540 43 1MB
English Pages 97 [114] Year 2000
Table of contents :
Cover
Other titles in this series
Title page
Author’s apology
Contents
Acknowledgments
Introduction
Tamagawa numbers
Tamagawa numbers. Continued
Continuous cohomology
A theorem of Borel and its reformulation
The regulator map. I
The dilogarithm function
The regulator map. II
The regulator map and elliptic curves. I
The regulator map and elliptic curves. II
Elements in ?₂(?) of an elliptic curve ?
A regulator formula
Bibliography
Index
Back Cover
Volume 11
C R M
CRM MONOGRAPH SERIES Centre de Recherches Mathématiques Université de Montréal
Higher Regulators, Algebraic KTheory, and Zeta Functions of Elliptic Curves Spencer J. Bloch
American Mathematical Society
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Selected Titles in This Series Volume 11 Spencer J. Bloch Higher regulators, algebraic Ktheory, and zeta functions of elliptic curves 2000 10 James D. Lewis A survey of the Hodge conjecture, Second Edition 1999 9 Yves Meyer Wavelets, vibrations and scaling 1998 8 Ioannis Karatzas Lectures on the mathematics of ﬁnance 1996 7 John Milton Dynamics of small neural populations 1996 6 Eugene B. Dynkin An introduction to branching measurevalued processes 1994 5 Andrew Bruckner Diﬀerentiation of real functions 1994 4 David Ruelle Dynamical zeta functions for piecewise monotone maps of the interval 1994 3 V. Kumar Murty Introduction to Abelian varieties 1993 2 M. Ya. Antimirov, A. A. Kolyshkin, and R´ emi Vaillancourt Applied integral transforms 1993 1 D. V. Voiculescu, K. J. Dykema, and A. Nica Free random variables 1992
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https://doi.org/10.1090/crmm/011
Volume 11
C R M
CRM MONOGRAPH SERIES Centre de Recherches Mathématiques Université de Montréal
Higher Regulators, Algebraic KTheory, and Zeta Functions of Elliptic Curves Spencer J. Bloch
The Centre de Recherches Mathématiques (CRM) of the Université de Montréal was created in 1968 to promote research in pure and applied mathematics and related disciplines. Among its activities are special theme years, summer schools, workshops, postdoctoral programs, and publishing. The CRM is supported by the Université de Montréal, the Province of Québec (FQRNT), and the Natural Sciences and Engineering Research Council of Canada. It is affiliated with the Institut des Sciences Mathématiques (ISM) of Montréal, whose constituent members are Concordia University, McGill University, the Université de Montréal, the Université du Québec à Montréal, and the Ecole Polytechnique.
American Mathematical Society Providence, Rhode Island USA
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The production of this volume was supported in part by the Fonds Qu´eb´ecois de la Recherche sur la Nature et les Technologies (FQRNT) and the Natural Sciences and Engineering Research Council of Canada (NSERC). 2000 Mathematics Subject Classiﬁcation. Primary 19F27; Secondary 14G10, 19D50. Abstract. Work of Borel on higher regulators for number ﬁelds is discussed. The Borel regulator for K3 of a number ﬁeld is described explicitly in terms of the dilogarithm function. A generalization, based on functions related to the dilogarithm and to the Dedekind ηfunction, leads to a regulator for K2 of an elliptic curve E over a number ﬁeld. Elements in K2 (E) analogous to cyclotomic units are described. The regulator is evaluated on these elements and the resulting values related to the value of the HasseWeil zeta function of E at s = 2 when E has complex multiplication. This regulator formula is worked out in detail for the case of E deﬁned over Q with complex multiplication by the ring of integers in an imaginary quadratic ﬁeld, when it takes a particularly simple form.
Library of Congress CataloginginPublication Data Bloch, Spencer. Higher regulators, algebraic Ktheory, and zeta functions for elliptic curves / Spencer J. Bloch. p. cm. — (CRM monograph series, ISSN 10658599 ; v. 11) Includes bibliographical references and index. ISBN 0821821148 (alk. paper) 1. Curves, Elliptic. 2. Ktheory. 3. Functions, Zeta. I. Title. II. Series. QA567.2.E44.B56 516.352—dc21
2000 00029969
Copying and reprinting. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 029042294 USA. Requests can also be made by email to [email protected]. c 2000 by the American Mathematical Society. All rights reserved. Reprinted in soft cover by the American Mathematical Society, 2011. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acidfree and falls within the guidelines
established to ensure permanence and durability. This volume was typeset using AMSTeX, the American Mathematical Society’s TEX macro system, and submitted to the American Mathematical Society in camera ready form by the Centre de Recherches Math´ematiques. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1
16 15 14 13 12 11
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Author’s Apology “The author apologizes for the long delay in publishing this monograph. The reader should understand that since this work was done, fundamental ideas of A. Beilinson, A. Suslin, V. Voevodsky, and others have totally transformed the landscape. Sometimes it is fun to drive around in a Model T Ford but one should be aware there are much faster cars on the road.”
v
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Contents Author’s Apology
v
Acknowledgments
ix
Lecture 0. Introduction
1
Lecture 1. Tamagawa Numbers
9
Lecture 2. Tamagawa Numbers. Continued
15
Lecture 3. Continuous Cohomology
23
Lecture 4. A Theorem of Borel and its Reformulation
29
Lecture 5. The Regulator Map. I
35
Lecture 6. The Dilogarithm Function
43
Lecture 7. The Regulator Map. II
51
Lecture 8. The Regulator Map and Elliptic Curves. I
61
Lecture 9. The Regulator Map and Elliptic Curves. II
69
Lecture 10. Elements in K2 (E) of an Elliptic Curve E
75
Lecture 11. A Regulator Formula
87
Bibliography
95
Index
97
vii
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Acknowledgments I should like to express my gratitude to Bill Messing and the Department of Mathematics at Irvine for inviting me to give these lectures. At the time of the lectures the whole relationship between the regulator and the zeta function of an elliptic curve was purely conjectural. Messing’s interest gave me the fortitude to push through the complicated business. I also want to acknowledge many fruitful conversations with D. Wigner about the dilogarithm function. It was he who ﬁrst wrote down the function D(x) (0.3.4) and worked out a number of its properties. Spencer J. Bloch Department of Mathematics The University of Chicago Chicago, IL 60637 USA
ix
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https://doi.org/10.1090/crmm/011/01
LECTURE 0
Introduction 0.1.
Let K be a number ﬁeld with ring of integers OK . Let O K ⊗ Z R = Rr 1 ⊕ C r 2 ,
so r1 and r2 are the numbers of real and complex places of K, and deﬁne ∗ → Rr1 +r2 = (1 , . . . , r1 +r2 ) : OK
(0.1.1)
∗ to be the composite of the inclusion OK → (OK ⊗Z R)∗ with the map ∗ r1 +r2 (OK ⊗ R) → R given by log   (resp. log  2 ) on the real (resp. ∗ complex) factors. The image (OK ) = L is known to be a lattice of rank r1 + r2 − 1, and the regulator RK is deﬁned by 1 (0.1.2) RK = √ Volume(L). r 1 + r2 ∗ /torsion, it is easy to show RK is the If 1 , . . . , r1 +r2 −1 is a basis of OK absolute value of the determinant of any (r1 + r2 − 1) × (r1 + r2 − 1) minor of the matrix i (j ) 1≤i,j≤r1 +r2 . The class number formula
lim(s − 1)ζK (s) =
(0.1.3)
s→1
2r1 +r2 π r2 RK h Dw
relates the residue at s = 1 of the zeta function ζK (s) to the regulator RK , the class number h, the number w of roots of 1, and the discriminant D of K. If one takes into account the functional equation satisﬁed by ζK (s), the class number formula can be rewritten ([Lic73]) −hRK . s→0 w Various conjectural generalizations of the class number formula have been proposed. One such, due to Lichtenbaum, would have lim ζK (s)s−(r1 +r2 −1) =
(0.1.4)
(0.1.5)
lim ζK (s)(s + m)−dm = ±
s→−m
#K2m (OK ) Rm (K) #K2m+1 (OK )tor
for any nonnegative integer m, where K∗ (OK ) denotes the Ktheory of Bass, Milnor, and Quillen, among others, Rm (K) is a suitable regulator, 1
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2
and (0.1.6)
0. INTRODUCTION
⎧ ⎪ m = 2n + 1 > 0 ⎨ r2 dm = r1 + r2 m = 2n > 0 ⎪ ⎩r + r − 1 m = 0. 1 2
0.2. Unfortunately, Lichtenbaum’s conjecture does not give the right value, even for m = 1 and K = Q. However, various results suggest some formula of this sort should hold. In particular Borel has computed the rank of the K∗ (OK ), proving 0 n = 2m (0.2.1) rk Kn (OK ) = dm n = 2m + 1. So rk K2m+1 (OK ) = order of zero of ζK (s) at s = −m. The indecomposable space I 2m+1 of the continuous cohomology 2m+1 (SL(OK ⊗Z R), R) Hcont
has dimension dm , and the Hurewitz map K∗ (OK ) → H∗ GL(OK ) gives a homomorphism (0.2.2)
r : K∗ (OK ) → Hom(I 2m+1 , R).
Borel shows further that r embeds K∗ (OK )/torsion as a lattice of maximal rank, and that for a suitable choice of basis (obtained by relating ∗ Hcont (SL(OK ⊗ R), R) to the cohomology of the compact dual symmet ric space of SL(OK ⊗ R)), r K2m+1 (OK ) has volume some rational multiple of (0.2.3)
π −dm lim ζK (s)(s + m)−dm = π −d(m+1) D1/2 ζK (m + 1). s→−m
The ﬁrst four lectures are devoted to Borel’s work, although we completely neglect the more diﬃcult aspect, the computation of rk K∗ (OK ), in order to focus on the adelicTamagawaWeil techniques for determining the volume of r K∗ (OK ) . Lectures 1 and 2 recall necessary ideas from Weil’s Princeton notes [Wei61], while Lectures 3 and 4 focus on Borel’s preprint [Bor77]. 0.3. The remaining lectures are more tentative, and the results are much less sweeping. They represent the author’s attempt to extend Borel type regulator results from number ﬁelds to elliptic curves over number ﬁelds. After all, Quillen has deﬁned higher Kgroups K∗ (X)
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0. INTRODUCTION
3
for any scheme X. One might hope for a computation of the rank d of K∗ (X) together with a regulator map r : K∗ (X) → Rd such that r K∗ (X) was a lattice of maximal rank whose volume was related to the value of the HasseWeil zeta function of X. Unfortunately, such results do not come by magic, and it should be pointed out that appropriate generalizations of the two basic tools in Borel’s arguments (viz., symmetric spaces and id`eles) are not at this time available. A conjectural generalization of the class number formula, due to Birch and SwinnertonDyer has impressive numerical evidence behind it. The idea is to describe the behavior of the HasseWeil zeta function of an elliptic curve E over a number ﬁeld K near the point s = 1. The role of the regulator is played by the matrix associated to the height pairing on points of the curve deﬁned over the given number ﬁeld. In particular, one expects that the order of the zero of the HasseWeil zeta function at s = 1 should equal the rank of E(K). Recently CoatesWiles proved that when E has complex multiplication by the ring of integers in an imaginary quadratic number ﬁeld k of class number 1, and when K = Q or k then the existence of points of inﬁnite order in E(K) implies the vanishing of the HasseWeil zeta function at s = 1. No one has been able to directly relate the height pairing matrix with the zeta function. Oddly enough, if one looks at s = 2 rather than s = 1, things get easier. For one thing, instead of K0 (E) or E(K), the regulator will involve K2 (E). It turns out to be possible to: (a) write down explicitly a regulator map K2 (EC ) → C (not R!); (b) write down interesting elements in K2 (EK ) analogous to cyclotomic units and related to points of ﬁnite order on E; (c) evaluate the regulator map on the interesting elements and relate the resulting mess to L(E, s), the HasseWeil zeta function, assuming E has complex multiplication. The simplest case is when E is deﬁned over Q and has complex multiplication by the ring of integers in an imaginary quadratic ﬁeld (necessarily of class number 1 since j(E) ∈ Q generates the Hilbert class ﬁeld). In the last lecture we write down an element U ∈ K2 (E)⊗Z Q, and show (0.3.1)
κRq (U ) = L(E, 2)
where Rq is the regulator (which depends “modularly” on the choice of q = e2πiτ ) and κ is a certain constant involving a Gauss sum, the imaginary part of τ , and the conductor of E.
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4
0. INTRODUCTION
Some comments on (a). The construction is purely transcendental and we may think of E as deﬁned over C. We deﬁne a bilinear map
(0.3.2) C(E)∗ ⊗Z C(E)∗ / p∈E Z f ⊗g
/ (f )− ∗ (g)
− where (f ) denotes the divisor of f , ni (ai ) = ni (−ai ), and mj (bj ) = ni mj (ai + bj ). ni (ai ) ∗ Thus any settheoretic function F : EC → A, A an Abelian group, induces a map F ∗ : C(E)∗ ⊗ C(E)∗ → A. It is natural to look for F ∗ such that F ∗ is a Steinberg symbol, f ⊗ (1 − f ) = 0 for any f . i.e. F ∗ Such an F factors through K2 C(E) and we obtain by composition F∗ (0.3.3) K2 (E) → K2 C(E) −→ A. Actually, it is necessary to back oﬀ a step and ﬁrst consider the number ﬁeld case. This means replacing K2 (EC ) by K3 (C) and writing down an explicit recipe for the Borel regulator, which is done in Lectures 5–7. We view K3 (C) as a direct summand of the relative Kgroup K2 (P1C , {0, ∞}) which can be thought of either as K2 of a degenerate elliptic curve or as K2 with compact supports of Gm . The analogue of C(E) becomes the semilocal ring R of functions on P1C regular at 0 and ∞, and K2 C(E) is replaced by K2 (R, I), I = ideal of {0, ∞}. The group structure on P1C \ {0, ∞} enables us to deﬁne
(f )− ∗ (g) ∈ Z p∈P1C \{0,∞}
for f ∈ 1 + I, g ∈ C(P1 )∗ . The function D : P1C → R,
x
dt t 0 was ﬁrst written down by D. Wigner, who showed it was continuous and singlevalued, vanished at 0, 1, ∞, and represented an Rvalued continuous 3cohomology class for the group SL2 (C). We show (0.3.5) D (f )− ∗ (1 − f ) = 0 (0.3.4)
D(x) = arg(1 − x) log x − Im
log(1 − t)
for f ∈ 1 + I. Results of Keune on generators and relations for relative K2 imply that D induces a map K2 (R, I) → R, and we may consider the composition (also denoted D) K3 (C) → K2 (R, I) → R.
