~1 = 1/2, X®C((1/x)) :::: W E9 x112c((1/x)), whence /\3(X®C((1/x))) :::: /\3(W) E9 /\2(W)®x1/Z. To prove (3) and (4), it then suffices to prove (a) and (b) below: (a) 1\ 3(W) has a 1-dim'l solution space, and irregularity 3. (b) 1\ Z(W) 181 x 1/2 has a 1-dim'l solution space, and irregularity 2. Since X has trivial determinant, we see that det(W) :::: x112c((1/x)). Denoting by L the rank one D.E. for eX, it follows from 3.4 .1.1 that W is a multiplicative translate of [6)*.1:. Since the assertions (a) and (b) are invariant under multiplicative translation, we may assume that w :::: [6)*.1:. (4.1.7) We now explain how to analyse the exterior powers of such a Kummer-induced W. It will be clearer if we consider a slightly more general situation. Fix a C-valued fibre functor w on D.E. (C((1/x))/IC). For any polynomial f(x) in IC[xl, define Lt(x) := ef(x)IC((1/x)) = the rank one D.E. for ef(x) over IC((1/x)), and denote by Lf(x) := the one-dimensionaliC-space w(Lf(x)). \jlf(x) := the corresponding character of I 00

.

In order to describe [d)*(.tf(x)). it is equivalent via descent theory to describe [d)*[d].,.(Lf(x)) with its canonical action of the covering group l-Id· Using the canonical isomorphism of functors [d]*[d)*(?):::: E9~''~-.. d [x

H

!;'x]*(?),

this amounts to making explicit the the natural action of the group l-Id on the d.-dimensional representation space W(cl, f(x)) := E9 ~e: ~'d Lf(~x) of I 00 . Clearly an element 1-1

E:

1-lcl maps Lf(~x) to Lt(~,~~x)• and l-Id induces the

identity. So there exists an eigenbasis {e!;' space W(d, f(x)),

E: Lf(~x)

} of this representation

Detailed Analysis of Exceptional Cases

127

'6(e~) = lJlt(~x)('6)e~ for '6 E: ! 00 ,

on which iJ.

E:

iJ.d acts by

[1J.](e~) := eiJ.~·

Thus W(d, f(x)) with its action of iJ.d corresponds via descent theory to [d]*(Lf(x)). For any "construction of linear algebra·· Constr, Constr(W(d, f(x))) carries an induced lAd action, and it corresponds via descent to Constr([d] * (Lf(x))). In order to avoid confusion, we will denote by I 00 (d) c I 00 the two inertia groups in question, and by P 00 (d) = P 00 their (common) wild inertia subgroup. Lemma 4 .1.7 .1 In terms of the descent dictionary, for any construction of linear algebra Constr we have ( 1) Constr([d] * (Lf(x)))P oo = Constr(W(d, f(x)))P oo (d) (2) Constr(W(d, f(x)))Pco (d) =Constr(W(d, f(x)))loo (d) ,and the action of I 00 on Constr([d]*(Lf(x)))Pco factors through its lAd quotient. (3) If X is a character of I 00 which is trivial on P 00 but which does not factor through the iJ.d quotient, then (Constr([d]*(Lf(x)))®x)lco = 0. (4) If X is a character of ! 00 which factors through the lAd quotient, then (Constr([d]*(Lf(x)))® x)loo = (( Constr(W(d, f(x)))loo (d) )® X)i-ld.

proof Assertion ( 1) is a tautology, since P 00 = P 00 (d) is a subgroup of l 00 (d). For (2), the point is that W(d, f(x)) is the direct sum of the characters

lJ;f(~x)·

For these characters on this form one has

lJlf(x)lJlg(x) = lJlf(x)+g(x). Therefore any Constr(W(d, f(x))) is a direct sum of characters of the form lJlg(x) where g(x) is of the form L:f(~ix). Since characters of the form lJlg(x) satisfy lJlg(x) is trivial on P 00 (d) #

g(x) is constant

#

#

l:.g(x) has slope zero

#

lJlg(x) is trivial on I 00 (d),

we obtain (2). Assertions (3) and (4) then follow immediately. QED We now turn to the explicit analysis of our [6]*!., which

128

Chapter 4

corresponds to W(5, x) with its 1-15 action. Pick a primitive sixth root of unity ~, and denote by {ei}i E: Z/5Z an eigenbasis of W(5, x) with ei E: L~ix, and the action of~ given by ~(ei) = ei+1· We will analyse the exterior powers of W(6, x). We already know that det(W) ~ x112c((1/x)), so it must be the case that det(W(5, x)) is the unique character of order two of 1-15· We can see this directly, since

~

cyclically permutes the ei, so maps

e1 "'e2"'e3Ae4Ae5Ae5 to e2"e3Ae4Ae5Ae5Ae1. Since W is self dual,so are its exterior powers. Therefore if 0 s. i s. 6, the wedge product pairing

t .,i(w) x /\6-i(w) ~ /\5(w) induces an isomorphism /\5-i(W) ~ (/\i(W))®x1/2. This cuts our work in half. However, for increased reliability we will not use it. We now systematically list the P 00 -invariants among the wedge products of the e(s in each 1\j, and give the action of

~

on these

invariants. A given wedge expression ei 1 A ... A eij with 1 s. i1 < i2 ... < ij s. 6 transforms under P 00 by the character .Pax for a =(~)i1 + ... + (~)ij, so it is is P 00 -invariant if and only if (~)i1 + ... + (~)ij =

0; otherwise its irregularity is 1. We write li1, .... ,ijl for ei 1 A ... A eij:

2

basis of (/\i(W(6, x)))Poo none. [1,4], [2,5], [3,6],

action of ~ here, and its eigenvalues none o;H r>~-+¥~-+-o;, eigenvalues -1-13

3 4

[1,3,5], [2,4,6], [1,2,4,5], [2,3,5,6], [1,3,4,5],

o;H r>~-+-¥~-+o;,

5

none.

1

eigenvalues ±1 eigenvalues 1-13·

o;~-+r>~-+o::,

none

Thus we obtain the following table (4.17.2) 1 2 3 4 5

Irr(/\i(W)) 1 2 3 2 1

dim((/\i(W))Ioo) 0 0 1 1 0

dim((/\ i(W) ® x 1/2)loo) 0 1 1 0 0

Detailed Analysis of Exceptional Cases

129

In particular, we see that (a) and (b) in the proof of 4.1.4 hold. This concludes the proof of the G2 theorem 4 .1.4. QED

4.2 The Spin(7), PSL(3) and SL(2)xSL(2)xSL(2) Cases We now turn to the remaining possible exceptional values of GO,der for hypergeometrics of rank eight. These can occur only for type (6,2) (or (2,6), by inversion). Both Spin(7) and PSL(3) are subgroups of S0(8) c 0(8), while (the image of) SL(2)xSL(2)xSL(2) is a subgroup of Sp(8), so in virtue of 3.6.1, a xS twist reduces us to computing GO,der for those X's with G c 0(8) or G c Sp(8). Our first observation is that if X has G c 0(8), then the question of whether or not G C S0(8) (i.e., whether or not detX is trivial) is invariant under twisting X in such a way that it stays self dual. This a general fact about even orthogonal groups.

Lemma 4.2.1 Suppose V is a symmetrically autodual Lie-irreducible D.E. (on any X/IC) of rank n: Ggal(V) C O(n). Let L be any rank one D.E. such that V® L is autodual. Then L ® 2 is trivial, and Ggal(V® L) c O(n). In particular, if n is even, det(V) ~ det(V® L). proof Since both V and V®L are self dual, their determinants have order 1 or 2, soL is of finite order. Denote by ')(. the character of 1T1 diff given by L, and by p the representation given by V. Since V is Lieirreducible, so is V ® L, and they define the same (once we fix a basis of the line w(L), so as to be able to identify w(V) with w(v®L)) representation of the open subgroup Ker( ')(.) of 1T 1 cliff By Lieirreducibility, there is a single (up to a !Cx factor) nonzero bilinear form < , > on w(V) = w(v® L) which is invariant by this open subgroup. By unicity, Ggal(V) c SO(w(V), ),and Ggal(V®L) C O(w(v®L), ). So for any '6

E:

1T 1 diff, both p(¥) and x(¥)p(¥) lie in O(w(v®L), ),

whence')(.(¥), being a scalar in O(w(v®L), ),is ±1. Therefore L®2 is trivial. If n is even, det(V®L)=det(V)®L®n ~ det(V). QED The next two lemmas show that if G c 0(8), then the Spin(7) case (resp. the PSL(3) case) is possible only for G c S0(8) (resp. G !.(3) (viewed as 3x3 matrices of trace zero) is the Cartan involution

c : x . . . -xt

So if we view 0(8) as the orthogonal group of the Killing form on 1>!.(3), the Cartan involution of 1>!.(3) (now viewd inside Lie(0(8)) by the adjoint representation) is Ad(C). But det(C) = -1, so any element of N inducing an outer automorphism must have det = -1. As N c S0(8) by its definition, every element of N induces an inner automorphism. And the on! y scalars in SO( 8) are ± 1. We remer k for Ia ter use that this same argument shows that the normalizer of PSL(3) in 0(8) is the semidirect product PSL( 3) IX {± 1, ± C} .] Therefore if G c SO(B) and GO,der = PSL(3), then G c ±PSL(3). Projection onto the ± 1 is a character of G, so a rank one object of d·t>, so an x 0 . So after an x 0 twist, we find an X with G = PSL(3). In virtue of the fact that one can lift projective representations of

1T

1 cliff of an

open, there exists a rank three D.E. V on IBm whose Ggal is SL(3) such that EncfJ(V) is X. Now the highest oo -slope of X is 1/6. Since the adjoint representation of SL(3) has a finite kernel, it follows from the next lemma that the highest oo-slope of Vis also 1/6. Since V has rank three, this is impossible [the multiplicity of a slope is always a multiple of its ex act denominator, ( cf. [ Ka- DGG], 2 .2 .7 .3)]. QED

Highest Slope Lemma 4.2.4 Let w be a IC-valued fibre functor on D.E.(IC((1/x))/IC), V a D.E. on IC((1/x)), p: I 00 ~ GL(w(V)) the corresponding representation. Suppose that G is a Zariski closed subgroup of GL(w(V 00 )) such that p(I 00 ) C G. Let/\: G ~ GL(d) be any representation of G with a finite kernel, say f, and denote by V 1\ the D.E. corresponding to the composite representation 1\op of I 00 . Then V

131

Detailed Analysis of Exceptional Cases and V 1\ have the same highest slope.

proof For any x

1

0, V has all slopes~ x if and only if p((I 00 )(x+))

and VI\ has all slopes~

X

if and only if p((Ioo)(x+)) c

r

= {e},

([Ka-DGG],

2.5.3.5). Since p((I 00 )(x+)) is connected ([Ka-DGG], 2.5.4.2), these two conditions are equivalent. QED

Lemma 4.2.5 If I(,,(., and regard x as a variable. In other words, denote by K the field()(>.), and by R the polynomial ring K[x) in one variable x over K. Then on the scheme (1!3m)R, both of our candidates X>. (A1, B1) := X>. (0, 1/2, ± x/2, ± ( 1 + x)/2, ± x; 1/3, 2/3), X>. (A1, B2) := X>. (0, 1/2, ± x/2, ± ( 1 + x)/2, ± x; 1/6, 5/6) make sense as free 0"-modules of rank eight endowed with integrable connections relative to the ground ring R. In order to prove that X>. (A1, B1) has GO

= PSL(3)

for every

Chapter4

138

x ;!. ± 113 mod Z, it suffices to prove that X>. (A1, B1) has dimGgal s. 8 for every x; by the specialization theorem it suffices to prove that X>. (A1, B1) has dimGgal s. 8 at the generic point. This is equivalent to showing that XA (A1, B2) has dimGgal > 8 at the generic point. By the specialization theorem it suffices for this to find a particular x where X>. (A1, B2) has dimGgal > B. For this, we take x= 1/3. Then XA(A1, B2)1>. (0, 1/2, ± 113, ± 113; 9f).

Therefore the Ggal of X>.(O, 1/2, ±1/3, ±1/3; ¢)is a quotient of that of X>.(O, 112, ±1/6, ±213, ±1/3; 116, 5/6), hence has lower dimension. But X>.(O, 112, ±1/3, ±1/3; %) has Ggal = 0(6) [being of type (6,0),not Kummer induced, and orthonally self dual with nontrivial determinant) which has dimension 15 > 8. QED Thus we find

PSL(3) Theorem 4.3.6 A hypergeometric X := X>.(

l;x]*~ of~

is also a

symplectic [3]-descent of [3)*X. By uniqueness, we infer that [x ~

1-->

l;x]*~

=K

:::: X. Comparing exponents at both 0 and oo, we see that

This being the case, it suffices to show that for any x ;f. ± 1/4 or ± 1/12 or ±5/12 mod Z, the first possibility Poss(O,O) := Xf..l(the six roots of 3z ±x mod Z, ±x; ±114)

=

has dimGgal s. 9. By the specialization theorem 2.4.1, it suffices to show

Detailed Analysis of Exceptional Cases

145

that Poss(O,O) has dimGgal i 9 for generic x. For this, it suffices to show that each of the other eight possibilities Poss(i,j) has dimGgal

2

10

for generic x. By 2.4 .1, it suffices to exhibit, for each of the other eight possibilities, a single numerical value Xi,j of x for which the corresponding specialized equation has dimGgal

2

10.