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0. INTRODUCTION
5
In Lecture 7 we do some number theoretic computations in the spirit of Kummer for cyclotomic ﬁelds, using D. Returning to the elliptic case, we write E = EC = C∗ /q Z , q = e2πiτ , Im τ > 0. Using the relation D(x−1 ) = −D(x), we show that the inﬁnite sum dfn (0.3.6) D(xq n ) = Dq (x) n∈Z
induces a continuous function Dq : E → R. In Lecture 9, we show Dq (f )− ∗ (1 − f ) = 0 for any elliptic function f , so Dq induces a map / K2 C(E) / R. K2 (E) 8 Dq
We may also consider the function J : C → R, J(x) = log x · log 1 − x. − One checks J (f ) ∗ (1 − f ) = 0 for f ∈ 1 + I, but the resulting map J : K2 (R, I) → R appears less interesting than D because K3 (C) maps to Ker J ⊂ K2 (R, I). Also the function ∞ ∞ n J(xq ) − J(x−1 q n ) (0.3.7) Jq = n=0
n=1 ∗
is welldeﬁned and continuous on C , but is not invariant under x → xq. Given elliptic functions f , g, however, we may choose liftings (f˜), (˜ g) to divisors on C∗ . It turns out that dfn Jq (f ⊗ g) = Jq (f˜)− ∗ (˜ g) is independent of liftings and Jq f ⊗ (1 − f ) = 0. The regulator map is √ dfn (0.3.8) Rq = Jq + −1Dq . Comments on (b); interesting elements in K2 (E). When E is deﬁned over K, there is an exact localization sequence
K2 (E) → K2 K(E) −−−−−−−→ K(x)∗ . tame symbol
x∈E
Suppose the points of order N , EN , are deﬁned over K, and let f and g be functions on E deﬁned over K whose divisors are supported on EN . Then one shows there is an expression {f, g}N · {fi , ci }, fi ∈ K(E)∗ , ci ∈ K ∗
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6
0. INTRODUCTION
lying in the kernel of the tame symbol. The regulator map Rq is trivial on symbols with one entry constant, so it is possible to calculate its value on such an expression without specifying the fi or the ci . Let f be the function with poles of order 1 at each nonzero point of EN and a zero of order N 2 − 1 at 0. Take g = ga to have a zero of order N at a ∈ EN and a pole oforder N at 0. The resulting element in Image K2 (E) → K2 K(E) is denoted Sa . When K is a number ﬁeld, the kernel of K2 (E) → K2 K(E) can be shown to be torsion, so Sa ∈ K2 (E)/torsion. Comments on (c): relation with L(E, 2). The idea is to imitate the following classical computation of Kummer. Let L(s, χ) =
∞ χ(n) n=1
ns
be a Dirichlet Lseries with conductor . Write 1 χ(r)e2πirk/ . r=0 −1
χ (k) = It is easy to see L(s, χ) =
−1
χ (k)
k=0
∞ e−2πink/ n=1
ns
so if χ is nontrivial we can write 1 − e−2πik/ L(1, χ) = − χ (k) log 2πi/ 1 − e k=0 −1
The point is that (1 − e−2πik/ )/(1 − e−2πi/ ) is a unit in OQ (e2πi/ ) , and such units generate a subgroup of ﬁnite index in the full group of units. Since ζQ (e2πi/ ) (s) is a product of the L(s, χ), the above formula expresses the relation between the regulator and the residue of the zeta function at s = 1. One does not see the class number; it appears as the index of the above cyclotomic units in the full group of units. The picture becomes even simpler if we consider a real quadratic extension K of Q, and let χ be its character. Then lims→1 ζK (s) = L(1, χ). The character χ is even and (1) χ (k) = χ(k)−1 χ so we ﬁnd (0.3.9)
χ(1) log U  lim ζK (s)(s − 1) = −
s→1
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0. INTRODUCTION
where
U=
k∈(Z/Z)∗
1 − e−2πik/ 1 − e−2πi/
7
χ(k)−1 ∗ ∈ OK
is the norm of a unit 1 − e−2πik/ ∈ OQ∗ (e2πi/ ) . 1 − e−2πi/ This formula should be compared with the formula (0.3.1) above.
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https://doi.org/10.1090/crmm/011/02
LECTURE 1
Tamagawa Numbers In this lecture and the next, we summarize the arguments given in [Wei61] to prove that the Tamagawa number τ (H) of the algebraic group H associated to the group of elements of reduced norm 1 in a division algebra D over a number ﬁeld k is 1. In keeping with the principle of saving time by focusing on the main ideas, we will discuss only the proof of Theorem 1.0.1. Let D∗ denote the group of invertible elements in D, and let Z ∗ ⊂ D∗ be the center. We view D∗ and Z ∗ as algebraic groups over k, and deﬁne G to be the quotient algebraic groupG = D∗ /Z ∗ . Then the Tamagawa number τ (G) = n, where n2 = [D:k]. The isogeny argument showing τ (G) = n ⇐⇒ τ (H) = 1 is given on [Wei61, pp. 48–50]. It will be omitted here. 1.1. The Tamagawa measure. Example 1.1.1. Let k be a number ﬁeld and let Ak denote the adeles of k. Deﬁne Haar measures dxν on the completions kν by dxν = dx dxν = idz ∧ d¯ z dxν = 1
kν = R kν = C kν nonArchimedean, Oν ⊂ kν ring of integers.
Oν
the Archimedean ones, For S a ﬁnite set of valuationscontaining dx deﬁnes a Haar measure on k × ν ν ν∈S ν ν∈S Oν , and hence (taking the limit over larger and larger S) a Haar measure wA on Ak . Recall k ⊂ Ak is discrete. Proposition 1.1.2. Ak /k is compact, and μk = A k /k ωA = Dk 1/2 where Dk = discrimimant k/Q. Proof. Let S = inﬁnite places of k, AS = ν∈S kν × ν ∈S / Oν . By the approximation theorem, Ak = AS + k, so Ak /k ∼ Oν = (k ⊗Q R/Ok ) × = AS /k ∩ AS ∼ ν ∈S /
9
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10
1. TAMAGAWA NUMBERS
where Ok ⊂ k is the ring of integers. This proves compactness, and shows μk = dxν . k⊗Q R /Ok ν∈S
If {aα } is a basis for Ok over Z and σβ : k → R (resp. σγ : k → C) are the distinct conjugacy classes of real (resp. complex) embeddings, 1 ≤ β ≤ r1 , 1 ≤ γ ≤ r2 , then one sets μk = det σβ (aα ), σγ (aα ), σγ (aα ) . Let M denote the matrix in parentheses. Then M t M = trk/Q (aα aα ) so μ2k = Δk .
1.2. More generally, let V be an algebraic variety of dimension n over k, x0 ∈ V a smooth point, ν a valuation on k, and ω a diﬀerential nform on V . Assume x0 and ω are deﬁned over k, and ω(x0 ) = 0. We choose local coordinates x1 , . . . , xn and deﬁne an isomorphism from a neighborhood of the origin in kνn to a (νadic) neighborhood of x0 on V (kν ) x → (x1 − x01 , . . . , xn − x0n ). We could use this isomorphism to deﬁne a measure dx1ν · · · dxnν near x on V , but this would depend upon the choice of local coordinates. The clever thing to do is to write ω = f dx1 ∧ · · · ∧ dxn near x0 and take the measure to be f ν dx1ν · · · dxnν . To check that this measure is independent of local coordinates, we can change coordinates one at a time and so reduce to considering a new set y1 , x2 , . . . , xn . We can then integrate a` la Fubini by ﬁxing values for x2 , . . . , xn and integrating ﬁrst with respect to x1 (resp. y1 ), thereby reducing to the case n = 1. Since for a ∈ kν f dx = a−1 f d(ax) we may assume y = x + a2 x2 + · · · , x0 = 0. In this case f dx = f
dx dy, dy
dy = dx, and dx/dy = 1 near 0, so we are done.
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1. TAMAGAWA NUMBERS
11
1.3. Let V be smooth of dimension n over a number ﬁeld k, and let ω be a nonvanishing nform on V also deﬁned over k. We can ﬁnd a ﬁnite set S of (bad) primes of k and a model VOk,S of V over the ring Ok,S of Sintegers, together with a nonvanishing nform ωS on VOk,S . If p ∈ / S, and Fq = Op /pOp , we can consider the sets V (Op ) and V (Fq ) of points of VOk,S with values in Op and Fq respectively. We view V (Op ) as a padic topological space. As such, it is a disjoint union of balls, and isomorphic to (p) × · · · × (p), n times
one for each point of V (Fq ). As above, ω induces a measure ωp on V (Op ). With notation as above, nonvanishing of ω means f  = 1 on the the xi ’s are local coordinates. By deﬁnition neighborhood where −1 dx = 1, so (p) dx = q , and Op (1.3.1) ωp = q −n · #V (Fq ). V (Op )
1.4. The Global Measure and Convergence Factors. A set of factors is a collection of positive real numbers {λν }, one for each place of k. These are a set of convergence factors if −1 λp ωp p
V (Op )
converges absolutely. One checks that this notion is independent of the model VOk,S chosen for Vk , and is independent of the choice of nonvanishing nform ω. Let V (Ak ) denote the adelevalued points of V (i.e. x ∈ V (Ak ) is a collection x = (xν ), xν ∈ V (kν ), xp ∈ V (Op ) for almost all ﬁnite primes p. Note that changing the model VOk,S changes V (Op ) only for a ﬁnite number of p, so V (Ak ) is independent of the choice of model.) Definition 1.4.1. Given a set of convergence factors {λν } and a nonvanishing nform ω on Vk , the Tamagawa measure (ω, {λν }) on V (Ak ) is deﬁned by passing to the limit over and larger ﬁnite larger −1 sets of places S of the product measure μ−n (λ ω ν ) on ν k ν V (kν ) × V (Oν ). ν∈S
ν ∈S /
Here μk is the volume of Ak /k as deﬁned in Proposition 1.1.2. It follows from the product formula that (ω, {λν }) = (cω, {λν }) for c ∈ k∗.
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12
1. TAMAGAWA NUMBERS
1.5. Restriction of Scalars. To understand Borel’s paper, one must understand the role of μk in the formula for the Tamagawa measure, and this involves restriction of scalars. Let K/k be a ﬁnite extension ﬁeld. Given V deﬁned over K, the restriction of scalars RK/k V is a variety over k together with a canonical identiﬁcation on T points for any kscheme T RK/k V (T ) = V (T × K). k
In particular (RK/k V ) × k¯ = k
Vσ
σ
where σ runs through embeddings K → k¯ over k, and V σ denotes the ¯ kvariety deﬁned via σ. In particular if n = dimK V and d = [K:k], dimk RK/k V = nd. The functor of points property implies (1.5.1)
V (AK ) = (RK/k V )(Ak )
(exercise!)
Given convergence factors {λw } for V over K, and a nonvanishing nform ω, it is natural to look for a nonvanishing ndform RK/k ω on RK/k V , and to expect λν = wν λw to be a set of convergence factors for RσK/k ω. The problem is that the natural diﬀerential to choose, ¯ depends on the ordering of the set of σ’s. To σ ω on (RK/k V )k √ remove this dependence, we write K = k(θ), Δ = i≤j (θσi − θσj ) for some ordering of the σ’s, and RK/k ω =
d √ n Δ ω σi . i=1 ∗
This is welldeﬁned up to a scalar in k , and is deﬁned over k. Moreover, the Tamagawa measure (RK/k ω, {λv }) is deﬁned, and under the identiﬁcation of (1.5.1) we have (ω, {λw }) = (RK/k ω, {λv }). 1.6. Algebraic Groups and Tamagawa Numbers. When V = G is an algebraic group over k, there is a nonvanishing leftinvariant nform ω unique up to scalars. For some G (e.g. G = special or projective linear group associated to a division algebra D over k) it will turn out that λν = 1 is a set of convergence factors for G. ω, (1) will be called the Tamagawa measure of G(Ak ), and the Tamagawa number τ (G) is deﬁned by τ (G) = ω, (1) . G(A k )/G(k)
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1. TAMAGAWA NUMBERS
13
Notice, in general, if {λv } is a set of convergence factors for an algebraic group G over k and ω is a leftinvariant diﬀerential, then ω, (λv ) is a left Haar measure on G(Ak ). Also, right translation by g ∈ G(k) changes ω to ψ(g)ω, ψ(g)∈ k ∗ , so the Tamagawa measure does not change (product rule) and G(A k )/G(k) makes sense.