Consider first what happens when we specialize x to 1/4. If i-,ej, write {0, ± 1} = {i, j, k}. Then Poss(i,j) specializes to Xl-!(±1/12, ±(1/12 + 1/3), ±(1/12- 1/3), ±(1/4 + i/3); ±(1/4 + j/3)) = Xl-!(±(114

+

i/3), ±(1/4

+

j/3), ±(1/4

+

k/3), ±(1/4

whose semisimplification is of the form Xl-!·(±(114 + i/3), ±(1/4 + i/3), ±(1/4 for 6 := 1/4

+

+

k/3);.0')

i/3); ±(1/4

+

+

j/3))

E9

E9 x°C[x, x-1] E9 x-°C[x, x-1] j/3. Therefore Ggal for this specialization admits as

quotient the Ggal of Xl-!·(±(114

+

i/3), ±(114

+

i/3), ±(1/4

+

k/3);.0'). The

six cases of i-,ej are (i,k) = (0,1), (0, -1), (1,0), (1, -1), (-1, 1), (-1, 0), and for these Xl-!·(±(1/4 + i/3), ±(1/4 + i/3), ±(1/4 + k/3);.0') is respectively Xl-!·(±1/4, ±1/4, ±5/12; .0') XI-!·(± 1/4, ± 1/4, ± 1/12; .0') Xl-!·(±5/12, ±5/12, ±1/4; .0') Xl-!·(±5/12, ±5/12, ±1/12; ¢) Xl-l·(± 1/12, ± 1/12, ±5/12; .0') Xl-!·(±1/12, ±1/12, ±1/4; .0'). Each of these is irreducible, not Kummer induced (the exponents are not stable by o. in C x, X:>.(O, -1/12,5/12, 1/12,7/12, 0; 9J). So it suffices to show that X:>.(O, -1112, 5/12, 1/12, 7/12, 0; 9J) has its

Chapter4

160

Ggal of dimension > 16. This is a hypergeometric of type (6,0) which is (automatically) irreducible and which is not Kummer induced (its exponents are not stable by ex; ~--+ ex; + 112 or by ex; ~--+ ex; + 1/3). It is symplectic (by the Duality Recognition Theorem 3.4) So its Ggal is Sp(6), which has dimension 21 > 16.

QED

Corollary 4.6 .17 .1 Suppose that x+y+2 = 0 mod Z, and that none of x, y, 2 is = 0 or ± 1/3 mod Z. Let J.l E: !C x, and let A = {x, y, 2, -x/2, -y/2, -2/2, 1/2 - x/2, 1/2 - y/2, 1/2 - 2/2}, B = {0, 1/3, 213},

Then Ggal(XJ.l(A, B))= f1X{1, a}, and for some>.. E: ex, there exists an isomorphism [2]*(XJ.l(cx;(s;

~j"s)) ~

X>..(x, y, 2; 52l')®T*X>..(x, y, 2; !2l').

proof This results formally from the preceding two lemmas.

QED

This corollary establishes the truth of 4.6.10. Thus we obtain

SL(3)xSL(3) Theorem 4.6.18 Hypergeometrics V of type (9,3) with G := Ggal = fIX {1, a} are precise! y those (isomorphic to one) of the form XJ.l(x, y, z, -x/2, -y/2, -2/2, 1/2 - x/2, 1/2 - y/2, 1/2 - z/2; 0, ± 1/3) for some J.lE:Cx, and for some x, y, z in !C which satisfy x+y+z = 0 mod l, and none of x, y, 2 is= 0 or ±1/3 mod Z. Hypergeometrics V of type (9,3) with GO,der

=r

are precisely the xS twists of these.

CHAPTER 5 Convolution of 1J-modules 5.1 Convolution of .U-modules; Generalities (5.1.1) Given a smooth C-scheme X/C, we denote by .DMOD(X) the abelian category of all sheaves of left .Dx-modules on X, by D(X; .D) its derived category, and by Db,holo(X) the full subcategory of D(X; J)) consisting of those objects K such that Xi(K) is holonomic for all i and such that Xi(K) vanishes for all but finitely many i. For morphisms f: X --+ Y between smooth separated C-schemes of finite type, one knows (cf. [Berl, [Borl, [Ka-Laul, [Me-SOJ) that these Db,holo support the full Grothendieck formalism of the "'six operations"'. Of these, we will need only L... and fl, both of which have fairly concrete descriptions. (The operations fJ and f* are defined as the duals of these, and are consequently less amenable to direct inspection.) (5.1.2) We will need f* primarily when f: X --+Y is smooth of relative dimension d; in this case one has f*K

= Rf*(K®c-x!J"x;y)[dl,

so

except for the dimension shift we are "'just"' talking about relative De Rham cohomlogy: xi-d(f*K)

= HiDR(X/Y,

K), with its Gauss-Manin connection.

The deep fact here is that for K a single holonomic left .Dx-module, each of the relative DeRham cohomology sheaves HiDR(X/Y, K), with its Gauss-Manin connection, is holonomic on Y. The other case off* we will need is when f: X --+ Y is the inclusion of a C-valued pointy Then f*C"x is the delta module Sy(5.1.3)

E:

Y(C).

For a general f :X --+ Y, and m on Y, fl:Jn. is defined as L

f 1m

= .Dx--+Y

® cllJy f- 1 m!dimX-

dimYl,

where .DX--+Y is the (J)X, c1J)y)-bimodule .Dx--+Y := DiffOps .DMOD(Xx C Y) (JTl, n) H Jflxn, which passes to nb,holo. If K and L are objects of .DMOD(X), we define their "exotic" tensor product, denoted K® !L, in terms of the diagonal map/::,: X

-->

XxcX, by

K® !L := /::, l(Kx L). (5.1.7) If G/C is a smooth separated C-groupscheme, we denote by the group law by productG : GxcG --> G. We define the convolution of objects of Db,holo(G) by (5.1.7 .1) (K, L) H K+L := (productG)*(KxL). The operation of convolution is associative, and the S-module Se supported at the identity of G is a two-sided identity object. [For if we

163

Convolution of SD-modules denote by 1..: e -+ G the inclusion of the identity, then for any Kin Db,holo(G), we have K = C}exK on excG ~ G, (1..xidG)*(C}exK)

= (1..*C}e)xK = SexK on GxcG.

Since the composite map (productG)o( 1.. x idG) is idG, the result follows.] If G is commutative, then convolution is commutative as well. (5 .1.8) In general, even if we start with two holonomic .:0-modules :m and n on G, viewed as objects of Db,holo(G) which are concentrated in degree zero, their convolution :m*n is "really" an object of Db,holo(G), and not simply a single holonomic .:0-module placed in degree zero. It is this "instability" of .:0-modules themselves under convolution that makes Db,holo the natural setting. (5 .1.9) The following formal properties of convolution are quite useful. ( 1a) If 'P: G-+ H is a homomorphism of smooth separated Cgroupschemes of finite type, then (jl*(K*L) ~ ((jl*K)*('P*L). This results from the fact that ((jlX(jl)*(KxL) any (jl) and the fact that productw((jlx(jl)

= ((jl*K)x((jl*L)

= (jl

0

(valid for

productG ((jl being a

homomorphism). In the special case when His the trivial group, this becomes: Denote by 'IT: G -+Spec(C) the structural map. Then for any two objects K, L in Db,holo(G), we have 'IT*(K*L) ~ ('IT*K)®c('IT*L). (1b) If 'P : G -+ G is a homomorphism, then for any two objects K, Lin Db,holo(G), we have 'P!(('P*K)*L) ~ K*('P!L). This is base change for the following commutative diagram, whose outer square is cartesian (verification left to the reader): GxG

T

GxG idx (jl

prodG I

l G

l

(j)Xid

GxG (jl

l G.

prodG

Chapter 5

164 (1c) If tp : G

--+

H is a homomorphism, then forK in nb,holo(G) and L in

Db,holo(H), we have ..)

H

x>...

So with these notations, we can rewite the above FT formula as j * FT(j * inv * Jn)

= ( prz) * ((j * inv * Jn)(x)eXY).

Using smooth base change and the projection formula, we find (prz)*((j*inv*Jn)(x)eXY)

=

= ( prz) * ( pq +(j * inv * Jn) .® (product)+ .t)

Chapter 5

166

= ( prz) * J* ((pq +inv * :ffi) ® -module :m on Gm, we have inv *j * FT(j * :ffi) ~ :m* (inv *j * l:).

proof The Key Lemma applied to inv*:m gives j*FT(j*:ffi) Because inv: Gm

--+

= (inv*:ffih(j*l:).

Gm is a group homomorphism, we get

inv*j*FT(j*:ffi)

= (inv*inv*:ffi)*(inv*j* l:) = :ffi*(inv*j* 1:).

QED

5.3 Convolution of Hypergeometrics on Gm We begin by explaining the heuristic motivation. Let P, Q, R, and S be four nonzero polynomials in !C[t]. Recall the hypergeometric differential operators Hyp(P, Q) := P(xd/dx) - xQ(xd/dx), Hyp(R, S) := R(xd/dx) - xS(xd/dx), and the associated 1>-modules on Gm X(P, Q) := 1>/J:lHyp(P, Q), X(R, S) := 1>1 J:lHyp(R, S). A formal series f(x) := Le.nxn is killed by Hyp(P, Q) if and only if its coefficients an satisfy the two-term recurrence relation P(n)an = Q(n-1)an-1· Similarly, a formal series g(x) := Lbnxn is killed by Hyp(R, S) if and only if its coefficients bn satisfy the two-term recurrence relation R(n)bn

= S(n-1)bn-1·

Thus if f(x) := Le.nxn and g(x) := Lbnxn are formal series solutions of Hyp(P, Q) and of Hyp(R, S) respectively, then their ""convolution"" (hg)(x) := Le.nbnxn is visibly a formal solution of Hyp(PR, QS). This suggests that, at least under reasonable hypotheses, one should have X(P, Q)*X(R, S) ~ X(PR, QS) as 1>-modules on Gm.

Convolution Theorem 5.3 .1 Suppose that P, Q, R, and S are four

167

Convolution of 1:>-modules

nonzero polynomials in C[tl, such that the two polynomials PR and QS have no common zeroes mod Z, i.e., whenever (PR)(o:.) = 0 = (QS)(I)), ..!J.(oc.'s, '{s; r.·s, &'s)

provided that x:>..j.J.(oc.'s, 'fs; r.·s, &'s) is irreducible, i.e., provided that no element of the set {oc.'s, '{s} is congruent mod Z to any element of the set {r>'s, &'s}.

Corollary 5.3.2.1 All irreducihle hypergeometrics on IBm can be built out of &1 and X1CD; ¢),using only the following operations on holonomic .D-modules on IBm: (1) convolution H (T 0 .)*:Jn

(2) :Jn (3)

(4)

:m

:m

~--> :Jn®x& ~-->

inv*:Jn.

Specializing the Key Lemma 5.2 .3 and its Corollary 5 .2 .3 .1 to the case of hypergeometrics, we find Proposition 5.3.3 Let X := X:>..(oc.1, ... , «n; r>1, ... , r>m), with no oc.i- r>j in Z. If no r> j lies in Z, then j*FT(j*inv*X) ~ X:>..(O, «1, ... , «n; r>1, ... , r>m). If no «i lies in Z, then

Convolution of ;t'l-modules

169

Corollary 5 .3 .3 .1 All irreducible hypergeometrics can be built out of the delta module S1 on 43m using only the the following operations on holonomic .D-modules on 43m: (1) m ~--+ j*FT(j*inv*:Jn) m (3) m

1-+

inv *j * FT(j * :Jn)

1-+

cr*m

(4) m

1-+

m®xs

(5) m

1-+

inv*m.

5.4 Motivic Interpretation of Hypergeometrics of type (n,n) In our earlier discussion of the determination of (Ggal)D,der for irreducible hypergeometrics of type (n,n), X:>.( 2 he integers. For any integer cc. we have

j 1.. ([ql*X:>.((cc. -1)/q, (cc. -2)/q, ... , (cc. -n)/q: ~ 1 , ~ 2 )):::: .b/.bL for L the operator on A 1 L := an -

(qn-2/:>.)xq-n(D + « - n- q~1)(D + « - n- q~2)·

Special Case (8, 2), q = 2r+1, r

l

4, cc. = r+5,

~1

= 0,

~2

=

1/Z

j! * ([q] *X>. ((r+ 4)/(2r+ 1), (r+3)/(2r+ 1), ... , (r-3)/(2r+ 1): 0, 1/2)) :::: .b/ .bL for L the operator on A 1 L := aS -

((2r+1)6/:>.)x2r-7(D + r -3)(0 -7/2).

This X has Ggal = Spin(7) in SO(S), by 4.4.1. Writings := r-3, we find

Corollary 6 .1.S For any integer s

1 1, and any nonzero constant f..l, L := as - f..lX2s-1(o + s)(D -7/2) has Ggal = Spin(7) inside SO(S).

6.2 Fourier Transforms of Kummer Pulllbacks of Hypergeometrics: A Remarkable Stability The following result shows that we can obtain (a Kummer pullback of) any (sufficiently general) hypergeometric of type (n, m)

Chapter 6

182

with n > m as the Fourier Transform of (a Kummer pullback of) a hypergeometric of type (n, n). We will see later the importance of this sort of ""reduction to the RS case"".

Theorem 6.2 .1 Suppose that n > m 1 0 are integers. Put d Suppose we are given a hypergeometric of type (n, m), X := XA(..(o:(s;

tlj's)*MX1('6, '6).

By definition, MX 1 ('6, ¥) is the convolution X 1 ('6; .0') *X 1 ( .0'; '6), which by 5.3.5 sits in a short exact sequence of .b-modules on 111m 0-+ s1 -+ X 1 ('6; .0')*X1(.0'; '6)-+ x'd'C)'-+ 0. Convolving this exact sequence with MX:>..(o:i's; tlj's) yields, via 5.3.4.1, the required short exact sequence. QED

Chapter 6

188

(6 .3 .10) In order to formulate the next result, it will be convenient to introduce the operator Cancel on both hypergeometrics and on modified hypergeometrics which "cancels" the exponents mod l common to numerator and denominator. Given X := X:>..(oc;1, ... '..(..( d, then 1, din, and an isomorphism

'f I IBm

~ [x ~--+ ~x]*('f

I IBm)·

Since 'f I IBm is irreducible, this isomorphism allows us to descend 'f I IBm through the d-fold Kummer covering, whence k := Swanco('f) + Swano('f) is divisible by d. But din, and gcd(n,k) = 1. Therefore d= 1. QED Class (3) The subclass of (2) consisting of Fourier sheaves which are lisse on IBm and which have all co- breaks < 1. Indeed for Fourier sheaves, the condition "all co- breaks 5. 1" is stable by Fourier transform. 'f # 9- in this class, then 'f has co- break a/(a+ b) > 0 with multiplicity a+ b # # 9- has 0-break alb> 0 with multiplicity b.