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https://doi.org/10.1090/crmm/011/03
LECTURE 2
Tamagawa Numbers. Continued 2.1. Let D be a central division algebra over a number ﬁeld k, with [D:k] = n2 . Let D∗ denote the algebraic group over k of invertible elements of D. Let Z ∗ ⊂ D∗ be the center, so Z ∗ ∼ = Gm,k , and let G = D∗ /Z ∗ . For convenience, we will write Ik = Gm (Ak ), the id`ele group. We want to prove τ (G) = n. We will sketch the argument ﬁrst, postponing proofs. Proposition 2.1.1. A set of convergence factors for D∗ is provided by
1 λv = 1 − N (v)−1
v inﬁnite v ﬁnite, N (v) = #(residue ﬁeld Ov ).
Note that {λv } does not depend on D. Since Z = Center(D) is also a division algebra, we get measures ωD∗ and ωZ ∗ with the same convergence factors. This implies that λv = 1 is a system of convergence factors on G = D∗ /Z ∗ . We write ωG for the Tamagawa measures. Consider now the maps  
D∗ (Ak ) − → Ik − → R∗t N
where N is the reduced norm of the division algebra, and   is the id`ele module (normalized in the usual way, so k ∗  = 1). We can, if we like, view N  as a map D∗ (Ak )/D∗ (k) → R∗+ . Proposition 2.1.2. Given 0 < m ≤ M , the set of points x of D∗ (Ak )/D∗ (k) determined by m ≤ N (x) ≤ M is compact. The key fact we need is ∞ Theorem 2.1.3. Let F be a function on R+ such that the integral F (t) dt/t converges absolutely. Then 0 ∞ dt F (N (x))ωD∗ = ρk F (t) , t 0 ∗ ∗ D (A k )/D (k)
where ρk = 0 and ρk depends only on k (and not on D or n). 15
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16
2. TAMAGAWA NUMBERS. CONTINUED
Granting, for a moment, these results, the Tamagawa number computation is easy. In fact, the exact sequence 0 → Z ∗ → D∗ → G → 1 admits local sections (because Z ∗ ∼ = Gm ) so we get 0 → Z ∗ (Ak ) → D∗ (Ak ) → G(Ak ) → 1 0 → Z ∗ (k) → D∗ (k) → G(k) → 1. Given a function F on R+ as above, we can thus write F (N (x))ωD∗ = ωG F (N (yz))ωZ ∗ . D ∗ (A)/D ∗ (k)
G(A)/G(k)
Z ∗ (A)/Z ∗ (k)
(We are using here the fact that the convergence factors chosen for total space D∗ , base G, and ﬁbre Z ∗ are compatible.) By Theorem 2.1.3 ∞ dt F (N (x))ωD∗ = ρk F (t) t 0 D ∗ (A)/D ∗ (k) F (N (yz))ωZ ∗ = F (N (y) zn )ωZ ∗ Z ∗ (A)/Z ∗ (k)
Z ∗ (A)/Z ∗ (k) ∞
= ρk 0
dt ρk F (N (y)t ) = t n
∞
n
F (t) 0
dt t
Comparing these values, we ﬁnd ωG = n = τ (G). G(A)/G(k)
This completes the proof of Theorem 1.0.1. 2.2. Some Proofs. Proof of Proposition 2.1.1. We can ﬁnd a ﬁnite set of places ∗ ∗ ∼ of D∗ such that DO / S. S of k and a model DO = GLn,Op for all p ∈ p k,S By (1.3.1), it will suﬃce to show that the product 2 (2.2.1) (1 − N (p)−1 )−1 · N (p)−n · # GLn k(p) p∈S /
converges absolutely, where k(p) is the residue ﬁeld of Op . For convenience, write q = N (p). # GLn k(p) = (q n − 1)(q n − q) · · · · · (q n − q n−1 ) = q n (1 − q −1 ) · · · · · (1 − q −n ). 2
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2. TAMAGAWA NUMBERS. CONTINUED
17
Convergence of (2.2.1) follows from convergence of the zeta function of k, ζk (s), for Re s > 1. Proof of Proposition 2.1.2. Let D (without the ∗) denote the additive group of the division algebra. The map x → (x, x−1 ) identiﬁes D∗ (A) as a closed subspace of D(A) × D(A). Let C ⊂ D(A) have measure > max(M n , m−n ), C = C +(−C) = {c1 −c2  c1 , c2 ∈ C}. For a ∈ D∗ (A), the automorphisms x → ax, x → xa of D(A) have modules N (a)n (N (a) is the reduced norm; its nth power is the norm in the algebra D). Since the measure of D(A)/D(k) is 1 (because of the factor μ−n in Deﬁnition 1.4.1) we ﬁnd for m ≤ N (a) ≤ M the sets a−1 C, k Ca do not map onetoone to D(A)/D(k), i.e., there exist α, β ∈ D∗ (k) with c = aα and c = βa−1 in C . But βα = c c ∈ C 2 ∩ D∗ (k) = ﬁnite set, since D(k) is discrete and C 2 is compact. Hence (aα, α−1 a−1 ) = (c , (βα)−1 · c ) ∈ (C × (ﬁnite set) · C ) ∩ D∗ (A) = S. The set S on the right is compact, so we have a compact set S in D∗ (A) such that for all a ∈ D∗ (A), m ≤ N (a) ≤ M , there exists α ∈ D∗ (k) with aα ∈ S. 2.3. Start of Proof of Theorem 2.1.3—The Zeta Function. With notation as in Theorem 2.1.3, ωD∗ and dt/t are Haar measures on D∗ (A) and R∗+ , and N  is a homomorphism, so it follows from Proposition 2.1.2 that ∞ dt (2.3.1) F (N (x))ωD∗ = c F (t) t 0 ∗ ∗ D (A)/D (k)
for some c independent of F . Via a zeta function argument we will exhibit an F for which both sides of (2.3.1) can be calculated. On the real vector space D(k ⊗Q R) we ﬁx a positive deﬁnite quadratic form f and a polynomial function P , and deﬁne Φ0 : D(k ⊗Q R) → C Φ0 (x) = P (x)e−f (x) . Choose a model for D over Ok,S for some ﬁnite set S of ﬁnite primes, and take ΦS : D kp → C p∈S
to be any locally constant function with compact support. For p ∈ / S, deﬁne Φp to be the characteristic function of D(Op ). Then Φp Φ = Φ0 · ΦS · p∈S /
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18
2. TAMAGAWA NUMBERS. CONTINUED
is a function on D(A). Definition 2.3.1. The zeta function of D with respect to the chosen function Φ, Z Φ (s), is deﬁned by Φ Z (s) = N (x)s Φ(x)ωD∗ . D ∗ (A)
Proposition 2.3.2. Z Φ (s) is absolutely convergent for Re s > n, and Φ lim+ Z (s) = ρk Φ(x) dxA , s→n
D(A)
where dxA denotes the Tamagawa measure on D(A) (with respect to which the measure of D(A)/D(k) = 1), and ρk is a nonzero scalar independent of D and n. Proof. Change notation a bit and take for S a ﬁnite set including the inﬁnite primes such that DOp ∼ = Mn,Op (n × n matrices) and Φp = characteristic function of D(O ), for p∈ / S. With this notation, Φ = p ΦS · p∈S / Φp . We now ﬁx a translationinvariant diﬀerential ω on D∗ ω = N (x)−n dx11 ∧ dx12 ∧ · · · ∧ dxnn for some basis xij of D over Ok,S . We write dxS = μ−n dx11 ∧ dx12 ∧ · · · ∧ dxnn , k viewed as a measure on D( ν∈S kν ) = DS . For p ∈ / S, let ωp denote ∗ −1 −1 the measure on D (kp ) deﬁned by (1−N (p) ) ω. Deﬁne for Re s > n Φ N (xv )s−n ΦS (xS ) dxS ZS (s) = v 2
∗ DS v∈S
= the same integral taken over DS . N (xp )sp Φp (xp )ωp , Zp (s) = D ∗ (kp )
so Φ
Z (s) =
(2.3.2) Since
(2.3.3)
Op
−1 −1
(1 − N (v) )
ZSΦ (s)
Zp (s).
p∈S /
v∈S
dxp = 1, we ﬁnd Φ ZS (n) =
ΦS (xS ) dxS = DS
D(A k )
Φ(x) dxA k .
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2. TAMAGAWA NUMBERS. CONTINUED
19
On the other hand, since Φp is the characteristic function of D(Op ) = Mn (Op ), we ﬁnd Zp (s) = (1 − q −1 )−1  det Xs−n p (dX)p Mn (Op )∗
where Mn (Op )∗ denotes the set of matrices with nonzero determinant, and q = N (p). To compute Zp (s), write U = GLn (Op ) and decompose into cosets Mn (Op ) = U A where A denotes an element of the set of matrices ⎛ d ⎞ π 1 αij ⎜ ⎟ ... ⎜ ⎟ ⎜ ⎟ . ⎜ ⎟ . . ⎜ ⎟ ⎜ ⎟ ... ⎝ ⎠
0
π dn with π ∈ Op prime, d1 ,. . . , dn running through all ntuples of integers ≥ 0 and αij running through a complete set of representatives for Op mod π dj . We ﬁnd (using the invariance of the measure) −1 −1 s (2.3.4) Zp (s) = (1 − q )  det Ap  det X−n p (dX)p A
= (1 − q −1 )−1
= (1 − q −1 )−1
A
= (1 − q −1 )−1
UA
U
 det X−n p (dX)p
 det Asp
A
 det Asp
(dX)p U
n q −sd1 +(−s+1)d2 +···+(−s+n−1)d −1 × (1 − q ) · · · (1 − q −n ) (d ) i
= (1 − q −1 )−1
(1 − q −1 ) · · · (1 − q −n ) (1 − q −s )(1 − q −s+1 ) · · · (1 − q −s+n−1 )
so Zp (n) = (1 − q −1 )−1 . Using lims→1 (s − 1)ζk (s) = ρk = 0, ∞, together with (2.3.2), (2.3.3) and (2.3.4), we easily deduce Proposition 2.3.2. We now write A = Ak , and ﬁx a character χ : A → Circle such that x → χ(x·) identiﬁes A with its Pontryagin dual, and such that k is its own perpendicular under this pairing. For the construction of such a χ, see [Wei67, p. 66]. Let Tr : D(A) → A denote the reduced trace,
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20
2. TAMAGAWA NUMBERS. CONTINUED
and deﬁne χD : D(A) → Circle, χD (x) = χ Tr(x) . Then χD identiﬁes D(A) with its Pontryagin dual and D(k) = D(k)⊥ . The Fourier transform ψ of Φ is deﬁned by (2.3.5) ψ(y) = Φ(x)χD (xy) dxA . D(A) One shows that there exists a ﬁnite set S of places such that ψ(y) = ψS (yS ) · p∈S / ψp (yp ), where ψp = characteristic function of D(Op ), and ψS is described analogously to ΦS above. In particular, the zeta function Z ψ (s) is deﬁned. For a ∈ D∗ (A), d(ax)A = N (a)n dxA , and χD (axa−1 ) = χD (x). Substituting in (2.3.5), it follows that the Fourier transform of x → Φ(ax) is N (a)−n ψ(xa−1 ). Also we have the Poisson summation formula Φ(x) = ψ(x). (2.3.6) x∈D(k)
x∈D(k)
Sketch of proof of (2.3.6). If G is a locally compact group, G∗ the dual group, f a “nice” function on G, fˆ its Fourier transform on G∗ , then fˆ(0) = G f dμ. Apply this to G = D(k) (discrete), G∗ = D(A)/D(k), f = Φ. Note that the measure of G∗ = D(A)/D(k) is 1, so the dual measure on D(k) gives every point mass 1. Thus fˆ(0) = x∈D(k) Φ(x). On the other hand, the Fourier transform of the function F on G∗ deﬁned by F (¯a) = x∈D(k) ψ(x + a) is given by F (u)u, y dμ = u, y dμ ψ(x + u) F (y) = G∗
D(A)/D(k)
=
x∈D(k)
ψ(u)u, y dxA = Φ(−y). D(A)
Hence F(y) = f (−y), F = fˆ, and fˆ(0) = x∈D(k) ψ(x).) Deﬁne a function f on R+ by ⎧ ⎪ ⎨1 f (t) = 12 ⎪ ⎩0
if t < 1 if t = 1 if t > 1.
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2. TAMAGAWA NUMBERS. CONTINUED
21
We have f (t) + f (t−1 ) ≡ 1. Deﬁne f± : D∗ (A) → R by f+ (x) = f (N (x)) f− (x) = f (N (x)−1 ),
and set Z±Φ (s)
= D(A)∗
f± (x)N (x)s Φ(x)ωD∗ ,
so Z Φ = Z+Φ + Z−Φ . Note the integral in Z+Φ converges absolutely and uniformly for all s. We can use Poisson summation to write Φ s Z− (s) = f− (x)N (x) Φ(xa) ωD∗ D ∗ (A)/D(k)∗
f− N 
s
=
f− N 
s
=
a∈D ∗ (k)
Φ(xa) − Φ(0) ω
a∈D(k)
N (x)
−n
s−n
ψ(bx ) − Φ(0) ω −1
b∈D(k)
=
−1
f− (x)N (x) ψ(bx ) ω ∗ b∈D (k) − f− (x)N (x)s N (x)−n ψ(0) − Φ(0) ω
= D ∗ (A)
f+ (y)N (y)n−s ψ(y)ω dt f (t−1 ) ts−n ψ(0) − ts Φ(0) −c t R+
where c is the same constant as in (2.3.1). This gives Φ(0) ψ(0) ψ Φ − . Z− (s) − Z+ (n − s) = c s−n s From Proposition 2.3.2 we get ρk Φ(x)dxA = ρk ψ(0) = lim Z Φ (s) s→n = lim (s − n) Z−Φ (s) + Z+Φ (s) s→n = lim (s − n) Z−Φ (s) s→n = lim (s − n) Z−Φ (s) − Z+ψ (n − s) = cψ(0). s→n
Assuming Φ chosen so ψ(0) = 0 (easy), we get c = ρk . In particular c is independent of n and D. This completes the proof of Theorem 1.0.1.
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https://doi.org/10.1090/crmm/011/04
LECTURE 3
Continuous Cohomology 3.1. For G a Lie group and A a continuous Gmodule, it is natural to conceive of a cohomology theory H ∗ (G, A) analogous to the EilenbergMacLane theory for discrete groups but taking into account the topology of G. It turns out that there is a nice theory, which ∗ we will denote Hcont (G, A) and refer to as the continuous cohomology ∗ of G. In fact (under reasonable hypotheses) the groups Hcont (G, A) can be computed using diﬀerentiable, continuous, or even measurable cochains. For example, we will see in Lecture 7 that the imaginary part of the dilogarithm of the crossratio deﬁnes a measurable function D : SL2 (C) × · · · × SL2 (C) → R
(3.1.1)
4 times
satisfying the homogeneous cocycle identity (3.1.2) (−1)i D(x0 , . . . , xˆi , . . . , x5 ) = 0, 3 (SL2 (C), R). An even simpler example: the and hence a class in Hcont 1 real part of the logarithm deﬁnes a class in Hcont (GL1 (C), R).
3.2. Let G be a real Lie group, K ⊂ G a closed compact subgroup. Let C q (G/K, R) denote the space of C ∞ diﬀerential qforms on G/K. We topologize C q by fi dx1 ∧ · · · ∧ dxq → f dx1 ∧ · · · ∧ dxq if and only if fi → f in the C ∞ topology. G acts on G/K by left translation, and hence on C q by (gω)(x) = ω(g −1 x), and it turns out that C q is injective in a suitable category of continuous Gmodules. We can consider the continuous hypercohomology of G acting on the de Rham complex C(G/K, R). As usual there are two spectral sequences, denoted E ∗ and E, abutting to Hcont G, C • (G/K, R) : q p,q G, C p (G/K, R) E1 = Hcont (3.2.1) p p,q G, H q C • (G/K, R) E2 = Hcont Thus E1p,q = (0), p > 0, and
E10,q = H q (C • (G/K, R)G ) = H q (g, k; R) dfn
23
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24
3. CONTINUOUS COHOMOLOGY
where g, k are the Lie algebras of G and K and the relative Lie algebra cohomology H q (g, k; R) can be deﬁned to be the cohomology of the complex of leftinvariant diﬀerential forms on G/K. On the other hand, q by de Rham H q (C • (G/K, R)) ∼ = Htop (G/K, R), the cohomology of the topological space G/K. Thus (3.2.2)
p q p+q E p,q (g, k; R). 2 = Hcont. (G, Htop (G/K, R)) =⇒ H
3.3. Several Special Cases of Interest. Case I. G compact, K = G. We conclude (3.3.1)
p Hcont (G, R) = (0),
p > 0.
Case II. G compact, connected, K arbitrary. Using Case I, we conclude (3.3.2) H q (G/K, R) ∼ = H q (g, k; R). top
Case III. G arbitrary connected, K ⊂ G maximal compact. Then G/K ∼ = Euclidean space, so (3.3.3) H p (G, R) ∼ = H p (g, k; R). cont
Here are some simple examples we will use later. Example 3.3.1. G = V = ﬁnite dimensional Rvector space. We apply (3.3.3) with k = (0), to get ∗ ∗ Hcont (V, R) ∼ (V ∗ ), V ∗ = HomR (V, R). = R
Example 3.3.2. G = T ∼ · · × C∗ , a torus. We have an = C∗ × · exact sequence n times 1 n 0 → (S ) → T → Rn → 0 ∗ ∗ ∗ (T, R) ∼ (Rn , R) ∼ (Rn )∗ . and since S 1 is compact, Hcont = Hcont = R
Example 3.3.3. Let B ⊂ SL2 (C) be the subgroup of upper triangular matrices. There is an exact sequence 0 / C / B i / C∗ / 1 ∗ ∗ and Hcont (B, R) ∼ (C∗ , R). Indeed, there is a spectral sequence = Hcont p q p+q (C∗ , Hcont (C, R)) =⇒ Hcont (B, R) Hcont
α ∈ R∗ with α = 1 and consider conjugation by the matrix Fix ccα 0 0 α−1 . This acts trivially on the continuous cohomology of B, and also acts trivially on C∗ viewed as a subgroup of B. The action on C is by multiplication by α2 . It follows that the above spectral sequence degenerates as claimed.
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3. CONTINUOUS COHOMOLOGY
25
Example 3.3.4. Take G = SL2 (C), K = SU2 (C), G/K = Poincar´e upper half space. Since dim G/K = 3, the ratio of two translationinvariant 3forms is a translationinvariant function, i.e., constant, so 3 dim Hcont (G, R) ≤ 1. But the volume form on G/K is Ginvariant, and 3 the diﬀerential on C 0 (G/K, R)G is zero, so Hcont (G, R) ∼ = R. 3.4. Now let G be a semisimple connected linear algebraic group deﬁned over Q, and let g = Lie algebra. We can use the existence of ∗ compact real forms of G(C), [Hel62], to compute Hcont (G, C). Namely we can decompose g(R) = K ⊕ p where K and p are the positive and negative eigenspaces of some Cartan involution, and K is the Lie algebra of a maximal compact subgroup K ⊂ G(R). Then √ gu = K ⊕ −1 p ⊂ g(C) is the Lie algebra of a maximal compact Gu ⊂ G(C). The two pairs of RLie algebras K ⊂ gu and K ⊂ g(R) have isomorphic complexiﬁcations (namely K ⊗R C ⊂ g(C)) so (need∗ less to say, the previous discussion of Hcont (G, R) is also valid for ∗ Hcont (G, C)) (3.4.1) H ∗ (G(R), C) ∼ = H ∗ (g(R), k; C) ∼ = H ∗ (g(C), k ⊗R C; C) cont
∗ ∼ (Gu /K, C). = Htop = H ∗ (gu , k; C) ∼ (3.3.2)
Similarly, dropping K and just using gu ⊗R C ∼ = g(R) ⊗R C, we get ∗ (3.4.2) H ∗ (g(C), C) ∼ (Gu , C). = Htop To conform with Borel’s notation, we now quotient out by compact groups acting on the left and deﬁne X = K\G(R),
Xu = K\Gu .
We get a commutative square ∗
∗ Htop (Gu , C) o α H ∗ (g(C), C) (3.4.2) O O
(3.4.3)
σ∗
λ∗
∗ Htop (Xu , C) o
α ¯∗
(3.4.1)
∗ H ∗ (g(C), K ⊗ C; C) ∼ (G, C). = Hcont
The maps σ ∗ , λ∗ are the evident ones, α∗ is obtained by restricting invariant diﬀerential forms on G to Gu , and α ¯ ∗ is deﬁned analogously.
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26
3. CONTINUOUS COHOMOLOGY
Example 3.4.1. Let k be a number ﬁeld of degree d over Q, and let Gn = Rk/Q SLn,k . Thus Gn (R) = SLn (k ⊗Q R) = SLn (R)r1 × SLn (C)r2 Gn (C) = SLn (C)d . With notation as above, Kn = SOrn1 × SUrn2 and Gn,u = SUdn , so Xn = (SOn \ SLn (R))r1 × (SUn \ SLn (C))r2 Xn,u = (SOn \ SUn )r1 × SUrn2 . ∗ The cohomology of these spaces is known. E.g., Htop (SUn , Q) is an exterior algebra with one primitive generator in degree 2m + 1 for ∗ 1 ≤ m < n. Similarly Htop (SOn , Q) is an exterior algebra with generators in degree 2m + 1 for m odd (at least for n → ∞). If n is odd, H∗ (SOn , Q) → H∗ (SUn , Q). H ∗ (SOn \ SUn , Q) is also an exterior algebra, and there is an isomorphism of algebras ∗ ∗ H ∗ (SOn \ SUn , Q) ⊗ Htop (SOn , Q) → Htop (SUn , Q).
In the context of Example 3.4.1, if we write gn = Lie(Gn ), kn = Lie Kn , etc., and take n odd, we obtain a diagram of algebra isomorphisms ∗ Htop (Gn,u , C) o O
α∗
H ∗ (gn , C) O
(3.4.4) #
∗ Htop (Kn , C) ⊗ H ∗ (Xn,u , C) o
H ∗ (gn , kn ; C) ⊗ H ∗ (kn , C).
An = A∗ is a graded algebra, the indecomposables of degree n−1 i n−i dfn n are deﬁned by I n (A∗ ) = An /( i=1 A · A ). The above discussion shows ∗ Proposition 3.4.2. I 2m+1 Hcont (Gn , C) has dimension (for n large relative to m and odd ) m odd, ≥ 1 r2 dm = r1 + r2 m even, ≥ 1. If
n≥1
∗ (G∞ , C) is an exterior Moreover, passing to the limit as n → ∞, Hcont algebra with dm generators in dimension 2m + 1.
3.5. and
Note
H ∗ (gn , C) ∼ = H ∗ (gn , Q) ⊗ C ∗ ∗ (Gn,u , C) ∼ (Gn,u , Q) ⊗ C. Htop = Htop
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3. CONTINUOUS COHOMOLOGY
27
We want to study the behavior of α∗ on these Qstructures. To this end, let 2m+1 P 2m+1 (Gn,u , Q) ⊂ Htop (Gn,u , Q)
P 2m+1 (gn , Q) ⊂ H 2m+1 (gn , Q) denote the spaces of primitive elements. By the structure theory of these cohomology algebras, P 2m+1 − → I 2m+1 so dim P 2m+1 = d = [k:Q] ∼ =
(assuming m < n). Deﬁne Lm (gn , Q) and Lm (Gn,u , Q) to be the dth exterior powers of the corresponding primitive spaces, d P 2m+1 . Lm = Proposition 3.5.1. α∗ Lm (gn , Q) = (πi)d(m+1) D−(2m+1)/2 Lm (Gn,u , Q), where D denotes the discriminant of k/Q. Proof. Consider ﬁrst the multiplicative group Gm , Gm (C) = C∗ and Gm,u = S 1 . Let = Lie(Gm ). H 1 (, Q) is generated by dt/t and top ∼ top H1 (Gm (C), Z) = H1 (Gm,u , Z) by a loop γ around the origin. Since dt/t = 2πi, we have γ 1 α∗ H 1 (, Q) = (πi)Htop (Gm,u , Q).
Now let Gn = SLn,Q and replace Gm by Tn ⊂ Gn a maximal torus. Let Tn,u = Tn (C) ∩ Gn,u be the maximal compact subtorus. The ﬁbration Gn,u Tu Tu \Gn,u gives rise to a spectral sequence, whose 5term exact sequence of terms of low degree 1 1 (Gn,u ) → Htop (Tu ) − → H 2 (TU \Gn,u ) 0 → H 1 (Tu \Gn,u ) → Htop τ
2 → Htop (Gn,u )
gives a transgression isomorphism (H 1 (Gn,u ) = H 2 (Gn,u ) = (0)) 1 Htop (Tu , Q) − → H 2 (Tu \Gn,u , Q). ∼ τ
=
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28
3. CONTINUOUS COHOMOLOGY
A similar situation obtains on the level of Lie algebras, so writing h = Lie(T ) we get a commutative diagram of isomorphisms H 1 (h, Q) τ
α∗
1 / πiHtop (Tu , Q) τ
H 2 (gn , h, Q)
α∗
/ πiH 2 (Tu \Gn,u , Q).
We will use two facts without proof. Firstly, H 2m (Tu \Gn,u , Q) is generated by products of elements in H 2 [Bor53], and secondly in the spectral sequence q p+q E2p,q = H p (Tu \Gn,u , Htop (Tu , Q)) =⇒ Htop (Gn,u , Q) 2m,1 P 2m+1 (Gn,u , Q) corresponds to a subquotient E∞ of 1 H 2m (Tu \Gn,u , Htop (Tu , Q))
[Ler51]. Of course, similar facts are also true for the Lie algebras. Since α∗ is compatible with the multiplicative and spectral sequence structures, we ﬁnd α∗ H 2m (gn , h; H 1 (h, Q)) ⊆ (πi)m+1 H 2m Tu \Gn,u ; H 1 (Tu , Q) . By the second fact, we have α∗
P 2m+1 (gn , Q) α∗
⊂
⊂ 2m,1 E∞ (gn , Q)
/ P 2m+1 (Gn,u , C)
2m,1 2m,1 / (πi)m+1 E∞ (Gn,u , Q) ⊂ E∞ (Gn,u , C)
so
2m,1 (Gn,u , Q) ∩ P 2m+1 (Gn,u , C) (3.5.1) α∗ P 2m+1 (gn , Q) ⊂ (πi)m+1 E∞ = (πi)m+1 P 2m+1 (Gn,u , Q). Now consider the group Gn = Rk/Q SLn,k . If ω2m+1 is a generator of P 2m+1 (gn , Q), we have seen that (3.5.2) Rk/Q ω2m+1 = D−(2m+1)/2 σ ω2m+1 σ
is a generator for Lm (gn , Q) =
d
P 2m+1 (gn , Q).
denotes the (Here σ runs through all embeddings of k → C, and σ ω2m+1 obvious element in the Lie algebra cohomology of Gn,C = σ σ SLn,k .) The proposition follows from (3.5.1) and (3.5.2).