If

225

The t'-adic Theory On breaks themselves, the rule is oo-break x > 0 for 'f # 0-break x/(1-x) > 0 for~0-break y > 0 for 'f # oo-break y/(1+y) > 0 for 'f. This rule is order- preserving., so we can read biggest 0- breaks in terms of biggest oo- breaks. In addition to dim'f 0 + dim~o = Swan 0 ('f) + Swan 00 0. We say that a constructible ii5rsheaf 'f' on

J>.l

is a

tame pseudoreflection sheaf if it satisfies the following three conditions:

(TPR1) 'f' is everywhere tame, i.e., for every t

E: 11'1, 'f'(t) is a tame representation of I(t). (TPRZ) for some (or equivalently for every) nonempty open set

j: U

--+

A 1 on which j * 'f' is lisse, we have 'f' ::::: j *j * 'f'.

(TPR3) For every s in A 1(k) at which 'f' is not lisse, 'f' (s)/'f' s is onedimensional (i.e., local monodromy at sis by tame pseudoreflections). (7.9.2) We say that a tame pseudoreflection sheaf 'f' is irreducible if in addition it satisfies (lrrTPR) for some (or equivalently for every) nonempty open set

j: U

--+

A 1 on which j * 'f' is lisse, j * 'f' is irreducible (as rep'n of

(7 .9.3)

1T

1).

Any nonconstant irreducible tame pseudoreflection sheaf is

an irreducible Fourier sheaf, and its set S of points of nonlissity in A 1 is a finite nonempty subset of A1(k).

Theorem 7.9 .4 Let 'f' be a nonconstan t irreducible tame pseudoreflection sheaf on A 1, with set S

= {s1,

... , sr} of points of

nonlissity in A 1(k) = k. Then (1)~ := NFTijl('f') is an irreducible Fourier sheaf which is lisse on 6m of rank r := Card(S). ( 2) The restriction of to

E9 i= l, ... ,r

~ ( oo)

to the wild inertia group P( oo) is isomorphic

J:.ljl(siy) ·

(3) ~( oo) is not Kummer induced. ( 4) ~ is not Kummer induced. (5) If p > 2r + 1, ~ is Lie-irreducible.

proof Since 'f' is irreducible Fourier, so is ~. Since 'f' ( oo) is tame, ~ is lisse on 6m, and ~(oo) ::::: E9i=l, ..,r FTijlloc(si, oo)('f'(si)/'f' s / By hypothesis each 'f'(si)/'f' s· is of the form J:."' ·(x-s·)· so 1

~(oo):::::

"'1

E9i=1,.,r

1

.r.ljl(siy)®Lxi(y)·

This proves (1) and (2). Assertion (3) trivially implies (4), and (1) and (4) together imply (5). Assertion (3) holds because the distinct J:.ljl(siy)

227

The € -adic Theory each occur with multiplicity one, thanks to the following lemma.

Lemma 7.9 .5 Let M be an

~-adic I( co )-representation whose restriction to P( oo) is a direct sum of r distinct characters .l:q.(siy) with

multiplicities ni. If M is Kummer induced of degree m, then m divides every ni. In particular, if gcd(all ni) = 1, M is not Kummer induced.

proof As I(oo)-representation, M is

EBi=i ....,r

.l:q.(siy)®(tame of dim. ni). If

M is [m)*X for some I(oo)-representation X, then (cf. the proof of 7 .8 .2 .3) X is a direct factor of [m) * M, so of the form X ~ EBi=i,.,r .l:q.(siym)®(tame of dim. di i ni), whence [m)*X is EBi=i, ..,r .l:q.(siy)®(tame of dim. mdi)·

QED

This lemma proves (3), and so concludes the proof of the theorem. QED

Theorem 7.9 .6 Let ':f be a nonconstant irreducible tame pseudoreflection sheaf on A1, with setS = {s1, ... , sr} of points of nonlissity in A 1(k) = k. Supppose that ( 1) among the numbers si E: k, there are no relations of the form Si - Sj = Sk - Sn except for the trivial ones (i=j and k=n) or (i=k and j=n). (2) p > 2r + 1. Then G := Ggeom for '3- := NFTq.(':f) has GO,der = SL(r).

proof Since p > 2r

+ 1, we know that 2r + 1. Then G := Ggeom for ~ := NFTljJ('f) has GO,der = SL(r) or Sp(r) or SO(r).

proof This time the hypothesis (1) of nonrelations and the torus trick 1.0 forces Lie(GO,der) to contain the full maximal torus of Sp(r), and we apply 1.2. QED

7.10 Examples We continue to work over an algebraically closed field k of characteristic p > 0. (7 .10 .1) (Lefschetz pencils) Start with a smooth connected projective variety X c lPN over k of dimension n 2 2, and consider a Lefschetz pencil of hyperplane sections Xn Ht of X, with associated fibration f:

X~

11'1 (i.e., r1(t)

= XnHt)·

If p ~ 2 or if n-1 is odd, the quotient

sheaf on 11'1 Evn-1 :=Rn-1f* iO~/(the constant sheaf Hn-1(X, iQ~)) is, when restricted to any A1 c 11'1, an irreducible tame pseudoreflection sheaf (cf [De-WI]). (7 .10 .2) (Supermorse functions) This example is a slight generalization of a Lefschetz pencil of relative dimension n -1 = 0. Let C be a complete smooth connected curve over k, fa nonconstant rational function on C, D the divisor of poles of f. View f as a finite flat morphism f:C-D~A1,

whose degree we denote deg(f). We suppose that the differential df is not identically zero (i.e., f is not a p'th power), and den~te by Z c C- D the scheme of zeroes of df on C- D. We putS := f(Z(k)) c A1(k). Then f makes C - D - Z a finite etale connected covering of A 1 - S. Consider the sheaf f * Ot on A 1. The trace morphism (for the finite flat morphism f) is a surjective map Tracef : f* iQ~ ~ iQ~, whose

229

The t'-adic Theory

restriction to the subsheaf iOt off* iOt is multiplication by deg(f). Thus 'f := Kernel of Tracef : f * iOt

-+

iOt

is a direct factor off* G:lt of generic rank deg(f) - 1.

Lemma 7.10 .2 .1 If deg(f) < p, 'f is a Fourier sheaf. proof Since 'f is a direct factor of a sheaf (namely f* iOt> which is the direct image of its restriction to A1 - S, 'f shares this property. Since 'f ( oo) is tame (because deg(f) < p), we have HO(A1, 'f®l: Ot, 2n-1 and if the condition *(p, n-1) holds (cf. 7.1), then Ggeom for ~(1/2)

I IGm is Sp(n-1).

7.14 Orthogonal Examples We now give examples where the Sato-Tate law is that given by the orthogonal group. We work over a finite field k of characteristic p ~ 2, and denote by X2 the character of order two of kx.

0-Example(1) Let h(x)

€ k(x) be an odd nonzero rational function which is holomorphic at oo and all of whose poles have order prime to

p. Let g(x) be a nonzero rational function. Let j : A 1 - S ---> A 1 be the inclusion of the open set of A 1 where h is holomorphic, and kA1- S- T---> A1- S the inclusion of the open set of A 1 - S where g is invertible. Let ')( be a multiplicative character of k x, of even order r, such that at any zero or pole of g(x) in T, the order of zero or pole of g there is not divisible by r. Suppose that there exists an odd rational function L(x) such that L(x)r = g(x)g(-x). Take ~( 1/2)

1 := j,.(l:q,(h)®k*l:X(g)),

~ := NFTq,(1). Then

is a lisse sheaf on IGm which is orthogonally self-dual (by the

argument in SL-Example(4) above) of rank N = Card(S( k)) + Card(T( k)) + ~geom poles of h in A 1 (order of pole of h) and pure of weight zero.The trace function of ~( 1/2) is t € E >-->

-(Card(E))-11 2

Lx in E -S(E)-T(E)

~E(tx

+ h(x))')(E(g(x)).

If p > 2N+1, and N ~ 2, then Ggeom for ~(1/2) is O(N). To see this,

note that Ggeom lies in O(N), so it must be either SO(N) or O(N), by the paucity of choice in 7.5.3.1. In fact Ggeom is O(N), because the local monodromy of~ around zero is, by 7.5.3.1, a reflection. (The odd rational function L(x) necessarily has odd oo -valuation, so from the equation g(x)g(-x)

'f

= L(x)r

we infer that ord 00 (g)

= (r/2)xodd,

so

: 2N+1, p does not divide 2nN1(1n-di)N2(1n-dl), and either N f/ {7,8} or In- dl ~ 6, then Ggeom for ~(1/2) is either SO(N) or O(N), by the paucity of choice in 7.7 .6. If in addition n > d, then

i

is lisse on A 1 , whence Ggeom

has no nontrivial prime-to-p quotients, so Ggeom is SO(N). Then

det(~(l/2))

is a geometrically trivial character of order one or

two, so it is either trivial or it is (the pullback to A 1 of) "( -1)deg ", the unique character of order two of Gal(kSeP/k). The question of which one is arithmetic, and in a given example can be decided by computing the determinant of Frobenius on ~( 1/Z)o and seeing whether it is 1 or -1. If we compute this sign ± 1 and choose an N'th root e of it, then we can directly apply Deligne's general result to the slightly twisted sheaf (e)deg®~(i/2) with Ggeom = SO(N). [Another course of action: simply

Chapter 7

248

replace the given ground field k by its quadratic extension kz, and directly apply Deligne's general result to n, then Ggeom is O(N). To show that Ggeom is O(N), we make a series of reductions to a case where it is obvious. First of all, since Ggeom is either SO(N) or O(N), and "simple".

(8 .1.6)

Consider the special case when X/k is a smooth,

€-adic Hypergeometrics

253

geometrically connected curve. Then an object K of Dbc(X, Ot) is perverse if and only if XiK = 0 fori ;~: -1, 0, x-1K has no nonzero punctual sections, XOK is punctual. We call a perverse object K '"nonpunctual" if ftOK = 0. If 'f is a lisse sheaf on an open nonempty open set j: U -+ X, then the middle extension j! * ('f[ 1]) is none other than (j *'f)[ 11. It is for this reason that we adapted the terminology '"middle extension'" for sheaves of the type j* 'f with 'f lisse on U. The dual D(j!*('f[1))) of such a middle extension is related to the naive dual D(j* 'f) := j*('fv) defined in 7 .3.1 by D(j, .. ('f[1))) = j, .. -+ >..x]*Hyp(*, ..(!, ;j;; x's; p's)(n+m),

D(Hyp>..(!, ..(!, ..(*, ..(!, ..(*, ..(*, ..(-+ exp(21Ti-+

exp(21Ti~x))

of Z

~

1T1((1Gm)an).

Having all the ')(.'s and p's of finite order is analogous to having all the . ( !, .(!, .(!, ..(!, ljl; x·s; p's) :::: j*K:>..(!, ljl; x·s; p's) :::: Rj*X'A(!, ljl; x·s; p's), whence j!X:>..(!, ljl; x·s; p's) is an irreducible Fourier sheaf on A1. (4) If Hyp:>..(!, ljl; x·s; p's) is defined over a finite subfield ko of k, then X:>.(!, ljl; x's; p's) is pure of weight n + m - 1. (5) The natural map Hyp:>..(!, ljl; x's; p's)--+ Hyp:>..(*, ljl; x's; p's) is an isomorphism. (6) If n > m, the sheaf X:>.(!, ljl; x's; p's) is lisse of rank non l!lm. As !(D)representation it is tame, isomorphic to EB distinct x·s l: X 181 Unip(multo( x)). As I(oo )-representation it has Swan conductor the direct sum

=1,

and is isomorphic to

ffi

(dim. n-m, brk. 1/(n-m)) EEl distinct p"s l:p®Unip(mult 00 (p)). (7) If n < m, the sheaf X:>..(!, ljl; x's; p's) is lisse of rank m on l!lm. As !(D)-representation it has Swan conductor direct sum (dim. m-n, brk. 1/(m-n))

ffi

=1,

EEl distinct

and is isomorphic to the

x·s

l:X ®Unip(multo(x)).

As I( oo )-representation it is tame, isomorphic to EEl distinct p"s J:p®Unip(multoo(p)). (8) If n = m, the sheaf X:>. ( !, l\1; X 's; p 's) is lisse of rank n on l!lm - {:>..}, from which it is extended by direct image. I(>..) acts by tame pseudoreflections of determinant l: 1\(x-:>..)• for 1\ .- TT iP/TT i Xi· As !(D)-representation it is tame, isomorphic to EEl distinct x·s l:x ®Unip(multo(x)). As I( oo )-representation it is tame, isomorphic to EEl distinct p's l:p®Unip(multoo(p)). proof We proceed by induction on n+m. If n+m = 1, the theorem is obvious by inspection. Suppose the theorem has already been proven universally for all (no, mo) with no + mo < n+m; we must prove it universally for (n, m). Notice that assertions (2) and (3) follow from assertions (1), (6), (7) and (8). We are thus "reduced" to proving assertions (1) and (4) - (8) ..By

Chapter 8

264

multiplicative translation, we may assume )>. = 1. By multiplicative inversion, we may suppose that n i m. For any tame character /\, (1) and (5) - (8) hold for Hyp1 (!, ljl; x's; p's) if and only if they hold for J:/\®Hyp1(!, ljl; x's; p's)::::: Hyp1(!, ljl; Ax's; 1\p's). For (4), we have this same equivalence for any tame character 1\ of finite order. Suppose first that 0 < n i m. Then by picking 1\ to be the inverse of one of the x's we may assume that one of the x·s is trivial, say 'X.n = 11.. To emphasize this, we will write our object of type (n, m) in the form Hyp1 (!, ljl; 11., ')(.'s; p's), where now there are n-1 listed x's in addition to 11. Since {11, the x's} and the p's are disjoint by hypothesis, none of the p's is trivial. Now apply the general formula relating convolution with Fourier Transform (j*J:lji)[1]*!K::::: j*FTljl(j!inv*K) to the hypergeometric object K of type (n-1, m) K := Hyp1(!, ljl; X's; p's). We find Hyp1(!, ljl; 11., x's; p's) ::::: j*FTljl(j!inv*K). ::::: j*FTljl(j!inv*Hyp1(!, ljl; X's; p's)) ::::: j*FT~(j!HYP1(!, ~; p's; x's)). Since none of the p's is trivial, we have by induction that j!HYP1(!, ~; p's; x's)

= j!X1(!,

~; p's; X's)[1]

::::: j*X1(!, ~; p's; :X.'s)[1] is (an irreducible Fourier sheaf)[1]. Thus we find Hyp1(!, ljl; 11., x's; p's)[-1] ::::: ::::: j*FTlJI(j*Xl(!, ~; p's; i's))

= j*NFTljl(j*X1(!,

~; p's; X's)).