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https://doi.org/10.1090/crmm/011/05
LECTURE 4
A Theorem of Borel and its Reformulation 4.1. Let Gn be as before (to ﬁx ideas, we stick to Gn , though the setup is valid more generally). Let Γn ⊂ Gn (Q) be some torsionfree arithmetic subgroup. We have a commutative square (4.1.1) β∗
H ∗ (gn , C) O
/ H ∗ (Gn (R)/Γn , C) O σ ∗
λ∗ ∗ Hcont (Gn (R), C) ∼ = H ∗ (gn , Kn ; C)
β¯∗
/ H ∗ (Xn /Γn , C) = H ∗ (Γn , C)
where β ∗ comes from interpreting the Lie algebra cohomology as the cohomology of rightinvariant forms on Gn . Such a form on Gn (R) descends to a form on Gn (R)/Γn . Similarly, classes in H ∗ (gn , Kn ; C) arise from rightinvariant forms which are biinvariant under Kn , hence descend to forms on Xn /Γn = Kn \Gn (R)/Γn . Viewed as a map ∗ (Gn (R), C) → H ∗ (Γn , C), β¯∗ : Hcont
β¯∗ simply restricts continuous cohomology classes on Gn (R) to the subgroup Γn . At this point we need a very diﬃcult theorem of Borel [Bor74]. Theorem 4.1.1. For m < (n − 1)/4, the map m β¯∗ : Hcont (Gn (R), C) → H m (Γn , C)
is an isomorphism for any arithmetic subgroup of Gn (Q). We will have nothing to say about the proof of this result. Corollary 4.1.2. With hypotheses as above, β¯∗ : H m (gn , C) → H m (Gn (R)/Γn , C) is also an isomorphism. Proof. If Γn is torsionfree, Γn ∩ Kn = (1) and we have a spectral sequence q H p Γn , Htop (Kn , C) =⇒ H p+q (Gn (R)/Γn , C), 29
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30
4. A THEOREM OF BOREL AND ITS REFORMULATION
which will be isomorphic to the spectral sequence H p gn , kn ; H q (kn , C) =⇒ H p+q (gn , C) for p < (n − 1)/4. For Γn arbitrary we can ﬁnd Γ ⊂ Γn torsionfree, so we get / H m (Γ , C) H m (Γn , C) Qh QQ Q QQQ QQQ QQQ β¯∗
β¯∗
m Hcont (Gn (R), C),
whence β¯∗ is an isomorphism.
4.2. We would like to study the behavior of the Qstructure on H (gn , C) and H m (Gn (R)/Γn , C) under β ∗ . For convenience, let Yn = Gn (R)/Γn . Recall Lm (gn , Q) denotes the dth exterior power of the subspace of primitive elements in H 2m+1 (gn , Q). Let Lm (Yn , Q) denote the dth exterior power of the quotient space of indecomposables of H 2m+1 (Yn , Q). For n > 4d(2m + 1) + 1, m
∼ =
→ Lm (Yn , Q) ⊗ C. β ∗ : Lm (gn , Q) ⊗ C − Theorem 4.2.1. With the above hypotheses, β ∗ Lm (gn , Q) = ir2 ζk (m + 1)Lm (Yn , Q). Proof. Granted that β ∗ is an isomorphism, the assertion is independent of which arithmetic Γn we choose. We may assume given a family of torsionfree Γq , q ≥ n, such that Γq ∩ Gr = Γr , r ≤ q. (We view Gr ⊂ Gq in the usual way.) We may therefore stabilize and take n as large as we please. We take n > 4d(m+1)2 , odd, and Γn torsionfree. Suppose β ∗ Lj (gn , Q) = sj Lj (Yn , Q) for some sj ∈ C, 1 ≤ j ≤ m. Recall from Lecture 3 we have generators Rk/Q ω2s+1 for Ls (gn , Q). Deﬁne j ηj = Rk/Q ω2s+1 . s=1
A simple argument in linear algebra (omitted) shows that β ∗ (ηj ) ∈ s1 · · · sj H d(j
2 +2j)
(Yn , Q).
If we exhibit compact subvarieties Zj ⊂ Yn of dimension d(j 2 +2j) such that j ∗ r2 (j 2 +2j) 0 = β (ηj ) ∈ i ζk (s + 1) · Q, 1 ≤ j ≤ m, Zj
s=1
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4. A THEOREM OF BOREL AND ITS REFORMULATION
31
then using that ir2 ∈ ir2 (2j+1) Q, it follows that sj ∈ ir2 ζk (j + 1)Q as desired. Let D be a division algebra of dimension (j + 1)2 over k and trivial at the archimedean places. Let H ⊂ D∗ be the subgroup of elements of reduced norm 1. View H as an algebraic group over k, and let H = Rk/Q H . We know [Wei61] that H (Ak )/H (k) is compact. Let Af ⊂ Ak denote the ﬁnite adeles, and let U ⊂ H (Af ) denote some nonempty open compact subset. By the strong approximation theorem H (Ak ) = H (k ⊗ R)U H (k). Let T = U · H (k ⊗ R) ∩ H (k). T is an arithmetic subgroup, and we have a ﬁbration U → H (Ak )/H (k) → H (k ⊗ R)/T. This implies that H(R)/(any arithmetic subgroup) is compact. We embed H → SLn,k and hence H → Gn by the regular repre2 sentation of H on D ∼ = k (j+1) , and take the subvariety Zj = H(R)/(H(R) ∩ Γn ). We have dim Zj = d(j + 1)2 − d = d(j 2 + 2j). Note that β ∗ (ηj )Zj is a form of maximal dimension. To see that it is nontrivial, it suﬃces to look on the level of Lie algebras, i.e., invariant diﬀerential forms. Since the derivative in this complex is zero, it suﬃces to show ηj hj = 0 where hj = Lie(H). Let ηj = js=1 ω2s+1 on gn . It suﬃces to show ηj hj = 0, hj = Lie H . We might as well work over C, and H (C) ∼ = SLj+1 (C). The embedding in SLn is via the regular representation, which is conjugate to j + 1 copies of the standard representation so we can think of the embedding as ρ : SLj+1 → SLn ⎛ ⎜I ⎜ ρ(A) = ⎜ ⎜ ⎝
j+1
⎞
...
⎟ ⎟ ⎟ ⎟ ⎠
A A
We can factor this embedding SLj+1
(f1 ,...,fj+1 )
/ (SLn )j+1
multiply
/ SLn
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32
4. A THEOREM OF BOREL AND ITS REFORMULATION
% $I A with A in the ith place. Since the ω2s+1 are where fi (A) = I primitive, we have multiply∗ (ω2s+1 )
j+1
∼
cohomologous
i ω2s+1
i=1
∗
so ρ (ω2s+1 ) ∼ (j + 1)ω2s+1 on SLj+1 . Finally ρ∗ (ηj ) ∼ (j + 1)j ηj on SLj+1 and this is nontrivial. Since we only wish to calculate Zj ρ∗ (ηj ) up to a rational factor, we are free to replace β ∗ (ηj )Zj by any volume form deﬁned over Q. In other words, we take on H some j 2 + 2jform ω which is nonzero, translationinvariant, and deﬁned over k. We deﬁne 2 ω = D−(j +2j)/2 σ ω σ
on H. Look at the ﬁbration as above U → H(AQ )/H(Q) → H(R)/T. Let ωf and ω∞ denote the measures on U and H(R)/T induced by ω. 2 Notice that ω∞ is obtained by integration with respect to ir2 (j +2j) ω on H(R)/T , the factor of i arising from the normalization dxν = i dz ∧ d¯ z in Example 1.1.1. Finally, using (1.3.1) and 2 # SLj+1 k(p) = q (j+1) −1 (1 − q −2 ) · · · (1 − q −j−1 ) we ﬁnd
ωf ∈ ζk (2)−1 · · · ζk (j + 1)−1 · Q.
U
Since
ωf · U
ω∈Q
ω∞ = H(R )/T
H(A Q )/H(Q )
we conclude
ω∞ ∈ ζk (2) · · · ζk (j + 1)Q H(R )/T
and
ω ∈ ir2 (j
2 +2j)
ζk (2) · · · ζk (j + 1)Q.
H(R )/T
This ﬁnishes the proof of Theorem 4.2.1.
4.3. It remains to relate this computation to the continuous cohomology, and ﬁnally to the Ktheory. We combine (3.4.4) and (4.1.1)
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4. A THEOREM OF BOREL AND ITS REFORMULATION ∗ Htop (Gn,u , C) o O
∗ Htop (Kn , C) ⊗ H ∗ (Xn,u , C) O id⊗α ¯∗
α∗
∼ =
≈
(4.3.1)
≈
33
H ∗ (gn , C) o
≈
∗ Htop (Kn , C) ⊗ H ∗ (gn , Kn ; C) id⊗β¯∗
∼ =
≈
β∗
∗ H (Gn (R)/Γn , C) o
≈
∗ (Kn , C) ⊗ H ∗ (Xm /Γn , C) Htop
Let μ = β ∗ α∗ −1 , μ ¯ = β¯∗ α ¯ ∗−1 . By Proposition 3.5.1 and Theorem 4.2.1, (4.3.2) μLm (Gn,u , Q) = π −d(m+1) id(m+1)+r2 D1/2 ζk (m + 1)Lm (Yn , Q). We have also isomorphisms H ∗ (gn , Kn ; C) ∼ = H ∗ (Xn /Γn , C) ∼ =
∼ =
∗ Hcont (Gn (R), C)
/ H ∗ (Γn , C)
Viewing μ ¯ as a map H ∗ (Xn,u , C) → H ∗ (Γn , C), and writing I 2m+1 (Γn , Q) for the quotient space of indecomposable elements, dm Lm (Γn , Q) = I 2m+1 (Γn , Q) (dm as in Proposition 3.4.2), we ﬁnd from (4.3.1) (4.3.3)
μ ¯Lm (Xn,u , Q) = (πi)−d(m+1) D1/2 ir2 ζk (m + 1)Lm (Γn , Q).
Remark 4.3.1. For k totally real and m odd, dm = 0 so I 2m+1 (Kn , Q) ∼ = I 2m+1 (Gn,u , Q). In this case μ is deﬁned over Q, so π −d(m+1) D1/2 ζk (m + 1) ∈ Q. The map μ ¯ : H ∗ (Xn,u , C) → H ∗ (Γn , C) is not compatible with the real structures. Indeed, if gn (R) = Kn ⊕ pn , the tangent space to Xn,u is ipn , whereas that of Xn is pn . It follows that the map jΓ = im μ ¯ in dimension m is compatible with the real structure. Rewriting Borel’s result with jΓ instead of μ ¯, jΓ Lm (Xn,u , Q) = π −d(m+1) D1/2 ζk (m + 1)Lm (Γn , Q). Let Γn = SLn (Ok ), Γ = SLn (Ok ). Recall that Kp (Ok ) is the pth + homotopy group of an Hspace BGL(O with the same homology as k) BGL(Ok ) . There is, therefore, a Hurewitz map (4.3.4)
Kp (Ok ) → Hp (Γ, Z)
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34
4. A THEOREM OF BOREL AND ITS REFORMULATION
and Kp (Ok ) ⊗ Q is identiﬁed with the dual of I p (Γ, Q). We can restate Borel’s result as follows: Theorem 4.3.2. K2m (Ok ) is torsion, and K2m+1 (Ok ) has rank dm . Moreover, the dual of the map jΓ deﬁned above (regulator map) embeds K2m+1 (Ok )/(torsion) as a lattice of maximal rank in the primitive homology I2m+1 (Xn,u , R). The volume of this lattice, computed with respect to a basis of I2m+1 (Xn,u , Q) is a rational multiple of π −d(m+1) D1/2 ζk (m + 1). Remark 4.3.3. The functional equation for ζk shows lim ζk (s)(s + m)−dm ∈ π −d(m+1)+dm D1/2 ζk (m + 1)Q
s→−m
so the volume of the above lattice is a rational multiple of π −dm lim ζk (s)(s + m)dm . s→−m
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https://doi.org/10.1090/crmm/011/06
LECTURE 5
The Regulator Map. I 5.1. In the next three lectures, we will write down explicit formulas for the regulator map D : K3 (C) → R. These formulas will subsequently be generalized to the case of an elliptic curve over C. We show in Lecture 7 that on the level of cohomology D gives a 3 measurable 3cocycle on SL2 (C), whose class in Hcont (SL2 (C), R) is 2 × (class of volume form on SU2 \ SL2 (C)). 3 Also in Lecture 7 for k = Q(ζ), ζ = 1 prime cyclotomic, we will associate to each ζ i an element [ζ i ] ∈ K3 (k)/torsion. We will show [ζ i ]+ [ζ −i ] = 0 and we will calculate explicitly the regulator associated to the basis [ζ], [ζ 2 ], . . . , [ζ (ell−1)/2 ] of K3 (k) ⊗ Q, comparing it to ζk (2). (Note that unlike the classical situation for units, one has K3 (Ok ) K3 (k) with torsion kernel so “integrality” plays no role.) The calculation will also be later generalized to the context of elliptic curves. 5.2. We begin by recalling some algebraic Ktheory which will be useful. Let C be a smooth projective curve deﬁned over an algebraically closed ﬁeld k. Let T ⊂ C be a ﬁnite set of points (possibly empty). K∗ (C) (resp. K∗ (T )) will denote the homotopy groups of the classifying space of the Quillen category of locallyfree sheaves on C (resp. on T = T Spec(k)). K∗ (C) = π∗+1 BQC K∗ (T ) = π∗+1 BQT . K∗ (C, T ) will denote the homotopy of the ﬁbre FC,T of the map BQC → BQT , K∗ (C, T ) = π∗+1 FC,T . Similarly we write CT for the spectrum of the semilocal ring of functions on C regular at all points of T , and deﬁne K∗ (CT ) and K∗ (CT , T ). 35
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36
We deﬁne
5. THE REGULATOR MAP. I
restrict dfn SK∗ (C, T ) = Ker K∗ (C, T ) −−−−→ K∗ (CT , T ) .
Proposition 5.2.1. There is a relative localization sequence & · · · → Kn (C, T ) → Kn (CT , T ) → Kn−1 k(p) → Kn−1 (C, T ) → · · · p∈C−T
Of particular interest to us is the following special case: Corollary 5.2.2. There is an exact sequence & k ∗ → SK1 (C, T ) → 0 0 → K2 (C, T )/SK2 (C, T ) → K2 (CT , T ) → C−T
The proposition follows from a more general result. Proposition 5.2.3. Let X be a regular noetherian scheme, and let Y , Z ⊂ X be closed subschemes. Assume Y ∩ Z = ∅. Then there is a long exact localization sequence · · · → Kn (X, Z) → Kn (X − Y, Z) → Kn−1 (Y ) → Kn−1 (X, Z) → · · ·
where Kn (Y ) denotes the Ktheory of the category of coherent OY modules. Proof. By a theorem of Quillen, the ﬁbre HX,Y of BQX → BQX−Y has homotopy groups K∗ (Y ). We consider the diagram of ﬁbrations FX−Y,Z O
/ BQX−Y O
/ BQZ
FX,Z
/ BQX O
/ BQZ O
HX,Y
/ ∗.
and deduce a ﬁbration HX,Y → FX,Z → FX−Y,Z . The corresponding homotopy sequence is the desired one. Example 5.2.4. Suppose T = ∅. By a theorem of Quillen, there is a resolution of sheaves
0 → K 2C → iη∗ K2 k(C) → ip∗ k(p)∗ → 0 p∈C
where the left hand sheaf is the Zariski sheaf associated to K2 on C, the middle is the constant sheaf with value K2 k(C) , and the right
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5. THE REGULATOR MAP. I
37
hand is the skyscraper sheaf with stalk k(p)∗ at p. Taking cohomology, one ﬁnds tame 0 → Γ(C, K 2 ) → K2 k(C) −−→ k(p)∗ → H 1 (C, K 2 ) → 0. p
Comparison with Corollary 5.2.2 shows SK1 (C) ∼ = H 1 (C, K 2 ) K2 (C)/SK2 (C) ∼ = Γ(C, K 2 ). In fact we have similar isomorphisms for all the Ki . Example 5.2.5. Let C = P1k , T = {0, ∞}. By yet another fundamental result of Quillen Kn (P1 ) ∼ = Kn (k) ⊗Z K0 (P1 ) k
∼ = Kn (k) · [OP1 ] ⊕ Kn (k) · [k(point)].
The restriction map Kn (P1k ) → Kn ({0, ∞}) ∼ = Kn (k)⊕Kn (k) is zero on Kn (k) · [k(point)] and maps the factor Kn (k) · [OP1 ] diagonally. From this one deduces an isomorphism (5.2.1) Kn (P1 , {0, ∞}) ∼ = Kn+1 (k) ⊕ Kn (k) k
as follows: the sequence Kn+1 (P1 ) → Kn+1 ({0, ∞}) → Kn (P1 , {0, ∞}) → Kn (P1 ) → Kn ({0, ∞}) yields 0 → Kn+1 (k) → Kn (P1 , {0, ∞}) → Kn (k) → 0. But K∗ (P1 , {0, ∞}) will have a K∗ module structure, so we can split the above by mapping α ∈ Kn (k) to α · x, where x ∈ K0 (P1 , {0, ∞}) is the class of the residue ﬁeld k(1) at the point 1 ∈ P1 \ {0, ∞}. We now analyze the localization sequence multlinegap0pt
(5.2.2) · · · → Kn (k) → Kn (P1 , {0, ∞})
p∈P1 \{0,∞} Kn−1 (k) → · · · . → Kn (P1{0,∞} , {0, ∞}) → For n = 0, the map
K0 k(p) ∼ = Z → K0 (P1 , {0, ∞})
◦ Z denote the kernel of the sends 1 to the class of k(p). Letting “degree map”
Z → Z, p∈P1 \{0,∞}
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38
5. THE REGULATOR MAP. I
◦ we obtain from (5.2.1) a map ρ : Z → k ∗ which can be described
◦ as follows: given (. . . ap . . . ) ∈ Z there exists a function f on P1 such that (f ) = ap (p). Then ρ(. . . ap . . . ) = f (0)f (∞)−1 ∈ k ∗ . Clearly ρ is surjective. For arbitrary n we use the compatibility of the localization sequence with the K∗ (k)module structure to deduce 1 Kn (k) → Kn (P , {0, ∞}) Coker P1 \{0,∞}
multiply ∼ = Coker Kn (k) ⊗ k ∗ −−−−→ Kn+1 (k) dfn
= Kn+1 (k)indecomposable .
We now go back to the localization sequence and consider the case n = 2. Since K2 (k) is known to be generated by symbols K2 (k)ind = (0), the analogue of Example 5.2.4 becomes (5.2.3) 0 → K3 (k)ind → K2 (P1{0,∞} , {0, ∞})
tame ∗ −−→ k(p) → K2 (k) ⊕ k ∗ → 0. p=0,∞
◦
k(p)∗ denote the kernel of the obvious norm mapFinally, let
ping k(p)∗ → k ∗ both for Example 5.2.4 (5.2.3). In both cases
and ◦ the image of the tame mapping lands in k(p)∗ , so we get (5.2.4) 0 → Γ(C, K 2 ) → K2 k(C)
◦ tame −−→ k(p)∗ → H 1 (C, K 2 )◦ → 0 p∈C
(5.2.5)
0 → K3 (k)ind → K2 (P1{0,∞} , {0, ∞})
◦ tame −−→ k(p)∗ → K2 (k) → 0. p=0,∞
We stress again the analogy between K2n+1 (k)ind and dfn K2n (k)decomposable = Image K2n−1 (k) ⊗ k ∗ → K2n (k) on the one hand, and Γ(C, K 2n ) and SK2n−1 (C) = H 1 (C, K 2n ) on the other hand.
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5. THE REGULATOR MAP. I
39
5.3. To simplify notation, let R denote the semilocal ring at 0 and ∞ on P1 , and let I ⊂ R be the ideal of {0, ∞}. When we want to specify a particular ground ﬁeld k, we write Rk , Ik . For now, we take k = C. We deﬁne a map
x∈P1 Z = x∈C ∗ Z (5.3.1) (1 + I)∗ ⊗Z k(P1 )∗ / x=0,∞
f ⊗g
/ (f − ) ∗ (g)
as follows: Let t be the standard parameter on P1 (t(a) = a, a = 0, 1, ∞). Since f (0) = f (∞) = 1, we can write f= (t − αi )di , di = 0, αidi = 1. Let f − = (t − αi−1 )di . Write g = c (t − βj )ej , c ∈ C, βj ∈ C. e=− ej = ord∞ g. Finally let (f ) =
di (αi ),
(f − ) =
di (αi−1 )
denote the divisors of these functions. We take (5.3.2) (f − ) ∗ (g) = di ej (αi−1 βj ) + e(f ) + e(f − ). i,j βj =0
We remark now for future use that there is a simpler analogue of this construction for elliptic curves E. Let k(E) denote the function ﬁeld of E (over some algebraically closed ﬁeld k) and deﬁne
(5.3.3) k(E)∗ ⊗ k(E)∗ / p∈E Z / (f )− ∗ (g) f ⊗g where if (f ) = di (ai ), (g) = ej (bj ) we have (5.3.4) (f )− = di (−ai ), (f )− ∗ (g) = di ej (bj − ai ). From work of Keune (see below) and from the wellknown presentation of K2 of a ﬁeld due to Matsumoto, we are led to consider the symbols f ⊗ (1 − f ) ∈ (1 + I)∗ ⊗ C(P1 )∗
for f ∈ 1 + I, f ≡ 1
f ⊗ (1 − f ) ∈ C(E)∗ ⊗ C(E)∗
for f ∈ C(E)∗ , f ≡ 1.
We suppose now given a function D : C∗ → R (resp. DE : E → R).
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40
5. THE REGULATOR MAP. I
Definition 5.3.1. The
function D (resp. DE ) induces a homomorphism C ∗ Z → R (resp. E Z → R). D (resp. DE ) is called a relative Steinberg function (resp. Steinberg function) for {0, ∞} ⊂ P1C (resp. for E) if D (f )− ∗ (1 − f ) = 0, all f ∈ 1 + I, f ≡ 1 (resp. D (f )− ∗ (1 − f ) = 0, all f ∈ C(E)∗ , f ≡ 1. Proposition 5.3.2. Let R ⊂ (1 + I)∗ ⊗ C(P1 )∗
(resp. RE ⊂ C(E)∗ ⊗ C(E)∗ )
denote the subgroup generated by the symbols f ⊗ (1 − f ). Then there is a natural homomorphism K2 (R, I) →
(1 + I)∗ ⊗ C(P1 )∗ R
(resp. an isomorphism C(E)∗ ⊗ C(E)∗ ). K2 C(E) → RE Corollary 5.3.3. A relative Steinberg function (resp. a Steinberg function on E) induces a homomorphism K2 (R, I) → R (resp. K2 C(E) → R). Proof of Proposition 5.3.2. The assertion for E is exactly the wellknown Matsumoto presentation for K2 of a ﬁeld. In the relative case we use a presentation due to Keune [Keu78] and valid for K2 (R, I) for any semilocal ring R and any ideal I in the Jacobson radical. Generators are elements a, b with a ∈ R, b ∈ I , or a ∈ I, b ∈ R. (Following a suggestion of Stienstra, we change Keune’s notation slightly so our a, b = Keune’s −a, b.) Relations are (5.3.5)
a, b + b, a = 0
(5.3.6)
a, b + a, c = a, b + c − abc
(5.3.7)
a, bc = ab, c + ac, b
where all symbols are assumed to make sense, i.e., to have at least one element in I. We map (1 + I)∗ ⊗ C(P1 )∗ R a, b → (1 − ab) ⊗ b ≡ (1 − ab) ⊗ a−1
K2 (R, I) →
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5. THE REGULATOR MAP. I
41
so a, b + b, a → (1 − ab) ⊗ ab ≡ 1 ⊗ 1 a, b + a, c → (1 − ab) ⊗ b + (1 − ac) ⊗ c ≡ (1 − ab) ⊗ a−1 + (1 − ac) ⊗ a−1 = (1 − ab − ac + a2 bc) ⊗ a−1 ≡ (1 − ab − ac + a2 bc) ⊗ (+b + c − abc) = Image of a, b + c − abc a, bc → (1 − abc) ⊗ bc = (1 − abc) ⊗ b + (1 − abc) ⊗ c = Image of ac, b + ab, c. Remark 5.3.4. It seems likely that (1 + I)∗ ⊗ C(P1 )∗ K2 (R, I) ∼ = R but I have not pursued this point.
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https://doi.org/10.1090/crmm/011/07
LECTURE 6
The Dilogarithm Function 6.1. The purpose of this lecture is to introduce the dilogarithm function and to derive from it a relative Steinberg function D(x), following [Blo78]. For an Abelian group A, A˜ will denote A/torsion. We have an exact sequence (6.1.1)
exp(2πi·)⊗Id 1⊗Id ˜∗ − 0→C −→ C ⊗ C∗ −−−−−−−→ C∗ ⊗Z C∗ → 0
obtained by tensoring the standard exponential sequence exp(2πi·)
0→Z− → C −−−−−→ C∗ to0 1
with C∗ . We have also for any ﬁeld F an exact sequence (6.1.2)
→ F ∗ ⊗Z F ∗ − → K2 (F ) → 0 0 → B(F ) → A(F ) − λ
S
where A(F ) = free Abelian group on generators [f ], f ∈ F , f = 0, 1, and λ[f ] = (1 − f ) ⊗ f . Of course B(f ) = Ker λ. Lemma 6.1.1. For a ∈ C \ {0, 1}, the expression ( ' 1 dfn log(1 − a) ⊗ a [a] = 2πi ( ' dt −1 a ∈ C ⊗Z C ∗ log(1 − t) + 1 ⊗ exp 2πi 0 t is welldeﬁned independent of the choices of branches for the multivalued functions, so we get a commutative triangle
(6.1.3)
s ss ss s s sy s
A(C)
C ⊗Z C ∗
JJJ JJJλ JJJ J%
exp(2πi·)⊗Id
/ C ∗ ⊗ C∗
Proof. The assertion of the lemma should be interpreted as fola lows: a value for the functions log(1 − a) and 0 log(1 − t)dt/t, and hence for [a], is determined by the choice of a path γ from 0 to a. The 43
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44
6. THE DILOGARITHM FUNCTION
expression [a] is independent of γ. To see this, the key conﬁguration to be considered looks like ta
6
2
t
0

1
t
1
C
where C denotes a small circle about 1, and the two paths are γ = 1 +2 and γ = 1 + C + 2 . We have ( ( ' 1 1 log(1 − a) ⊗ a log(1 − a) ⊗ a − =1⊗a 2πi 2πi eval. eval. via γ via γ ( ( ' a ' a −1 dt −1 dt 1 ⊗ exp log(1−t) − 1 ⊗ exp log(1−t) 2πi 0 t eval. 2πi 0 t eval. via γ via γ a −1 dt as C → 0 = 1 ⊗ a−1 . −−−−−→ 1 ⊗ exp 2πi 2πi 0 t '
One must still consider possible problems at 0 due to the fact that log(1 − t) dt/t has a residue at 0 for a nonprincipal branch of the logarithm. However if C is a small circle about 0 and log(1 − t) = 2nπi + principal branch log(1 − t) for t inside C we get dt −1 dt = exp −n =1 exp log(1 − t) 2πi C t C t
so these problems disappear as well. Corollary 6.1.2. The expression D(x) = arg(1 − x) log x − Im 0
x
dt log(1 − t) t
gives a welldeﬁned function C → R.