The key point is that NFTlJI(j*X1(!, ~; p's; x's)) is irreducible Fourier. This shows first that Hyp1(!, ljl; 11, X's; p's)[-1] is an irreducible middle extension sheaf on Gm, thus proving (1) and providing the required X1(!, ljl; 11, x's; p's). [Because the Euler characteristic of a convolution is the product of the Euler characteristics of the convolvees (cf 8.1.8), always X-+

s/x]*~) = 0 for all s, so the

perverse object '.f[1]*!~[1] is of the form X[1] for some sheaf X on IBm. Because X[1] is perverse, X has no punctual sections. QED Returning to Hyp:>..(!,

.p; ')('s; p's) and Hyp:>..(*, .p; ')('s; p's), we can

be much more precise about their structure.

Corollary 8.4.6.2 Suppose that the ')('s and p's are not identical. Then the perverse object Hyp:>..(!, .p; x's; p's) is nonpunctual, i.e., Hyp:>..(!, ..(!,

.p; ')('s; p's) on IBm with no nonzero punctual sections.

proof If n=O or m=O, this is proven in 8.4.2. By multiplicative translation, we may assume:>.. = 1. The Kloosterman expression

269

€-adic Hypergeometrics Hyp(!, lj>; ;(s; p's) := Kl(lj>, ')(s)[1]* !inv*Kl(~, p's)[1l, and 8.4 .6 .1 then give the existence of a sheaf X 1 (!, lj>; ")(s; p's) on 113m with no punctual sections such that Hyp(!, tj.>; ·x.,'s; p's) = X1(!, tj.>; ;(,'s; p's)[1].

QED

Corollary 8.4 .6 .3 Suppose that the ;..(!, tj.>; 11, ; Hyp1(!, Y.,; 11; 11) -'> &1(-1) -'> 0.

proof By the fundamental relation 8.1.12 between Fourier Transform and convolution, denoting by j: l!lm --" A1 the inclusion, we have (j*!.y_.)[1]*!K ~ j*FT.p(j!inv*K). Applying this to K := inv* j* !.~[1], we find

Chapter 8

270

Hypl(!, ..p; 11; 11) ~ j*FT..p(jjj*.l:;j;HD Now FT..p(.l:;j;"®K) = [x

1-+

= j*FT..p(.l:;j;"®j!Q~)[l].

x+l]*FT..p(K) for any K on Al, and it is proven

in [Ka-PES, Prop. AZ] (cf. 2.10.1(1) for the .U-module analogue) that FT..p(j!O~) = Rj* 0~. Thus we find j*FT..p(.l:;j;"®j!O~)[l] ~ j*[x ~---+ x+1]*Rj*Q~[1] = Rk*Q~[l].

QED

Corollary 8.4.8.1 of Lemma 8.4.8 For any ')(., the perverse object Hypl ( !, ..p; ')(.; ')(.) sits in a short exact sequence of perverse sheaves on IBm

o __. .l:xlll __. Hyp 1 (!, ..p; x; x> __. s1 (-1) __. o.

proof Simply tensor with J: X.

QED

Apply 8.4.8 to calculate HypA.(!, ..p; x's; p's)*!Hypl(!, ..p; 11.; 11). The convolution HypA.(!, ..p; x's; p's)*!Q~ is the constant sheaf with value V := HDc(IBm, HypA.(!, ..p; X's; p's)). The convolution Hyp),.(!, ..p; x's; p'sh!Sl(-1) is Hyp),.(!, ..p; x's; p's)(-1), so we have the asserted short exact sequence 0--> V[l]--> HypA.(!, ..p; 11, ')(.'s; 11, p's)--> HypA.(!, ..p; X's; p's)(-1)--> 0. QED We now develop some of the immediate consequences of the cancellation theorem. Cancellation Theorem bis 8.4.9 Given arbitrary x's and p's, and a tame character /\, denote by W the one-dimensional Orvector space W := HO(IBm, HypA.(!, ..p; 1\-lx·s; A-lp's)). In the categor~ Perv(IBm), HypA.(*, ..p; 1\, x's; /\, p's) sits in a short ex act sequence

proof This is the dual statement, with the dual proof.

QED

Semisimplification Theorem 8.4.10 Suppose that the x's and p's are disjoint. Let r 1 1, and let 1\ 1• ... , 1\r be r not necessarily distinct tame characters. In the category Perv(IBm) over the algebraically closed field

t'-adic Hypergeometrics

271

!JJ; t\1, ... , Ar, ')(.'s; t\1, ... , Ar, p's) !JJ; t\1, ... , Ar, ')(.'s; t\1, ... , Ar, p's) are each isomorphic

k, the semisimplifications of Hyp:>.(!, and of Hyp:>.(*,

to the direct sum HypA(!,

!JJ; x's; p's) EB

ffii=1, ... ' r

J:t\i[1].

proof This is obvious by the cancellation theorem and the fact (8 .4 .2) that if the ')(. 's and p's are disjoint, then Hyp:>.(!, !JJ; ')(.'s; p's)::::: Hyp:>.(*, !JJ; X's; p's) is simple.

QED

Corollary 8.4.1D.1 The perverse object Hyp:>.(!, lj;; ')(.'s; p's) is simple if and only if the x·s and p's are disjoint. Theorem 8.4.11 Suppose that the x's and p's are not identical. Write Hyp:>.(!, !JJ; ')(.'s; p's) = X:>.(!, lj;; ')(.'s; p's)[1]

!JJ; x's; p's) on Gm with no nonzero punctual sections. !JJ; ')(.'s; p's) is the following: (1) If n > m, the sheaf X:>.(!, !JJ; x's; p's) is lisse of rank non Gm· As !(D)for a sheaf X:>.(!,

The local monodromy of X:>.(!,

representation it is tame, isomorphic to ffi distinct ;.(!,

ffi

ffidistinct p's

J:p®Unip(mult 00 (p)).

!JJ; ')(.'s; p's) is lisse of rank m on Gm· As

!(D)-representation it has Swan conductor = 1, and is isomorphic to the direct sum (dim. m-n, brk. 1/(m-n))

ffi

ffidistinct x·s

J:'X. ®Unip(multD(')(.)).

As I( oo )-representation it is tame, isomorphic to ffidistinct p's J:p®Unip(multoo(p)). (3) If n = m, the sheaf X:>.(!, !JJ; ')(.'s; p's) is lisse of rank non Gm- {:>.}, from which it is extended by direct image. I(:>.) acts by tame pseudoreflections of determinant J:t\(x-:>.)• fort\ := TT i'X./TT iPi· As !(D)-representation it is tame, isomorphic to ffidistinct x·s J:'X. ®Unip(multo(x)). As I( oo )-representation it is tame, isomorphic to ffidistinct p's l:p®Unip(mult 00 (p)).

Chapter 8

272

proof Suppose first n ~ m. Then it follows immediately from the cancellation theorem 8.4.7 (and 8.4.2 in the case when the ")(.'sand p's are disjoint) that XA (!, lj.l; ")(.'s; p's) is lisse of rank max(n, m) on IBm, and that the description claimed for its local monodromy at zero and oo is correct up to semisimplification. Similarly, if n= m, we see that the sheaf XA(!, lj.l; ")(s; p's) is lisse of rank non IBm- {A}, and that the description claimed for its local monodromy at zero and oo is correct up to semisimplification. To see that the description claimed for its local monodromy at zero and oo is absolutely correct, we must see that (universally, so after any!_/\ twist) the local monodromy at zero or at oo has at most a single unipotent Jordan block. This is a consequence of the fact that H1c j*'f->

yi(O)®so

EfJ

'fi(oo)®soo -> 0,

gives an injective map ('fi

H1c(1Bm, 'J').

Since HO(IBm, 'J') injects into either 'J'I(O) or 'J'I( 00 ), it follows that 'J'I(O) and

-rH 00 ) each have dimension at most one. If n

sheaf

1'

= m,

it remains to examine the local monodromy at

A

of the

:= XA (!, lj.l; x's; p's). Denote by k :IBm- {A} -> IBm the inclusion.

By the Cancellation Theorem 8.4.7 and 8.4.2,

1' is lisse on IBm-

{A} of

rank n, and it has a one-dimensional drop at A. Since 1' has no nonzero punctual sections, either 1' has pseudoreflection local monodromy at A and

1' :::: k*k*'J', or k*k*'J' is lisse on IBm. In this latter case, k*k*'J'

1' is tame at both zero and oo. But comparing the local monodromies of 1' at zero and oo, we see that 1' cannot be a successive extension of !_A.'s. Therefore 1' has pseudoreflection local monodromy at A, and 1':::: k*k*'J'. Because ")(.(IBm, 'J') = -1, 1' must be tame at A. The must be a successive extension of !_A.'s, becaue we already know that

determinant of the tame local monodromy at it is at 0 and oo, as indicated. QED

A

is determined by what

C-adic Hypergeometrics

273

Corollary 8.4 .11.1 Suppose that the x's and p's are not identical. Then there exists a middle extension sheaf X:>. ( *, ljl; ')(.'s; p's) on IBm such that HypA ( *' .(!, lJJ; x·s; p's).

proof The existence is given by the theorem. The x·s (resp. the p's) with their multiplicities are the precisely the tame characters which occur in the 1(0)-semisimplification of 'f as !(D)-representation (resp. in the I( oo )-semisimplification of 'f as I( oo )-representation). The uniqueness of:>.. results from the the following general lemma.

278

Chapter 8

Translation Lemma 8.5.4 (compare 3.7.7 and [Ka-GKM, 4.1.5)) Let 'f be an irreducible middle extension sheaf on 6m whose Euler characteristic ")(.(6m, 'f) is nonzero. Suppose that for some

A E:

kx there

exists an isomorphism 'f ~ [x 1-+ AX)* 'f. Then A is a root of unity of order dividing ")(.(6m, 'f). In particular, if ")(.(6m, 'f) = -1, then A = 1, i.e., 'f is isomorphic to no nontrivial multiplicative translate of itself. proof We first show that A must be a root of unity. If 'f is not lisse on 6m, then its finite set S of points of nonlissenesss on 6m is stable by s 1-+ AS, hence A is a root of unity of ord~r dividing Card(S). If 'f is lisse on 6m, but A is not a root of unity, then by Verdier's lemma [Ver, Prop. 1.1) 'f is tame at both zero and oo, whence ")(.(6m, 'f) = 0, contradiction. Once A is a root of unity, say of order N, then because 'f is irreducible it descends through theN-fold Kummer covering, and hence ")(.(6m, 'f) is divisible by N. QED Rigidity Corollary 8.5.5 Let 'f be an irreducible middle extension sheaf on 6m whose Euler characteristic ")(.(6m, 'f) = -1. Then the isomorphism class of 'f is determined up to (a unique) multiplicative translation by the isomorphism classes of the 1(0) and I( oo )semisimplifications of the tame parts of the local monodromy of 'f at zero and oo. Rigidity Corollary bis 8.5 .5 Let 'f be an irreducible middle extension sheaf on 6m whose Euler characteristic ")(.(6m, 'f) = -1. (1) If 'f is not lisse on 6m, the isomorphism class of 'f is determined by the following three data: the I(O)-semisimplification of 'f, the I( oo )-semisimplification of 'f, the unique point A E: k x where 'f is not lisse. (2) If 'f is lisse on 6m, the isomorphism class of 'f is determined by the following two data: the I(O)-semisimplification of 'f, the I( oo )-semisimplification of 'f. proof The first assertion is an immediate consequence of the first

279

f-adic Hypergeometrics

Rigidity Corollary, since fixing the unique point of nonlisseness in 4lm rigidifies the situation entirely. The second assertion is a bit more delicate. We know that 'f is a hypergeometric X:A(!, \jl; ')(.'s; p's) of type (n, m) with n ~ m. By inversion, we may suppose that n > m. Then 1' as I( oo )-representation is of the form 1' :;:; T E9 W = (tame of rank m) E9 (rank n-m, all breaks 1/(n-m)). Since W is the unique wild irreducible I( oo )-constituent of 1', it is an intrinsic invariant of the the I( oo )-semisimplification of 'f. So it suffices to show that W .. detects .. multiplicative translations. This is proven in [Ka-GKM, 4.1.6, (3)]. QED 8.6 Local Rigidity (8.6.1) We continue to work over an algebraically closed field of characteristic p ~ ~. with ~-adic representations of I( oo). We first note the following variant of Grothendieck"s local monodromy theorem [Se-Ta, Appendix]. Theorem 8.6 .2 Suppose that (W, p) is an irreducible I( oo )representation. Then an open subgroup of I( oo) acts as scalars. If detW is of finite order, then p (I ( oo)) is finite. proof Clearly the first statement implies the second. To prove the first, denote by t~ : I(oo) -+ Z~(1) the canonical projection defined by the ~-power Kummer coverings. Recopying the beginning of the proof of the local monodrom y theorem, one shows that there exists an endomorphism N E: EndiQ~ (W)( -1) such that on a sufficiently small open subgroup p('6) = exp(t~(¥)N)

r

of I(oo), we have

for every '6 in f. This N is unique, and by unicity it is I( oo )-equivariant. By irreducibility, N is scalar. QED

Local Rigidity Theorem 8.6.3 Let V and W be I(oo)-representations, each of the same rank d 2 1 with all breaks = 1/d. Then (1) If :A E: kx and W :;:; [x ,..... :>..x]*W, then :A (2) If d

1

= 1.