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6. THE DILOGARITHM FUNCTION
45
Proof. Consider the diagram (deﬁning Φ) 0
/ B(C)
0
/C )∗
/ A(C)
λ
/ C ∗ ⊗ C∗
Φ
/ C ⊗ C∗
(6.1.4)
/ C ∗ ⊗ C∗
(real part)⊗Id
R ⊗ C∗
log  
Id ⊗ log 
Ro
multiply
R⊗R
The composition
(multiply) ◦ (Id ⊗ log  ) ◦ (real part) ⊗ Id ◦ : A(C) → R
is easily seen to be [a] → D(a), so D is welldeﬁned on C \ {0, 1}. Taking D(0) = D(1) = 0 gives a continuous extension to all of C. 6.2. This construction has an important rigidity property. To formulate it, let A denote the ring of holomorphic functions, say, on some ﬁxed disk about 0 in C. Let A0 ⊂ A denote the subset of functions a(z) such that a(z) and 1 − a(z) ∈ A∗ . Let A(A) = free Abelian group on generators [a(z)], a(z) ∈ A0 , and deﬁne B(A) by the exact sequence 0 → B(A) → A(A) → A∗ ⊗Z A∗ λ[a(z)] = 1 − a(z) ⊗ a(z). λ
Let Ω1A denote the Amodule of holomorphic 1forms on the disk, and consider the commutative diagram B(A)
/ A(A)
ΦA
A˜∗
EE EE EE dlog EE"
1⊗Id
Ω1A
A
/ A ⊗Z A∗
v vv vv v vv vz v Id ⊗ dlog
where ΦA and A are deﬁned precisely as Φ, above, and dfn db (Id ⊗ dlog)(a ⊗ b) = a . b
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46
6. THE DILOGARITHM FUNCTION
But (Id ⊗ dlog)A ([a(z)]) *+ −1 a(z) 1 dt log(1−a)⊗a + 1⊗ exp = (Id ⊗ dlog) log(1−t) 2πi 2πi 0 t =
a (z) a (z) 1 1 log 1 − a(z) − log 1 − a(z) = 0. 2πi a(z) 2πi a(z)
This implies:
) ∗ ⊂ A˜∗ . Lemma 6.2.1. With notation as above, ΦA B(A) ⊆ C
D(X) , be as in Corollary ni 6.1.2 and∗suppose Corollary 6.2.2. Let 1 − ai (z) ⊗ ai (z) = e in A ⊗ A∗ ). ni [ai (z)] ∈ B(A) (i.e., Then ni D ai (z) ≡ const. Let fz denote an analytic family of meromorphic functions on P1 parametrized by z in some small disk about 0. We assume that neither fz nor 1 − fz has multiple roots for any z, and that moreover fz (0) ≡ fz (∞) ≡ 0. We may then write fz = c(z)
e t − βj (z) j j∈J
d t − αi (z) i 1 − fz =
with
di = 0,
αi (z)di = 1
i∈I
where t is the standard parameter on P1 , and di , ej = ±1. Note that either αi (z)−1 βj (z) ≡ 1, or αi (z)−1 βj (z) = 1 for any z. The expression (1−fz )− ∗(fz ) =
i,j αi =βj =0
with e = −
di ej [αi (z)−1 βj (z)]+e
di ([αi (z)]+[αi (z)−1 ]),
i
ej therefore makes sense in A(A) as long as no αi (z) = 1.
Lemma 6.2.3. Let fz be as above. Then (1 − fz )− ∗ (fz ) ∈ B(A). Proof. Let I, J be the index sets for the roots of 1 − fz , fz respectively. Let K ⊂ I, J be the common set of poles. For k ∈ I we have αk (z) = βk (z), dk = ek = −1. Write I = K J , J = K J . Then
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6. THE DILOGARITHM FUNCTION
47
λ (1 − fz )− ∗ (fz ) (1 − αi−1 βj )di ej ⊗ αi−1 (1 − αi−1 βj )di ej ⊗ βj = i∈I j∈J
×
(1 −
i∈I,j∈J βj =0 −1 dk ej αk βj ) ⊗ αk−1
k∈K,j∈J j=k
(1 − αi−1 αk )di dk ⊗ αk k∈K,i∈I i=k
×
−1
e (−αi )di ⊗ αi i∈I
= (−α) ⊗ α.) We (We use here that (1 − α) ⊗ α (1 − α−1 ) ⊗ α evaluate these products (6.2.1) ((1 − αi−1 βj )di ej ⊗ αi−1 ) i∈I (αi − βj )di ej ⊗ αi−1 j∈J (αidi e ⊗ αi−1 ) · = i∈I
=
i∈I j∈J
(αidi e ⊗ αi−1 )
i∈I
(c ⊗ αIdi )
i∈I
because for i ∈ I , f (αi ) = 1 so (αi − βj )di ej = c−di f (αi )di = c−di . j∈J
Next (6.2.2)
−1 di ej di ej (1 − αi βj ) ⊗ βj ) = (βj − αi ) ⊗ βj = 1. i∈I,j∈J βj =0
Also (6.2.3)
j∈J βj =0
i∈I
(1 − αk−1 βj )dk ej ⊗ αk−1 (1 − αi−1 αk )di dk ⊗ αk k∈K,j∈J j=k
=
= =
k∈K,i∈I i=k
+ −di dk di dk i∈I (−αi ) i∈I (αk − αi ) i=k i=k ⊗ αk d (e+dk ) d e αk k j∈J (αk − βj ) k j k∈K j=k d / . (−αk )d2k 1 − fz (x) k dk ⊗ αk · c lim dk e+d2k x→αk f z (x) α k∈K d ek (αkk ⊗ αk−1 )(c ⊗ αkdk ). k∈K
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48
6. THE DILOGARITHM FUNCTION
Combining (6.2.1)–(6.2.3) yields λ (1 − fz )− ∗ (fz ) = 1.
Theorem 6.2.4. The function D(x) in Corollary 6.1.2 is a relative Steinberg function (cf. Deﬁnition 5.3.1). Proof. Given f , g ∈ I, one can ﬁnd an analytic family fz with f0 = f , f1 = g. Moreover one can arrange that except for a ﬁnite number of values of z, say z1 , . . . , zn , the functions fz , 1 − fz have no multiple zeros or poles, and no zero or pole of 1 − fz is 1. It follows from Corollary 6.2.2 and Lemma 6.2.3 that (6.2.4) D (1 − fz )− ∗ (fz ) ≡ const., z = z1 , . . . , zn . We claim, however, that the function of z on the left is deﬁned and continuous at z = zi . Indeed D(0) = D(1) = 0 so we may simply write (notation as above) D (1 − fz ) ∗ (fz ) = di ej D(αi−1 βj ) + e di D(αi ) + D(αi−1 ) . i,j
i
We will see below (Lemma 6.2.5) however that D(x) = −D(x−1 ) so the second sum in the above expression drops out. Also we can extend D to a continuous function on all of P1C by taking D(∞) = D(0) = 0. Thus (6.2.5) D (1 − fz )− ∗ (fz ) = di ej D(αI−1 βj ) i,j
where the βj run through all zeros and poles of fz , including 0 and ∞. This expression is clearly continuous in z. It remains to show that the constant in (6.2.4) is zero. Fix f ∈ I and take fz = c(z)f where c(z) → 0. The βj are then ﬁxed, but the poles zeros and of 1 − fz coalesce. It follows from (6.2.5) that − D (1 − fz ) ∗ (fz ) → 0. We used in the above the following: Lemma 6.2.5. Let D be as in Corollary 6.1.2. Then D(x) = −D(x−1 ). Proof. We have dx 1 d(1/x) dx log(1 − x) + log 1 − = log(−x) . x x 1/x x Note x 2 1 dt Im = Im log(−x) + C + C log x log(−t) t 2
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6. THE DILOGARITHM FUNCTION
49
where C accounts for the ambiguity in the branch of log(−t). Thus $ 2 % 1 +arg(−x) log x+C +C log x D(x)+D(x−1 ) = − Im log(−x) 2 with C accounting for the ambiguity in the branches of log(−t) and arg(1 − x). From this we see D(x) + D(x−1 ) = C + C log x. However for x < 0 it is clear from the deﬁnition that D(x) = 0. Thus C + C log x = 0 for all x < 0, whence C = C = 0. We note also, while we are at it Lemma 6.2.6. (i) D(¯ x) = −D(x). In particular D ≡ 0 on R. (ii) D(1 − x) = −D(x). Proof. (i) is straightforward and won’t be used anyway, so we leave it to the reader. For (ii), integrate by parts to get ' ( x dt D(x) = arg(1 − x) log x − Im log(1 − x) log(x) + . log t 1−t 0 1 Take x = 1 − u in the integral and note 0 log(1 − u) du/u is real: ( ' 1−x du D(x) = arg(1−x) log x− Im log(1−x) log(x)− log(1−u) u 0 1−x du = − arg(x) log 1 − x + Im log(1 − u) u 0 = −D(1 − x). Remark 6.2.7. From (6.2.5) itis possible to deduce a stronger version of Corollary 6.2.2. Suppose ni λ[ai (z)]∈ A∗ ⊗ A∗ is invariant under the switch map a ⊗ b → b ⊗ a. Then ni D ai (z) ≡ const. This follows from 4(λ[z] + λ[z −1 ]) = z 2 ⊗ z 2 together with the existence of square roots in A and the polarization trick.
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https://doi.org/10.1090/crmm/011/08
LECTURE 7
The Regulator Map. II 7.1.
We have now a triangle / K2 (R, I)
K3 (C)
FF FF FF D FFF "
(7.1.1)
vv vv v vv D vz v
R
and a good functiontheoretic hold on D. As applications we consider the case of a prime cyclotomic ﬁeld k = Q(ζ), ζ = 1 and use D to obtain a regulator formula in the style of Kummer [BS66] for the value of the zeta function of k at s = 2. We also show how D leads to a measurable cohomology class in H 3 (SL2 (C), R) which is cohomologous to a multiple of the class arising from the volume form on the Poincar´e upper half space SL2 (C)/ SU2 . 7.2.
Consider the commutative diagram
◦ C∗ 0 / tor1 (C∗,C∗ ) / (1+I)∗ ⊗Z C∗ / ∗
p∈C
(7.2.1)
/0
/ K2 (C)
/0
s
0
/ C∗ ⊗C∗
/ K2 (R, I)
/ K3 (C)ind
/
◦ p∈C
C∗ ∗
where the bottom line is (5.2.5) for k =
C, the top line arises from ∗ tensoring the sequence 0 → (1 + I) → ◦C ∗ Z → C∗ → 0 with C∗ , and the map s is induced from the product structure in Ktheory, speciﬁcally from K1 (R, I) ⊗ K1 (C) → K2 (R, I). Taking the quotient yields the top line of 0 (7.2.2)
/
K3 (C )ind tor1 (C ∗ ,C ∗ )
O
θ
0
/ B(C)
/
K2 (R,I) {1+I,C ∗ }
T
/ C∗ ⊗Z C∗
/ K2 (C)
/0
λ
/ C ∗ ⊗ C∗
/ K2 (C)
/ 0.
O
η
/ A(C)
The bottom line is (6.1.2) with F = C. 51
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52
7. THE REGULATOR MAP. II
Proposition 7.2.1. For x ∈ C \ {0, 1} deﬁne 1 0 xt t ∈ K2 (R, I). η[x] = 6 , (t − 1)2 − xt2 t − 1 With this η, the diagram (7.2.2) is commutative. Proof. C∗ ⊂ K0 (P1C , {0, ∞}) and the divisor class map
◦ d: Z → C∗ p∈C ∗
is given as folows: For δ a divisor ﬁnd f on P1 such that (f ) = δ. Then d(δ) = f (0)f (∞)−1 . There is a commutative diagram K2 (R) O
(7.2.3)
tame symbol
◦ p∈C ∗
C∗
d⊗Id T
K2 (R, I) Note (7.2.4)
/
. η[x] =
/ C ∗ ⊗ C∗ .
xt2 t 1− , 3 2 (t − 1) − xt (t − 1) t − 1
/6
as a symbol. Since tame{f, g} =
(−1)ordy f ·ordy g
y∈P1
f (y)ordy g g(y)ordy f
a straightforward calculation yields (writing u = t/(t − 1), ζ 3 = 1) /6 . . /6 t xt2 1−xu3 , (7.2.5) tame 1− = tame ,u (t−1)3 − xt2 (t−1) t−1 1−xu2 = (−1)6 u=∞ + x2 u=x−1/3 + x2 u=ζx−1/3 + x2 u=ζ 2 x−1/3 − x3 u=x−1/2 − x3 u=−x−1/2 . A function f with divisor in u coordinates (f ) = 2(x−1/3 ) + 2(ζx−1/3 ) + 2(ζ 2 x−1/3 ) − 3(x−1/2 ) − 3(−x−1/2 ) is f=
(1 − xu3 )2 . (1 − xu2 )3
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7. THE REGULATOR MAP. II
53
Since u(0) = 0, u(∞) = 1, we ﬁnd f (0)f (∞)−1 = 1 − x. It now follows from (7.2.3) that Tη [x] = (1 − x) ⊗ x = λ[x] as claimed. Note that η determines a map θ : B(C) → K3 (C)ind /tor1 (C∗ , C∗ ).