2, and if there exists :x E: kx with V :;:; [x

~-->

:>..x]*W, then

280

Chapter 8

detV ::: detW. (3) If detV ::: detW, there exists a unique)..

E:

kx with V ::: [x

H

)..x]*W.

proof Assertion (1), as noted above, is proven in [Ka-GKM, 4.1.6, (3)]. If d ~ 2, then detW is tame, necessarily some I.x, so its isomorphism class is invariant by multiplicative translation, whence (2). Assertion (3) is trivial ford= 1, and in general the unicity in it results from (1). The existence for d ~ 2 is more delicate. Consider the canonical extensions (cf. [Ka-LG, 1.5]) of V and W to IBm. Both of them are necessarily hypergeometrics of type (d, D) (by the intrinsic characterization of hypergeometrics), say Vcan =X),.(!, ljl; x's; Rf), Wean= Xll(!, ljl; ~·s; Rf). Looking at determinants, we see that TTxi = TT~i· Replacing W by a multiplicative translate of itself replaces W can by the corresponding translate, so we may further assume that ).. = ll· By a further translation, we may suppose that ).. = ll = 1. The problem now is to show that for fixed ljl, the isomorphism class of the I(oo)representation of the Kloosterman sheaf Kl(ljl; x's) := X 1 (!, ljl, x·s; Rf) depends only on TT Xi· This is a special case of the following Change of Characters Theorem 8.6 .4 Let X).. ( !, ljJ; X' s; p' s) be a hypergeometric sheaf of type (n, m) with n ;t: m. If n > m (resp. if n < m), denote by W),.(!, ljl; X's; p's) the wild part of its I(oo)representation (resp. of its !(D)-representation). For fixed().., ljl), the isomorphism class of W)..(!, ljl; X's; p's) as I(oo)-representation (resp. as !(D)-representation) depends only on the tame character TTix/TTjPj and on the integer n-m. proof The proof proceeds by induction on the quantity In - mi. The statement is invariant under multiplicative translation, so we may assume that ).. = 1. The statement is invariant under multiplicative inversion, so we may suppose that n > m. The statement is also invariant under ®I./\. Given two W),.(!, ljl; x's; p's) and W)..(!, ljl; ,

we have Hypx(!, ..(!,

proof This is the Lemma with IJ. = -1.

we have ljJ; x·s; p's).

QED

Corollary 8 .7 .4 If n = m, then over an algebraically closed field k of characteristic p ~ t the isomorphism class of Hyp:>.. ( !, ljJ; ')(. 's; p 's) is independent of ljJ. Over a finite field

ko

of characteristic p

~

t, for any

hypergeometric Hyp:>..(!, ljJ; ')(.'s; p's) of type (n, n) which is defined over

ka,

and for any ll in

ko,

if we define

o.(!, ljJ; x's; p's) is geometrically self-dual (i.e., geometrically isomorphic to its dual), if and only if the following three conditions hold: (1) the set of x·s with multiplicity is stable under ')(. H (2) the set of p's with multiplicity is stable under p H p, (3) the product p(n - m) is even.

X,

proof By (8.4.2, 8.3.3), the dual of X:>.(!, ljJ; x's; p's) is X:>.(!,

;j;; x's; p's),

so the conditons are obviously sufficient. In order for X:>.(!, ljJ; ')(.'s; p's) and X:>.(!,

;j;; x's; p's) to be geometrically isomorphic, their local

monodromies at zero and oo must agree, whence (1) and (2). If (1) and (2) hold, then the dual of X:>. ( !, ljJ; ')(.'s; p's) is its multiplicative translate by (-1)n-m, to which it is isomorphic if and only if (-1)n-m = 1 in the field k. QED

284

Chapter 8

Parity Recognition Theorem 8.8.2 Suppose that the ')(s and p's are disjoint, and that XA(!, ljl; x's; p's) is self dual. Then on the open set of IBm where XA(!, ljl; x's; p's) is lisse, the (unique up to a iQ~ X-multiple, by irreduciblity) autodaulity pairing is alternating if and only if max(n, m) is even, n- m is even, and TI;x/TijPj = 11. Otherwise (i.e., if max(n, m) is odd, or or if n - m is odd, or if TI;x/TijPj :t 11) the pairing is symmetric.

proof If the generic rank max(n, m) is odd, the pairing has no choice but to be symmetric. Suppose now that max(n, m) is even. By multiplicative inversion, we may suppose that n l m. By multiplicative translation, we may suppose A = 1. If n = m, then the character 1\ := TI;X/TijPj is of order dividing two, because !./\(x-1) is the determinant of the (pseudoreflection) local monodromy at 1. If 1\ = 11, local monodromy at A is a unipotent pseudoreflection; as O(n) contains no unipotent pseudoreflections (cf. the proof of 3.4), the autoduality must be alternating. If 1\ is nontrivial, then the pairing must be symmetric; it cannot be alternating since Sp(n) c SL(n). If n - m l 1, then as I( oo )-representation we have (in the notations of the proof of 8.6.4) X1(!, ljl; x·s; p's) ~ W1(!, ljl; x·s; p's) EEl (tame), with W1(!, ljl; x's; p's) totally wild of Swan conductor 1, and (hence) I(oo)-irreducible and Jordan-Holder disjoint from the "tame" factor. Therefore the global autoduality of X1(!, ljl; x's; p's) must induce an autoduality of W 1 (!, ljl; x's; p's) as irreducible I( oo )-representation. Of course this local autoduality has the same sign as the global one which induces it. By 8.6.4, W1(!, .(!, ljl; '){.'s;;1) cannot be self-dual (by 8.8.1), so

we may suppose that pd is even. By 8.6.4, W is isomorphic to the l(oo)-representation attached to X;..(!, ljl; 1\, d-1 11's). So it suffices to show that if d 1 2, pd is even, and 1\ has order 1 or 2 then XA.(!, ljl; 1\, d-11l's) is self-dual, and the autoduality is alternating if and only if d is even and 1\ is trivial. This is a special case of the 8.8 .1 and 8 .8 .2 QED

8.9 Kummer Induction Formulas and Recognition Criteria Kummer Induction Theorem 8.9 .1 Let d

1 1 be an integer which is prime top, and denote by [d) the d'th power endomorphism of Gm.

Over an algebraically closed field k of characteristic p ;o: ~. for any hypergeometric Hyp;..(!, ljl; '){.'s; p's) there exists an isomorphism [d) .. HypA.(!, ljld; '){.'s; p's) :::: :::: Hyp).d(!, ljl; all d'th roots of all'){.'s; all d'th roots of all p's).

proof By multiplicative translation, we reduce to the case A. = 1. Since [d] is a homomorphism from 6m to itself, we have the convolution relation

286

Chapter 8 [d)*(K*!L):::: ([d)*K)*!([d)*L).

So we are reduced to the case where Hyp1(!, IJ.Id; ")(.'s; p's) is of type (1, 0) or (0, 1). Since [d)* and inv* (= inv*) commute, the (0, 1) case results from the (1, 0) case. Since every tame character ")(. has a d'th root, say ")(.

= ~ d,

our

(1, 0) hypergeometric Ly..d®.r."X. may be rewritten !.y..d®[d)*!.~. Applying [d)*, and using the projection formula, we get ldl* :::: ([dJ*>®.r.~. So we are reduced to showing that, denoting by 1\ 1· /\2, ... , 1\d the d tame characters of order d, we have an isomorphism ldl* :::: Kl(IJ.I; /\1, /\2, ... , /\d)· This is proven in [Ka-GKM, 5.6.2).

QED

Corollary 8.9 .2 (Kummer Recognition) Let d

1 1 be an integer which is prime top, and denote by [d) the d'th power endomorphism of IGm. Over an algebraically closed field k of characteristic p ;t: t, an

irreducible hypergeometric Hyp>..(!, lj.l; x's; p's) of type (n, m) is Kummer induced of degree d, i.e., of the form [d)*K for some Kin obc(IGm, Ot), if and only if the following three conditions are satisfied: ( 1) (2) all (3) all

d divides both n and m, there exists a set of n/d tame characters oc;'s such that the ")(.'s are the d'th roots of all the oc;'s, there exists a set of m/d tame characters ~·s such that the p's are the d'th roots of all the ~·s.

Moreover, if these conditions hold, then for any 1-1 E: k with 1-1 d = >.., there exists an isomorphism Hyp>..(!, lj.l; ")(.'s; p's) :::: [d)*Hypl-l(!, IJ.Id; oc;'s; ~·s).

proof If any (not necessarily irreducible) Hyp>..(!, lj.l; x's; p's) of type (n, m) satisfies (1), (2), and (3), then by the Kummer Induction Theorem above, for any 1-1 E: k with l-Id = >..,there exists an isomorphism Hyp>..(!, lj.l; x's; p's) :::: [d)*Hypl-l(!, IJ.Id; oc;'s; ~'s). Conversely, suppose that an irreducible Hyp>.. (!, lj.l; x's; p's) is of the form [d)*K for some Kin ohc(IGm,

Ot>·

Then [d)*K is perverse

£-adic Hypergeometrics

287

simple with X(6m, [d)*K) = 1. Since [d) is finite, K must be perverse simple, and X(6m, K) = X(6m, [d]*K) = 1. Therefore K is itself irreducible hypergeometric, so of the form Hypf..l(!, .(!, n. Then Hyp>. (!, , (2) {p's} = {all a+b 'th roots of «r.}. Moreover, if (1) and (2) hold, then Hyp>.(!, .)*Hyp1(!, . of X>.(!, . (!, .(x))*K for some Kin obc(6m, 0~), then K is perverse simple, lisse on 6m- {1}, and X(6m, K)

= X(6m,

[Bela,b)*K)

= 1.

Therefore K is itself hypergeometric of type ( 1, 1) with singularity at 1, i.e., K is Hyp1(!, .(!, ~Jl; x's; p's) :::: [Bela,b,:)*Hyp1(!, ~Jl; .} or inv*X>.(!, ~Jl; x·s; p's) I IBm-{>.} is Belyi induced of type (a,b) for some partition of n = a + b as the sum of two strictly positive integers; (3) X>.(!, ~Jl; ')(.'s; p's) I IBm-{>.} is the tensor product L®'J' of a lisse rank one L with a lisse irreducible addition det(X>.(!,

~Jl;

1' of rank n whose Ggeom is finite.

If in

x's; p's) I IBm-{>.}) is of finite order, then

X>.(!, ~Jl; ')(.'s; p's) I IBm- {>.}itself has finite Ggeom in this case (3). proof This is a special case of 7 .2 .6, ( 1), ( 6), and ( 8).

QED

Lemma 8.10.3 (Geometric Determinant Formula) Over an algebraically closed field k of characteristic p :t: t, let X>.(!, ~Jl; x's; p's) be an arbitrary hypergeometric of type (n, n). If TT i 'Xi det(X>.(!, ~Jl; ')(.'s; p's) I IBm-{>.}) :::: LA.

= TT iPi = A,

then

E-adie Hypergeometrics

289

If TI (X.i ~ TI iPi· then det(X:>..(!, ljJ; i.))~ X:>.(!, ljJ; Tip. is a unipotent

pseudoreflection, so det(X>.(!, ljJ; ')(.'s; p's) I 6m- {>.))is unramified at>..

Since}{:>..(!, ljJ; ')(.'s; p's) is everywhere tame, its determinant must be an

L /\• and we can compute 1\ as the determinant of local monodromy at zero. In the second case, det(X>.(!, ljJ; 'X.'s; p's) I 6m- {:>..))is everywhere tame, lisse on 6m - {>.) of rank one with nontrivial local monodromy at :>...So det(X>.(!, ljJ; ')(s; p's) I 6m- {>.))must be X>.(!, ljJ; TiiXi; Tiipi)· QED

Corollary 8.10.4 Over an algebraically closed field k of characteristic p

~ ~.let}{).(!,

ljJ; x's; p's) be an arbitrary hypergeometric of type

(n, n). Then det(X:>..(!, ljJ; ')(.'s; p's) I 6m- {>.))is of finite order (resp. trivial) if and only if both of the tame characters TI iXi and TI iPi are of finite order (resp. trivial).

8.11 Calculation of Ggeom for irreducible hypergeometrics (8 .11.1) Throughout this section we work over an algebraically closed field k of characteristic p ~ ~.

Theorem 8.11.2 (compare 3.5.8) Let X := }{>.(!, ljJ; ')(.'s; p's) be an irreducible hypergeometric of type (n, n). Suppose that p > n i 1 and that, X I 6m - {:>..} is neither Kummer induced nor Belyi induced nor inverse Belyi induced. Let 1\ := Tiix/TiiPi- Denote by G the group G := Ggeom for X I 6m - {:>..). Then (1) The group G is reductive. If both TiiXi and TiiPi are of finite order,

then G0

= GO,der.

Otherwise, G0

= 6mGO,der.

(2) The group GO,der is either {1}, SL(n), SO(n), or (if n is even) Sp(n). (3) If 1\ does not have order dividing 2, GO,der = {1} or SL(n). (4) If 1\ has exact order 2, GO,der (5) if 1\ = 'D., GO,der

= SL(n)

= {1}

or SO(n) or SL(n).

or (if n is even) Sp(n).

290

Chapter 8

(6) If 1\ is not of finite order, G = GL(n).

proof Local monodromy around >. is a pseudoreflection of determinant .r./\(x _ >.)·So if X I Gm- {>.} is Lie irreducible, the theorem is an immediate consequence of the Pseudoreflection Theorem 1.5. In view of 8.10 .2, the only other case is when X I Gm - {>.} is the tensor product .r. ® r of a lisse rank one .r. with a lisse irreducible r of rank n whose Ggeom is finite. In this case G0 is either {1} or Gm, depending on whether or not !., or equivalently detX

I Gm - {>.}, is of finite order. So

(1) through (4) hold (trivially) in this case. If 1\ is either trivial or of infinite order, then we cannot be in this case, for then local monodromy around >. is a either a unipotent pseudoreflection or is Diag(/\, 1, 1, ..., 1), no power of which is scalar. QED We can be more precise about the distinguishing the various Lieirreducible cases. (We will discuss in section 8.17 how to detect the case when GO,der is {1}.) Corollary 8.11.2.1 Notations and hypotheses as above, suppose further that GO,der ~ {1}. Then GO,der is SO(n) (respectively Sp(n)) if and only if there exists a tame character !; such that X®!.!;:= X>.(!, lJ!; !;')(.'s; !;p's) is self dual and its autoduality pairing is symmetric (resp. alternating).

proof Entirely analogous to that of 3 .5 .8 .1.

QED

In the case n~ m, we have Theorem 8.11.3 Suppose that X:=X>.(!, lJ!; x's; p's) is an irreducible hypergeometric of type (n,m), n ~ m, which is not Kummer induced. Let N:=max(n,m) be the rank of X, d := ln-ml, and G the group Ggeom for X. Suppose p > 2N + 1. If d < N, suppose also that p does not divide the integer ZN1(d)Nz(d) of 7.1.1. Then ( 1) G is reductive. If detX is of finite order (i.e., for n > m, if lT i 'X.i is of finite order; form> n, if lTjPj is of finite order), then GO=GO,der; otherwise GO = GmGO,der (2) If dis odd, GO,der is SL(N). If d = 1 then G ::) IJpSL(N). (3) If d is even, then GO,der is SL(N) or SO(N) or (if N is even) SP(N), or ln-ml=6, N=7,8 or 9, and GO,der is one of

f-adic Hypergeometrics

291

N=7: the image of G2 in its 7-dim'l irreducible representation N=8: the image of Spin(7) in the 8-dim'l spin representation the the the the N=9: the

image image image image image

of of of of of

SL(3) in the adjoint representation SL(2)xSL(2)xSL(2) in std®std®std SL(2)xSp(4) in std®std SL(2)xSL(4) in std®std SL(3)xSL(3) in std®std.

proof Since p > 2N + 1, X is Lie-irreducible, by 7.2.6 (4). So this theorem is just the special case a/ b = 1 I d of the Main ~- adic Theorem 7 .2 .7. The only extra remark is that if d = 1, then detX has break = 1 at either zero (if n < m) or oo (if n > m), hence detX has order divisible by p. QED Proposition 8.11.4 (Ofer Gabber) For N=8, neither of the two groups the image of SL(2)xSp(4) in std®std the image of SL(2)xSL(4) in std®std occurs as GO,der for a hypergeometric of type (8,2). proof Entirely analogous to that of 4.0.1. QED

The discrimination among the various possible cases is aided by Proposition 8.11.5 Hypotheses and notations as in 8.11.3 above, GO,der is contained in SO(N) (resp. in Sp(N)) if and only if there exists a tame character~ such that X®!.~:= X>.(!, lj.>; ~')(,'s; ~p's) is self dual and its autoduality pairing is symmetric (resp. alternating). Moreover, if pN is odd, then GO,der is contained in SO(N) if and only if there exists a tame such that X®!.~ X>.(!, lj.>; ~')(s; ~p's) has its

character~

Ggeom

C

SO(N).

proof Entirely analogous to that of 3.6.1.

QED

Lemma 8 .11.6 (Geometric Determinant Formula) Let X:=X:>.(!, q.; x's; p's) be an irreducible hypergeometric of type (n,m), n > m. Let A Then (1) if n- m

~

2, det(X) :::: I:/\.

292

Chapter 8

proof Since n > m, det(X) is lisse on ~m• tame at zero, and

( L 1\) v ® det( X) ex tends to a lisse sheaf on A 1_ If n - m 2 2, then det(X) is tame at oo as well (since X has all its

oo-slopes 1/(n-m) < 1), and hence (L/\)v ®det(X) is lisse on A1 and tame at oo, hence geometrically constant. If n - m = 1, then as I( oo )-representation X ~ (tame) ffi WA(!, .p; x's; p's), with W)..(!,

.p;

x's; p's) of rank one. By 8.5.4,

WA(!' lj;; l(.'s; p's):::: WA(!, lj;; 1\/(Tijp}; f?J) := [x

H

AX]*(Ly..)®(L/\/TTjp} ~ (tame)®[x

Therefore XA(!,

.p; /\;

H

AX]*(Ly..®L/\).

f?J)v®det(X) is lisse on A1 and tame at oo, so

geometrically constant.

QED

Corollary 8.11.6.1 Suppose that X:=XA(!,

.p;

l(.'s; p's) is an irreducible

hypergeometric of type (n,m), n > m. Then Ggeom c SL(n) if and only if n-m

z

2 and TTiXi = 11.

8 .11.7 Direct Sums and Tensor Products (compare 3 .8) We continue to work over an algebraically closed field k of characteristic p -' ~. Lemma 8 .11.7 .1 (compare 3.8 .1) Suppose that X and X' are irreducible hypergeometrics of types (n,m) and (n',m') respectively, whose generic ranks max(n,m) and max(n',m') are both 2 2. Suppose that there exists a dense open set j: U -+ ~ m• a lisse rank one L on U, and an isomorphism j*X

~

j*X'®L of lisse sheaves on U. Then

(1) (n,m) = (n',m') (2) If n = m, denoting by A (resp. A') the unique singularity of X (resp. X') in ~m• we have A = A·. (3) If (n,m) is not (2,1), (1,2) or (2,2), then Lis L/(. for some tame character /(.,and X

~

X'®L/(. on

~m-

proof Entirely analogous to the proof of 3.8 .1. QED

.f-adic Hypergeometrics

293

Proposition 8.11.7.2 (compare 3.8.2) Suppose that X1, ... , Xn are n ~ 2 irreducible hypergeometrics, with Xi of rank Ni

~

2. Suppose that

(1) ifNi = 2, Xi is of type (2,0) or (0,2). (2)for each i, denote by Gi c GL(Ni) the group Ggeom of Xi (restricted to some dense open U where it is lisse), and suppose that GiO,der is one of the groups SL(Ni), any Ni

~

2,

Sp(Ni), any even Ni

1

4,

SO(Ni), Ni =7 or any Ni

~

9,

S0(3), ifNi= 3 and no Nj = 2, S0(5), ifNi = 5 and no Nj = 4, .S0(6), ifNi = 6 and no Nj = 4, G2 c S0(7), ifNi = 7, Spin(7) c S0(8) ifNi= 8, and no Nj= 7. Suppose that for all i;< j, arid all «

E:

k x, there exist no isomorphisms

from Xi®!. 'X. to either X j or to its dual (X} v. Then group G := Ggeom of E9X·1 has GO,der = TTG·O,der and that of ®X·1 has GO,der = the image of 1 ' TTGiO,der in ®stdn·· 1

proof Entirely analogous to the proof of 3.8.2, using 8.5.4 in place of 3.7.7. QED Corollary 8.11.7.2.1 (compare 3.8.2.1) Let X := Kh(!, ljJ; x's; p's) be an irreducible hypergeometric of rank N 1 2. If N = 2, suppose that X is of type (2,0) or (0,2). Suppose that X is self-dual, and that Ggeom (resp. (Ggeom)O) is one of the groups G: Sp(N), if N even, SO( N), if N "' 4, 8 . G2 c S0(7), if N = 7, Spin(7) c S0(8) if N Let d

1

= 8.

2, and let 1-11, ... , 1-ld be d distinct. elements of k x. Then the

direct sum

294

Chapter 8 ffii X),./1-.li(!, . = 1. Denote by 6 the group Elgeom for X. Recall

that since X is pure (of weight n + m - 1), Elgeom is semisimple, so eO,der = eO. As above, put f\ :: n

j ')(.

i• a character Of ko X,

296

Chapter 8

r

:= TT j pj· a character of ko X'

A:= /\((-i)n-l)qn(n-1)/ZTfi,j (-g(;j;", P/Xi))

E:

iQ~.

We also pick a square root of q := Card(ko) so as to be able to speak of the Tate twist X((n+m-i)/2) of X, which is pure of weight zero. Lemma 8.13.2 Notations and hypotheses as above, suppose that X is geometrically self-dual. If (1\/f)(-i) = i (a condition which is always satisfied if either the autoduality is symplectic, or if p = 2, or if p is odd and we replace ko by its quadratic extension) then the Tate twist X((n+m-1)/2) is arithmetically self-dual with values in Ot· proof X:= Xi(!, for X((n+m-1)/2) meets Kin a single conjugacy class, denoted ~(E, t). The conjugacy classes ~(E, t) are equidistributed in the space Ki:l of conjugacy classes of K with respect to normalized Haar measure, in any of the three senses of equidistrbiution of ([Ka-GKM, 3 .5]).

proof By the above lemma, all the Frobenii of X((n+m-1)/2) lie in O(n). The obstruction to their lying in SO(n) is a character of order dividing two which is geometrically constant, hence is trivialized by any constant field extension of even degree. QED

Theorem 8.1:3.5 Hypotheses and notations as above, suppose that Ggeom is SL(n). Let OG E: iit be any solution of oc:-n

= A.

Then X®oc:deg is pure of weight zero, and has all its Frobenii in Ggeom· Fix an embedding of

iit

into C, and a maximal compact subgroup K of

the Lie group Ggeom(C). The conjugacy class of the semisimple part of each p(FrobE,t) for X®oc:deg meets Kin a single conjugacy class, denoted ~(E, t). The conjugacy classes ~(E, t) are equidistributed in the space Kl:l of conjugacy classes of K with respect to normalized Haar measure, in any of the three senses of equidistrbiution of ([Ka-GKM, 3.5]).

298

Chapter 8

proof This results from the arithmetic determinant formula (and Weil QED

II).

In a similar vein, we have the following slightly less precise result. Theorem 8.13.6 Hypotheses and notations as above, suppose that either (1) (Ggeom)O is SL(n) or Sp(n)

or (2) n is odd and (Ggeom)O is SO(n). Then G := Ggeom is of the form J..ldGO for some d, and there exists a constant ex. in iQ~ x with cx.-n = (root of unity of order dividing order of detGgeom)xA, such that X® ex. deg is pure of weight zero, and has all its Frobenii in Ggeom· Fix an embedding of iQ~ into C, and a maximal compact subgroup K of the Lie group Ggeom(C). The conjugacy class of the semisimple part of each p(FrobE, t) for X® ex. deg meets K in a single conjugacy class, denoted _,(E, t). The conjugacy classes _,(E, t) are equidistributed in the space Ktl of conjugacy classes of K with respect to normalized Haar measure, in any of the three senses of equidistrbiution of ([Ka-GKM, 3.5]). proof In all the cases listed, the normalizer of GO in GL(n) is IBm GO. The rest of the proof proceeds as in [Ka-MG, Cor. 16].

QED

8.14 Criteria for finite monodromy (8 .14 .1) Let C be a smooth geometrically connected curve over a finite field k of characteristic p "$. ~. and '.f a lisse iQ~ sheaf on C. Fix a geometric point~ of C®kk, and denote by 1 geom := 1T 1 (C®kk, ~) c 1T 1 arith := 1T 1 (C, ~) the geometric and arithmetic fundamental groups respectively of C, by P: 1T 1 arith -+ GL('f l;) 1T

299

£-adic Hypergeometrics the t-adic representation that '.f "is", and by Ggeom := the Zariski closure of p( 1T 1 geom), Garith := the Zariski closure of p(1T 1 arith). (8.14.2)

One knows that the radical of (Ggeom)O is unipotent (this is

Grothendieck's global version of the local monodromy theorem, cf. [DeWII, 1.3.8]). Thus if '.f is geometrically semisimple, its Ggeom is a semisimple group. Applying this to det('.f), we recover the fact that det('.f) is geometrically of finite order. Therefore a suitable twist '.f ® 1

f-adic Hypergeometrics 1

(1/n)[n(n-i)/2

+

Li, j

fP

309

l.

Now pick a common denominator N for all the xi's and Yj's, and an integer f such that pf

=

i mod N. In view of the definition of

f P'

it

= 0, i, ... , f-i, we have L·J 2 J 2 (1/n)[n(n-i)/2 + L·~ J· ) 1 . Now pd« is simply another element«' of lTt,.p Ztx ~ Aut((IQ/Z)not p). Since the hypotheses are Aut((Q/Z)not p)-stable, it suffices to

suffices to show that for every integer d

2::·1 1

+

prove universally that Li + Lj

2

(1/n)[n(n-i)/2

whenever the two subsets X := {xi, .... , xn}

and

+

2::i,/Yj- xi>)

Y := {yi, ... , Yn}

of (IQ/Z)not p are intertwined in (IQ/Z)not p· The verification of this is straightforward. The only properties of the function which will he used in the proof are = x for x in [0, i), = 1 - if x is not in Z. By renumbering, we may suppose that we are in one of the three following cases: (Case 1) 0

=x1

< Y1 < x2 < Y2 < ... ) i - (xi- yj>l

= Li,}Yj-

Xi) + Li>j 1 = nLjYj - nLixi + n(n-1)/2. In case 3, we have Li,}Yj- xi> = Li2}Yj- xi> + Li

= 2::i

1 }

= n2:: ·YJ· j

In case i, we have

i - (xi- yj>l + Li < }Yj - xi) - n2::.x· + n(n+1)/2 . 1 1

Chapter 8

310

2:· + L· + l:J.yJ· = n- 1 + l:.y· 1 1 J J = 2:.1 1 2 (1- x·) 1 J J - l:.x· 1 1" In cases 2 and 3, we have l:i + Lj = l:i (1- xi) + LjYj = n + LjYj - l:ixi. Comparing, we see see that the asserted inequality is in fact an equality in cases 1 and 3, and that it holds with a margin of 1 in case 2. QED

Combining this last result with the numerical criterion 8.16.8, we obtain Corollary 8.17 .2 .1 Let E be a finite field of characteristic p ~ t, and X := X1(!, ljl; x's; p's) a nonpunctual, geometrically irreducible hypergeometric defined over E of type (n, n) with n 1 1 and A = 1, whose local monodromy at both zero and co is of finite order. Fix an embedding "\. : Kp,nr c Op· Let x1, ··· • Xn, Yi· ··· • Yn E: (IQ/Z)not p be the 2n distinct elements of (IQ/Z)not p such that "~-""Xi

"~-"Pj

= Xxi, E = Xyj, E

fori for j

= 1, ... , n, = 1, ... , n.

Then X has finite Ggeom if for every OG the two subsets XOG := {OGx1, .... , OGXn}

and

E:

TT b·

Thus Cancel( Hyp).. (!, lj.o; 'X.!, ... , Xn; Pl• ... , Pm)) is an irreducible hypergeometric of type (n-r, m-r).

Theorem 9.3.2 Over an algebraically closed field of characteristic p, let HypA(!, lj.o; X(s; Pj's) := Hyp:A(!, lj.o; 'X.!, ··· • Xn; Pl• ··· • Pm) he an irreducible (i.e., no Xi is a p} hypergeometric of type (n, m). Let d

~

1 he an integer which is prime to p. Denote by {/\ 1• ... , /\d} all the

characters of 'Tfl(IBm) of order dividing d. Then we have isomorphisms of perverse objects on A 1 (1)

FTlj.o(j*[d]*HypA(!, lj.o; X!• ···, Xn; P1· ···, Pm)) ::::: ::::: j*[d]*Cancel(Hyp(-llm-n(d)dh.(!, lj.o; /\1, ... , 1\d, Pj's; X(s)).

(2)

j*[d]*Hyp)..(!, lj.o; Xi's; pJs) ::::: :::::FTili(j*[d]*Cancel(Hyp(-l)n-m(d)dh.(!, lj.o; /\1, ... , 1\d, Pj's; Xi's)) :::::FTIJ.o(j*[d]*Cancel(Hyp(-l)d•n-m(dldh.(!, lj.o; /\1, ... , 1\d, Pj's; Xi's)).

proof The isomorphism (2) is obtained from ( 1) by Fourier inversion. In

0 2 Examples, F.T.'s, and Hypergeometrics

327

order to prove (1), we will first establish the following

Lemma 9 .3 .3 Over an algebraically closed field of characteristic p, for any hypergeometric Hyp),.(!, q,; Xi's; Pj"s), and any integer d l 1 prime top, we have an isomorphism of perverse objects on ISm j*FTq,(j![d]*Hyp),.(!, q,; Xi's; Pj"s)) ::: ::: [d]*Hyp(-1)m-n(d)dh(!, q,; /\1, ... , 1\d, Pj"s; Xi's).

proof As recalled in 8 .1.12, for any object K we have (j*l:q,)[1]*!K::: j*FTq,(j!inv*K). We apply this to the object K := [d]*inv*HypA(!, q,; X(s; Pj"s) = inv*[d]*Hyp),.(!, q,; X(s; Pj's) and find j*FTq,(j![d]*Hyp),.(!, q,; Xi's; Pj's)) ::: ::: (j*l:q,)[1]*![d]*inv*Hyp),.(!, q,; X(s; Pj's) By the base change formula for convolution (8.1.10, 2(b)), for any two objects K and Lon ISm, and any nonzero integer d, we have K* !([d]*L) ::: [d]*(([d]*K)* !L). Thus we have j*FTq,(j![d]*Hyp),.(!, q,; Xi's; Pj's)) ::: :::

[d]*(([d]*(j*l:q,)[1])*!inv*Hyp),.(!, q,; Xi's; Pj's)).

It remains only to simplify the convolvees. By the inversion property (8 .3 .3) and the change of q, formula (8.7 .2), we have inv*Hyp),.(!, q,; Xi's; Pj's) ::: Hyp1fA(!,

;j;';

Pj's; X(s)

::: Hyp(-l)n-mh(!, q,; Pj's; X(s). By the Kummer Induction Theorem 8.9.1, we have [d]*(j*l:q,)[1]) := [d]*(Hyp1(!, q,; 11, ¢)::: ::: Hypl(!, q,l/d; /\1, ... '1\d; ¢), and by 8.7 .2, we have Hyp..(!, Y,.; 'Xi's; Pj's))

~

~ [d)*Cancel(Hyp(-1)m-n(d)d/A(!, Y,.; A1, ... , Ad, Pj's; Xi's)).

Since both of these perverse objects are semisimple, it suffices to show

0 2 Examples, F.T.'s, and Hypergeometrics

329

that they have isomorphic semisimplifications. For this, we argue as follows. We have a short exact sequence of perverse objects on A1 of the form 0--+V®So --+ j 1[d]"HypA(!, lJJ; Xi's; Pj's) --+ j,.[d]"HypA(!, lJJ; Xi's; Pj's) --+ 0, for some punctual sheaf V®So at zero. In view of the known structure of the local monodromy at zero of HypA(!, lJJ; X(s; Pj's), we see that V has dimension r := Card(R), R := {k in {1, ... , d} such that /\k is among the Xi). Taking the Fourier Transform of the above exact sequence, applying j", and using the lemma, we find a short exact sequence 0--+ (V®Q~[1])--+ [d]"HYP(-1)m-n(d)dh.(!, lJJ; /\1, ... , 1\d, Pj's; Xi's)--+ --+ j"FTlJJ(j,.[d]"HypA(!, lJJ; Xi, ···' Xn; P1· ···' Pm)) --+ 0. On the other hand, by the Semisimplification Theorem 8.4.10, the semisimplifiation of Hyp(-l)m-n(d)dh.(!, lJJ; /\1, ... , 1\d, Pj's; Xi's) lS

Efli in R l:./\i[1]

E9

Cancel(Hyp(-l)m-n(d)dh(!, lJJ; /\1, ... , 1\d, Pj's; Xi's)).

Pulling back by [d]", we find that the semisimplification of [d]"Hyp(-l)m-n(d)d/).(1, lJJ; /\1, ... , 1\d, Pj's; Xi's) is (iQ~[1W E9 [d]"Cancel(Hyp(-1)m-n(d)dh,(!, lJJ; /\1, ... , 1\d, Pj's; Xi's)).

Comparing this with the above short exact sequence, we conclude that the two perverse objects on Gm [d]"Cancel(Hyp,.(!, lji; 'Xi, ···, 'Xn; Pi• ···, Pm) is not Kummer induced of any degree di which divides d. Then j,.[d]*Hyp;>,.(!, lji; 'Xi, ···, 'Xn; Pi· ···' Pm) is perverse irreducible on A i, and consequently the isomorphism FTlji(j,.[d]"Hyp;>,.(!, lji; 'Xi,···, 'Xn; Pi• ···, Pm)):::: :::: j,.[d]*Cance1Hyp(-1)m-n(d)dh.(!, lji; t\1, is an isomorphism of perverse irreducibles on A i_ proof Indeed, since H := Hyp;>,.(!, lji; 'Xi, ... , 'Xn; Pi• ... , Pm) is perverse irreducible, and [d] is finite etale galois, either [d] * H is isotypical or H is induced from an intermediate covering. So the hypothesis insures that [d]*H is isotypical. It remains only to show that if [d]*H is isotypical, then it is irreducible. If [d] * H is isotypical, say k l 1 copies of an irreducible K, then since the isomorphism class of K is l-Id-invariant, K itself descends

through the cyclic covering [dl, to a perverse irreducible Ko. Therefore H is of the form Ko ® M, with M the k-dimensional representation of 1T

i (IBm) given by Hom(Ko, H). This M must be irreducible if H is to be

irreducible. But this M becomes trivial after [d]*, so it is a sum of l..t\'s.

G 2 Examples, F.T.'s, and Hypergeometrics Therefore we have k

= 1,

331

and hence [d]*H is perverse irreducible on

Gm. Taking its middle extension j*[d)*H to A1, we find that j*[d]*H and with it FTljJ(j*[d]*H) are perverse irreducible on A1.

QED

9.4 Reduction to the Tame Case (9 .4 .1) In the case when n > m and d = n-m is prime to p, we obtain a striking relation between hypergeometrics of "wild" type (n, m) and those of "tame" type (n-r, n-r). The above results give, in this case: Corollary 9.4 .2 Hypotheses as in the theorem, suppose that n > m and that d = n - m is prime top. Then (a) we have isomorphisms of perverse sheaves on A 1 (1)

FTljJ(j*[d]*Hyp),(!, lj.l; X1, ... , Xn; P1• ... , Pm)) :::: :::: j*[d]*Cancel(Hyp(-i)m-n(d)dh..(!, lj.l; /\1, ... , 1\d, Pj's; Xi's)).

(2)

j*[d]*Hyp),(!, lj.l; X(s; Pj's) :::: ::::FT4i'(j*[d]*Cancel(Hyp(-1)n-m(d)dh..(!, lj.l; /\1, ... , 1\d, Pj's; X(s)) ::::FTljJ(j*[d]*Cancel(Hyp(-1)d+n-m(d)dh..(!, lj.l; /\1, ... '1\d, Pj's; x(s)).

(b) If Hyp),(!, lj.l; Xi's; Pj's) is not Kummer induced, these are isomorphisms of perverse irreducibles. (c) If none of the Xi satisfies (Xi)d

= 11,

we may rewrite these

isomorphisms: (1)

FTljJ(Rj*[d)*Hyp),(!, lj.l; X1, ... , Xn; P1• ... , Pm)) :::: :::: j*[d)*Hyp(-1)m-n(d)dh..(!, lj.l; /\1, ... '1\d, Pj's; x(s).

(2)

Rj*[d)*Hyp),(!, lj.l; Xi's; Pj's) :::: ::::FT4i'(j*[d)*Hyp(-i)n-m(d)dh..(!, lj.l; /\1, ... , 1\d, Pj's; Xi's)) ::::FTljJ(j*[d]*(Hyp(-i)d•n-m(d)dh..(!, lj.l; /\1, ... , 1\d, Pj's; Xi's)).

proof This is just rewriting the previous results for d = n - m. QED

CHAPTER 10 .f -adic Exceptional Cases 10.0

In traduction This chapter is devoted to the exceptional possibilities for the group Ggeom of an irreducible ~-adic hypergeometric on IBm in characteristic p of type (n,m), n :t m, which is not Kummer induced. Let N:=max(n,m), d := In- mi. Suppose that p > 2N + 1 and that p does not divide the integer 2N 1 (d)N2(d) of 7 .1.1. Recall (8 .11.2-4) that the exceptional possibilities for GO,der can occur only for ln-ml=6, N=7,8 or 9: N=7: the image of 62 in its 7-dim'l irreducible representation N=8: the image of Spin(7) in the 8-dim'l spin representation the image of SL(3) in the adjoint representation the image of SL(2) x SL(2)x SL(2) in std® std® std N=9: the image of SL(3)xSL(3) in std®std. We will show that the cases in which these exceptional groups occur are "the same" as they were for hypergeometric .D-modules. Indeed, the proofs in the two cases are quite analogous. We will largely content ourselves with indicating these analogies, rather that giving the ~-adic proofs in complete detail. 10.1 The Gz and Spin(7) Cases Gz Recognition Theorem 10.1.1 Let k be an algebraically closed field of characteristic p > 15. Suppose that p does not divide the integer 2N1(6)Nz(6) of 7.1.1. Let')(., p be two tame iOrvalued characters of 1T 1 (ill m

® k) such that none of ')(., p, or ')(. p is the unique character 1\ 112

of exact order two. Then for any >.

E:

k x, and lJi any nontrivial Or

valued additive character of a finite subfield ko of k X:= X>.(!, !Ji; 11, ')(.,X, p,

p,

'X.p,

xp;

/\112>

has Ggeom = Gz. These are all the hypergeometric of type (7,1) with Ggeom = Gz. The hypergeometrics of type (7,1) with GO,der = Gz are precisely the tame !_ 1\ twists of these. proof Exactly as in the differential galois case (4.1), but using 8.11.5 instead of 3.6.1, the only nonobvious point is that such an X has Ggeom = Gz. To show this, we argue as follows. X is irreducible, and, being of type (7, 1), it is not Kummer induced. Since p > 15, X is Lie-

332

f-adic Exceptional Cases

333

irreducible. Visibly X is self-dual with trivial determinant. Exactly as in 9.1.1, we see that the only possibilities for (Ggeom)O are PSL(2), G2 , or S0(7). In all of these casess, every automorphism of (Ggeom>O is inner, so Ggeom c l!lm(Ggeom)O. Since Ggeom c S0(7), it contains no nontrivial scalars, so Ggeom

= (Ggeom)O. By

the same slope argument as in 9.1.1,

we can rule out the PSL(2) possibility. So Ggeom is either G2 or S0(7). To rule out the S0(7) possibility, it suffices to show that, denoting by j: illm

-+

IP1 the inclusion, we have ')(.(IP1, j*t\3(1{)) ~ 2 > 0.

By the Euler-Poincare formula, ')(.(IP1, j*t\3(1{))

=

= -Swano(t\3(1{))- Swan 00 (t\3(X))

+

dim(t\3(}t))lo

+

dim(t\3(X))Ico.

So it suffices to show that ( 1) dim(/\ 3(}t))lo ~ 5. (2) Swano(t\ 3(1{))

= 0.

(3) dim(t\3(}t))lco = 2. (4) Swan 00 (/\3(X))

= 5.

The proofs of these four assertions are entirely analogous to those of their differential galois theoretic avatars, using 8.6 .4 and 8.9 .1 instead of 3.4.1.1. Assertions (1) and (2) hold in any characteristic p. In proving (3) and (4), one needs that pis prime to 6, and that the relations of the form ~1 + ~2 + ~3 = 0, with the ~i three distinct sixth roots of unity in characteristic p are the two cases {~ 1• ~2, ~3} = {all the cube roots of 1}, (-~1, -~2, -~3}

= {all the cube roots of 1}.

We will now show that this is the case so long as p :c 2, 3, 7. Indeed, suppose we have any relation ~ 1 + ~2 + ~3 = 0 where each ~i is a sixth root of unity. Since either ±~i is a cube root of unity, we can rewrite this relation in the form ± W1 ± W2 ± W3 = 0 where the wi are cube roots of unity. Dividing by ±w3, we get a relation of the form

Chapter 10

334

=

± w 1 ± w2, where the Wi are cube roots of unity. We now analyze the possible cases. If w1 = 1, then its sign must be minus (lest w2 = 0), and the relation is 2 = ± w2, whence 2 is a 1

sixth root of unity. But 26 - 1 = 63 then the relation is 1

= ± 2w 1•

;e

0 (since p

;e

2, 3, 7). If w1 = w2,

and again 2 would be a sixth root of 1.

So the only possible relation has w1 and w2 the two nontrivial cube roots of unity, say w and w2. The relation is one of 1 1 1 1

= =-

=

= -

w + w2, w + w2, w - w2, w - wZ.

Of these, we claim that only the last one (which always holds so long as p ;z: 3) holds in characteristic p ;z: 2, 3, 7. Using the last one, the first three become - w- w2 = - w - w2 = - w + w2, - w - w2 = w - w2, each of which trivially implies that p = 2.

QED

Remark 10 .1.2 We can summarize the result of the preceeding calculation as the statement that N2(6) (cf 7 .1.1) is divisible only by the primes 2, 3, 7.

Spin(?) Recognition Theorem 10.1.3 Let k be an algebraically closed field of characteristic p > 17. Let

A

e: kx, and ljJ any nontrivial Q~­

valued additive character of a finite subfield ko of k. Suppose that p does not divide the integer 2N 1 (6)N2(6) of 7 .1.1. Let ')(., p,

~

be three

tame Q~-valued characters of rr1(1Bm®k) such that X:= XA(!, ljl;

x,

x, p, p,

l;,

~. Xpl;, Xp~; 11, /\1/2)

is irreducible and not Kummer induced. Then X has Ggeom equal to (the image in S0(8) of) Spin(7). These are all the hypergeometric of type (8,2) with Ggeom = Spin(7). The hypergeometrics of type (8,2) with GO,der

= Spin(7)

are precisely the tame I/\ twists of these.

t' -adic Exceptional Cases

335

proof Again the only hard point is that such an X has Ggeom = Spin(7). For this it suffices to show that, denoting by j: illm --+ IP1 the inclusion, we have ')(.(IP1, j.,t\4(}{))

2

2 > 0.

By the Euler-Poincare formula, ')(.(IP1, j.,t\ 4(}{)) =

= -Swano(t\ 4(}()) - Swan 00 (t\ 4(}())

+

dim(t\ 4(J-t))Io + dim(!\ 4(rt))loo.

So it suffices to show that (1) dim(t\4(X))Io 2 8. (2) Swan 0 (t\4(X)) = 0. (3) dim(t\ 4(J-t))loo = 4. (4) Swan 00 (t\4(rt))

= 10.

The proofs of these four assertions are entirely analogous to those of their differential galois theoretic avatars. Assertions (1) and (2) hold in any characteristic p. In proving (3) and (4), one needs that pis prime to 5, and that all relations of the two forms t'1 + t'2 + t'3 = 0, t'1 + t'2 + t'3 + t'4 = 0 with the t'i three (resp. four) distinct sixth roots of unity in characteristic p are exactly the same as in characteristic zero. Since -1 is a sixth root of unity, the relations in question can be rewritten to be of the forms t'1 + t'2 = t'3 • t'1 - t'z = t'3 - !;4. So this is guaranteed by the hypothesis that p does not divide the integer ZN 1 (5)N 2 (5). QED 10.2 The PSL(3), SL(2)xSL(2)xSL(Z), and SL(3)xSL(3) Cases, via Tensor Induction (10.2.1) We will give a unified treatment of these three cases by thinking systematically about tensor induction (cf. [C-R-MRT, 131, [Ev)) of ~-adic hypergeometrics. I am indebted to Ofer Gabber for making me aware of this point of view. [This same method could also be used in the differential galois case, where it would obviate the use of the specialization theorem.] 10.3 Short Review of Tensor Induction (10.3.1) Let us recall the basic setup (cf. [C-R-MRT, 131, [Ev]). Given a

Chapter 10

336

group 6 and a subgroup H of finite index n, consider the "wreath product" (H)n t> 7. Let >. e: k x, and ljJ any nontrivial Ot-valued additive character of a finite subfield ko of k. Let "X1, "X2, "X.3 be tame Ot-valued characters of 1T 1 (~m®k) none of which has order dividing three, and whose product is trivial: -x.1 X2"X3 = 11. Denote by /\112 := the unique tame character of exact order 2, /\113· /\2/3 := the two characters of exact order 3, {p 1 , ... , pg) := the ®2"nd roots of {")(.1, "X2, "X.3l

= {"Xi, "X2, "X.3l U {both square roots of

X i•

of

x2 , of x3 ).

(1) Any hypergeometric of type (3, 0) of the form X := K>. (!, lJ.I; "Xi, "X2, "X.3; ¢) has Ggeom = SL(3), and X®Li *X has Ggeom = the image in SL(9) of SL(3) X SL(3). (2) There exists, for some IJ. in kx, an isomorphism of lisse sheaves [2l®*(X) :::: XIJ.(!, lJ.I; Pi····, p9; 11, /\1/3· /\2;3). (3) There exists an isomorphism of lisse sheaves X®Li *X :::: [2]*XIJ.(!,

q,; p 1 , ... , pg; 11, /\ 1 ; 3 , /\ 2 ; 3 ).

(4) For any ~ in kx, XIJ.(!, ljJ; P1' ... , pg; 'D., /\113, Az/3) has (Ggeom> 0

=

the image in SL(9) of SL(3)xSL(3), and (Ggeom)O has index two in Ggeom· (5) The hypergeometrics of type (9, 3) with (Ggeom)O,der = the image in SL(9) of SL(3) x SL(3) are precisely the tame !./\-twists of these.

proof Exactly like those of the previous two theorems, now using i0.6.9 and imitating 4 .6 .10. QED

CHAPTER 11 Reductive Tannakian Categories 11.1 Homogeneous Space Recovery of a Reductive Group ( 11.1.1) In this section we work over an algebraically closed field K of characteristic zero. Suppose that we are given an integer n 2 1, an ndimensional K-vector space V, and an algebraic subgroup G of GL(V). We know that we can recover G from the the Tannakian category E := Rep(G) of its finite-dimensional K-representations and the fibre functor w := "forget the G-action" : Rep( G) -+ (fin.-dim'l K-spaces). ( 11.1.2) Suppose now that G is reductive. Following Ofer Gabber, we will explain how to recover G (more precisely, the conjugacy class in GL(n, K) of G(K)) from the Tannakian category E := Rep(G) using only the functor "G-invariants" : E -+ (fin.-dim'l K-spaces) W

>-+

wG := Home(11, W),

but without using a fibre functor. (11.1.3) Let us first describe the idea. Consider the space X := IsomK(Kn, w(V)). This is a left GL(w(V))-torsor by postcomposition, and a right GL(n, K)torsor by precomposition. The quotient space Y := G\X is then a right homogeneous space under GL(n, K). In terms of this homogeneous space we recover G (up to GL(n, K)-conjugacy) as the stabilizer in GL(n, K) of any chosen pointy e: Y. ( 11.1.4) How are we to turn this idea into a proof? And where will the hypothesis that G is reductive enter? First of all, we can construct the space X using the fibre functor w as follows. If we view X as the space of ordered bases (v1, vz, ... , vn) of w(V), then X sits as an open set in vn We will instead view X as the closed subscheme of vnx(Vv)n consisting of those tuples (v1, vz, ... 'vn; v1 v, ... 'vn v) satisfying the n2 equations VjV(Vi) = si,j· (This is just a longwinded way of saying that the vi are a basis of V and the Vj v are the dual basis of vv. For n= 1, it amounts to defining Gm by the equation xy= 1 rather than by the condition "x invertible".) This description has the merit of exhibiting X as an affine K-

352

Reductive Tannakian Categories

353

scheme of finite type, say X = Spec(A). The group G, being a subgroup of GL(V), certainly acts freely on X; indeed, for any K-algebra R, the group G(R) acts freely on the set X(R). Because G is reductive, the quotient Y := G\X exists and is affine, with coordinate ring B = AG. On K-valued points, we have Y(K) = G(K)\X(K). Since X(K) = Isom(Kn, w(V)) is a left GL(w(V))-torsor and a right GL(n, K)-torsor we see that Y(K) is a right homogeneous space under GL(n, K), and the stabilizer in GL(n, K) of any pointy E: Y(K) is (a conjugate of) G(K). To conclude this discussion, it remains to construct the ( 11.1.5) coordinate ring B of the affine scheme Y and the left action of GL(n, K) on B, using only the Tannakian structure of t: and the functor "Ginvariants". To clarify the discussion which follows, let us denote by V 1• ... , V n n copies of V, and by V 1 v, ..., V n v n copies of vv. We can construct X as the closed subscheme of the spectrum of the symmetric algebra S :=

® i=

l, ...,n

Symm*(w(Viv))®Symm*(w(Vi))

defined by the ideal I generated by the n2 elements fi,j - si,j· where fi,j is the G-invariant element of w(Viv)®w(Vj) = w(Vv)®w(V) = w(Vv®V) which is the canonical map 11. -+ vv ® V. This fi,j is just the function (v1, vz, ... 'vn; v1 v' ... 'Vn v) ~--+ vjv(vi), described in an invariant way. (11.1.6) Since G is reductive, and the ideal I is generated by the invariants fi,j - si,j· we have B = (S/I)G = sG/(the ideal in sG generated by all the fi,j - Si}• and sG is the IN Zn_graded algebra EB -x]*(Til*))*+(jl~""' ). . ""m proof This is just the inverse Fourier Transform of the standard

Fourier Universality

369

structure of Tannakian category on D.E.Wm/IC). QED Remark 12.3.9 For any« E: ex, the functor DA,B - t (fin. dim'l. IC-vector spaces) JTt ~---+ HO("TT*(JTt®e«X)) := H1DR(1Bm/IC, Jft®e«X) is a fibre functor. !It is the Fourier Transform of the usual fibre functor w« := fibre at« on D.E.(~Bm/IC).) We do not know how to construct explicit fibre functors on DA,B without invoking Fourier Transform. 12.4 The Tannakian Category DA,RS (12.4.1) Recall (cf. [Ber), [Bar)) that for X a smooth separated Itscheme of finite type, one has the notion of RS ("regular singular")

.b-

modules, and of the full subcategory nRS(X) consisting of the RS objects of Db,holo(X). One knows that nRS(X) is stable under the "six operations". In particular, if X is a smooth C-groupscheme G, then nRS(G) is stable under convolution. (12.4.2)

We now return to the A1 setting. We define DA,RS to be the

full subcategory of DA,B consisting of those objects which are RS. Given a short exact sequence 1 -.

o -. m

m2

-.

m3

-.

o

in DA,B• JTt2 is RS if and only if both Jft1 and JTt3 are RS. The subcategory DA,RS of DA,B is stable under convolution and under "dual", and contains the unit object 11. :=

j,~.,

. ""m

. Thus we find

Theorem 12.4.3 DA,RS is itself a Tannakian category with ( 1) "tensor product" given by convolution * +• (2) "unit object" 11. given by j (3) "dual" given by

m

v

:= ([x

1~..,

. ""m 1-+

,

-x]*(JTt*))*+(jl~"'

. ""m

).

Proposition 12.4.4 Let JTt be an object of DA,RS· Then JTt and FT:rrt have the same generic rank. In other words, the "dimension" of an object of the Tannakian category DA,RS is its generic rank. proof Since FT:ffi is a D.E. on IBm, say of rank r, for any « complex

E:

m as

ex the

Chapter 12

370

FTJn, placed in degree 0 and 1, has HO = 0, dimcH1 = r. Thus the generic rank of FTJn is -x(i Spec(k) the structural map,

i A 1 the inclusion of the point A1xkA1 the diagonal embedding, sum: A1xkA1 -> A1 the addition map (x, y)

H

j : 6m -> A 1 the inclusion. We will denote additive * and ! convolution by * * + and * !+ respectively. ( 12.9 .2) Recall (8 .1.7) that the Fourier Transform

x+y,

Chapter 12

380 FT

ohc(A1k, Ot)

is defined by FTljJ(K) := R(pr 2 )!(pr1 *(K)®.l.ljJ(xy))[1). It commutes with duality up to an additive inversion,

= FTljJ•[x

D•FTljJ

...... -xl*•D,

and is involutive up to a Tate twist of (-1) and an additive inversion: FTljJ·FT~(K)

= [x ...... -x]*(K)(-1).

One knows (cf [Br, 9.6]) that FTljJ[-1] interchanges tensor product and additive ! convolution: FT~(K* !+L)[-1] :::: FT~(K)[-1]®FT~(L)[-1]. Key Lemma 12.9.3 (compare Key lemma 12.2.3) ForK, L objects in obc(A1k, Ot), and any o;

E:

k, we have canonical isomorphisms in

obc(Spec(k), Oe) ( 1)

Rrr!(K®.l.ljJ(o::x)) :::: io; *(FTljJ(K))[-11,

(2)

Rrr!(FT~(K)®.l.~(-o::x)) :::: io; *(K)(-1)[-11,

(3)

Rrr!(K*!+L):::: Rrr!(K)®Rrr!(L),

(4)

io; *(K®L) :::: io; *(K)®io; *(L),

and a canonical isomorphism in ob c ~ (f*n>*!+(j*G"). Taking the Fourier Transforms of these isomorphisms and restricting to

m.

Fundamental Comparison Theorems

423

IGm, we get isomorphisms in Dbholo(IGm,IC) j * FT(:ffi) ::: j * FT(f!n) ::: j * FT(f * n). Visibly we have

m :::

j*FT(t 1

(prz) 1(pq +(n)®etf(xl),

j *FT(f * n) ::: (prz) * (pq +(n) ® etf(x)),

as results from base change via the diagram X x IG m

fxid

l l

------>

lf Ai

Aix!G m

prz

X

QED

!Gm.

14.i4 Examples (i) Suppose that n ~ i is an integer, and (xi, xz, ... , xn) are n functions on X which define a finite morphism from X to AnR (e.g., if X is given as a closed subscheme of AnR, one might take for the xi the coordinate functions in the ambient AnR)· Then by (the proof of) [Ka-Lau, 5.41, there exists a nonzero homogeneous polynomial inn variables Yj, F(Yi, Yz, ... , Yn)

E:

R[Yi, Yz, ... , YnL

with the following property: for any n-tuple (a1, ... , an) of elements of R such that F(a)

;