Remark 7.2.2. The above discussion was completely algebraic, and would be valid with C replaced by any algebraically closed ﬁeld. Example 7.2.3. Let ζ ∈ C be a primitive nth root of 1 for some n > 2, and let k = Q(ζ). Then n[ζ] ∈ B(k). Also η[ζ] ∈ K2 (Rk , Ik ), and it follows from a glance at the right hand side of (7.2.5) and the exact sequence
K3 (k) → K2 (Rk , Ik ) → k(p)∗ p∈P1k p=0,∞
that θ(n[ζ]) ∈ Image K3 (k) → K3 (C)ind / tor1 (C∗ , C∗ ) . The kernel of this map can be shown to be torsion, so θ(n[ζ]) ∈ K3 (k)/torsion. We want to compute the image of this element under the regulator map D. For simplicity we restrict ourselves to the case n = , an odd prime. Theorem 7.2.4. With notation as above, the elements θ([ζ]), θ([ζ 2 ]), . . . , θ([ζ (−1)/2 ]) form a basis for K3 (k)Q . Their image under the map ⊕(−1)/2
⊕(−1)/2 D K3 (k)Q → K3 (k ⊗Q R)Q ∼ −−−−−−→ R⊕(−1)/2 = K3 (C)Q
generates a lattice of maximal rank whose volume is given by 2−(−1)/23(−1)/4 L(2, χ) χ odd
where χ runs through the odd characters of (Z/Z)∗ , and L(s, χ) = χ(n)n−s is the Dirichlet Lseries. Before proving Theorem 7.2.4, we verify a compatibility. Lemma 7.2.5. The diagram A(C)
DD DD DD D DD "
η
R
/ K2 (R, I)
v vv vv v D zvvv
commutes, where the D on the left is the function D (Corollary 6.1.2) and that on the right is from (7.1.1).
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54
7. THE REGULATOR MAP. II
Proof. As a temporary expedient let D denote the D on the right. With reference to (7.2.5) we ﬁnd (replacing x by x6 ) 2 * 6 3 − u 1 − x ∗ (u) D η[x6 ] = 6D 1 − x 6 u2 = 6D (1 − x−3 ) + (1 + x−3 ) + (1) − (1 − x−2 ) − (1 − ζx−2 ) − (1 − ζ 2 x−2 ) , where ζ 3 = 1. Using D(z −1 ) = −D(z) = D(1 − z), we reduce to proving (7.2.6) 6D(x3 )+6D(−x3 )−6D(x2 )−6D(ζx2 )−6D(ζ 2 x2 )−D(x6 ) = 0. By Corollary 6.2.2 it suﬃces to show 6(x3 ) + 6(−x3 ) − 6(x2 ) − 6(ζx2 ) − 6(ζ 2 x2 ) − (x6 ) ∈ B(A). (This will imply the left side of (7.2.6) is constant. Letting x → 0 shows that constant is zero.) The requisite identity [(1 − x3 ) ⊗ x18 ] · [(1 + x3 ) ⊗ x18 ] · [(1 − x2 ) ⊗ x12 ] · [(1 − ζx2 ) ⊗ x12 ] · [(1 − ζ 2 x2 ) ⊗ x12 ] · [(1 − x6 ) ⊗ x6 ] = 1
is easily checked.
7.3. We now turn to the proof of Theorem 7.2.4. Notice that Borel’s theorem implies K3 (k)Q has rank ( − 1)/2, so it suﬃces to do the volume computation. (Note the Lseries has a product expansion, so the volume = 0.) By (7.2.5) ζi ∞ ζ im dv i (7.3.1) Dη[ζ ] = − Im . log(1 − v) = Im v m2 0 m=1 Write D(i) = Dη[ζ i ]. Note D(−i) = −D(i), D(0) = 0, D(i+) = D(i). Thus we can view D as an odd function on (Z/Z)∗ . Hence D= cχ χ. χ odd character of (Z/Z)∗
As representatives for the diﬀerent equivalence classes of embeddings k → C we can ﬁx one (ζ → ζ) and take as the others ζ → ζ j , 2 ≤ j ≤ ( − 1)/2. The volume of the lattice spanned by the elements η[ζ i ], 1 ≤ i ≤ ( − 1)/2, is then (−1)/2 R, R = det D(ij) 1≤i,j≤(−1)/2 .
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7. THE REGULATOR MAP. II
Lemma 7.3.1.
R=
−1 2
(−1)/2
55
cχ .
χ odd
Proof. Deﬁne Dj (i) = D(ij) so Dj = χ odd χ(j)cχ χ. The matrix D(ij) is the matrix of coeﬃcients expressing the functions Dj in terms of the basis for the odd functions on (Z/Z)∗ given by ⎧ ⎪ j = i, −i ⎨0 −1 1≤i≤ Si (j) = 1 . j=i ⎪ 2 ⎩−1 j = −i Since the odd χ form another basis for this space, R is necessarily a homogeneous function of degree ( − 1)/2 in the cχ . Clearly if some cχ = 0, R = 0 (the Dj won’t span), so R = C cχ  for some universal constant C. To compute C, consider the function F = χ odd χ. F (ij) = χ χ(i)χ(j). Ordering the χ, we can consider the matrix M = χ(i) χ odd . 1≤i≤(−1)/2
Then
F (ij) = M t M,
C =  det M 2 .
To compute det M , let θ be a generator of the ( − 1)st roots of 1, and let g ∈ (Z/Z)∗ be a generator. The numbers 1 ≤ i ≤ ( − 1)/2 form a set of coset representatives for {±1} ⊂ (Z/Z)∗ . Replacing this set by another in the deﬁnition of Mchanges det M by at most a sign, so we may consider instead det χ(i) χ odd,i∈S where S = {g j  1 ≤ j ≤ ( − 1)/2}. We have χ(g j ) = θ(2r+1)j
some r = r(χ), 0 ≤ r ≤
−3 , 2
so the determinant becomes det θ(2r+1)j 0≤r≤(−3)/2 = θ(−1)(+1)/8 det(θ2ri )0≤r≤(−3)/2 1≤j≤(−1)/2 1≤i≤(−1)/2 (−1)(+1)/8 (−1)/2 2j = ±θ (θ − θ2i ). i,j=1 i 0, with the vertex p in question lying at ∞. In this case geodesics through p are vertical lines and the Poincar´e metric looks like (dx ∧ dy∧ dz)/z 2 . Finiteness of volume follows from ﬁniteness of the ∞ integral z0 dz/z 2 . Suppose now that all four points lie at ∞. Given g1 , g2 , g3 , g4 ∈ P1C , let V (g1 , . . . , g4 ) denote the oriented volume of the tetrahedron they span. P1C
Lemma 7.4.1. V satisﬁes the cocycle condition, given g1 , . . . , g5 ∈ 5
(7.4.1)
(−1)i V (g1 , . . . , gˆi , . . . , g5 ) = 0.
i=1
Proof. By taking the “geodesic span”, the points g1 , . . . , g5 deﬁne a map Φ : Δ4 → H = H ∪ P1C , such that δΦ = (−1)i Δ(g1 , . . . , gˆi , . . . , g5 ) where Δ(g1 , . . . , gˆi , . . . , g5 ) denotes the geodesic tetrahedron with vertices g1 , . . . , gˆi , . . . , g5 . We can approximate Φ by maps Φα : Δ4 → H. Writing ω = volume form on H, the left hand side of (7.4.1) is approximated by ∗ Φα ω = Φ∗α dω = 0. δΔ4
Δ4
Theorem 7.4.2. V (z1 , . . . , z4 ) = ± 32 D({z1 , . . . , z4 }), where D is as in Corollary 6.1.2 and {z1 , . . . , z4 } =
(z1 − z2 )(z3 − z4 ) (z1 − z4 )(z3 − z2 )
is the crossratio. Lemma 7.4.3. 5
(−1)i D({z1 , . . . , zˆi , . . . , z5 }) = 0.
i=1
Proof. Notice that it suﬃces to show the left side is constant, because we can take, e.g., z1 = z2 and use the fact D(0) = D(1) = D(∞) = 0 to deduce that the constant is zero.
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58
7. THE REGULATOR MAP. II
To show that the desired expression is constant, we use (6.2.7). Notice (z1 − z3 )(z2 − z4 ) 1 − {z1 , . . . , z4 } = {z1 , z3 , z2 , z4 } = . (z1 − z4 )(z3 − z2 ) Thus
λ{z1 , . . . , z4 } = (z1 − z4 )(z3 − z2 ) ⊗ (z1 − z4 )(z3 − z2 ) · {terms of the form (zi − zj ) ⊗ (zk − z )±1 }
where i, j, k, are not all distinct. The ﬁrst term is symmetric and can be ignored. Consider a term (zi − zj ) ⊗ (zk − z )±1 . Since i, j , k , are not all distinct, there will be m ∈ {1, 2, 3, 4}, m = i, j , k , . One checks then that the term (zi −zj )⊗(zk −z )∓1 appears in the expansion m for (λ{z1 , . . . , zˆm , . . . , z5 })(−1) . It is occasionaly necessary to replace (zi − zj ) ⊗ (zk − z ) by (zk − z ) ⊗ (zi − zj )−1 , but this is permissible by symmetry. With these hints, the patient reader can check the details for himself. The key point is now Theorem 7.4.4. The set of measurable functions f : P1C × · · · × P1C → R 4 times
which are invariant underthe diagonal action of PGL2 (C) and satisfy the cocycle condition (−1)i f (z1 , . . . , zˆi , . . . , z5 ) = 0 forms a one dimensional Rvector space generated by D({z1 , . . . z4 }). For the proof, let PGL2 (C) and deﬁne a complex of Gmodules G1 =i+1 • i C by C = Morph (PC ) , R , i ≥ 0, where “Morph” means measurable functions, identifying those which agree a.e. For details about properties of C • see [Moo76]. Note in particular: Lemma 7.4.5. (i) If U ⊂ (P1C )i+1 is open with complement of measure zero, then C i ∼ = Morph(U, R). • (ii) C is a resolution of R by topological Gmodules, so H ∗ (G, R) ∼ = H ∗ (G, C • ) cont
(hypercohomology computed as in [Moo76]). ∗ ∗ (iii) H ∗ (G, C 0 ) ∼ (B, R); H ∗ (G, C 1 ) ∼ (T, R); = Hcont = Hcont R ∗=0 H ∗ (G, C 2 ) ∼ = 0 ∗>0 where T ⊂ B ⊂ G are the torus and Borel subgroups.
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7. THE REGULATOR MAP. II
59
Proof. (i) is clear, and (iii) follows from [Moo76, Theorem 6], together with (i). (Note G/B ∼ = P1 and G/T → P1 × P1 is open dense.) For (ii), the fact that functions in C i are only deﬁned a.e. makes it awkward to write down a contracting homotopy. Instead, let f ∈ C i be a cocycle for some i ≥ 1: (−1)j f (x0 , . . . , xˆj , . . . , xi+1 ) = 0 (a.e.), and choose y ∈ P1 such that f (x0 , . . . , xi ) =
i
(−1)j f (y, x0 , . . . , xˆj , . . . , xi )
j=0
for almost all (x0 , . . . , xi ). Taking g(x0 , . . . , xi−1 ) = f (y, x0 , . . . , xi−1 ) we get δg = f . For f ∈ C 0 , δf = 0 means f (x0 ) = f (x1 ), i.e., f ≡ const. 2 Lemma 7.4.6. (i) Hcont (B, R) = (0); 3 1 ∼ ∼ (ii) Hcont (G, R) = R = Hcont (T, R).
Proof. We already did these computations in (3.3). Lemma 7.4.7.
(i) The map δ R∼ → H 0 (G, C 2 ) ∼ = H 0 (G, C 1 ) − =R
is an isomorphism. (ii) The map δ 1 1 Hcont (B, R) ∼ → H 1 (G, C 1 ) ∼ (T, R) = H 1 (G, C 0 ) − = Hcont
is twice the natural restriction map. In paricular, it is surjective. Proof. (i) A Ginvariant map f : P1 × P1 → R is necessarily constant (a.e.), f ≡ C. Then δf (x0 , x1 , x2 ) = C − C + C = C. (ii) For g ∈ G, let g¯ ∈ B\G, g˜ ∈ T \G denote the images. The boundary map δ : Morph(B\G, R) → Morph(T \G, R) is given by (δf )(˜ g ) = p0 (f )(˜ g ) − p∞ (f )(˜ g ), where pj (f )(˜ g ) = f (j, g˜), j = 0, ∞. Let i : T → B be the inclusion. 0 1 The map p∗0 on cohomology coincides with i∗ . Let J = ( −1 0 ) ∈ G, so ∗ ∼ ∞J = 0, 0J = ∞. Conjugation by J acts on T = C by x → x−1 , so it acts by the identity on H 0 (T, R) and by −1 on H 1 (T, R). Since p∗∞ is given in cohomology by J ◦ i∗ , the lemma follows.
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60
7. THE REGULATOR MAP. II
Proof of Theorem 7.4.4. We compute with the spectral sequence p+q E1p,q = H q (G, C p ) =⇒ Hcont (G, R). From the lemmas, E21,1 = E20,2 = (0) $ δ % E23,0 ∼ → MorphG (P1 )5 , R = Ker MorphG (P1 )4 , R − q q Q Q q q
Q Q qH Q d3 Q 0 HH Q d2 HH Q sq Q q q j H
3, 0
p
E2 3,0 3 It follows that E23,0 ∼ → Hcont (G, R) ∼ = E∞ = R. Since D({z1 , . . . , z4 }) ∈ 3,0 3,0 ∼ E2 is nontrivial, we get E2 = R, the statement of the theorem.
From Lemma 7.4.1 and Theorem 7.4.4 we conclude V (z1 , . . . , z4 ) = CD({z1 , . . . , z4 }) for some constant C. Coxeter ([Cox35, p. 29]) computes the volume of the regular tetrahedron inscribed at ∞ to be √ 1 1 1 1 1 3 3 1 − 2 + 2 − 2 + 2 − 2 + ··· . V = 4 2 4 5 7 8 The crossratio of the four vertices can be taken to be e2πi/3 = √ 1 (−1 + −3) so 2 √ ∞ e2πin/3 3 1 1 1 = V = ±C · Im C 1 − 2 + 2 − 2 + ··· . 2 n 2 2 4 5 n=1 Thus C = 32 , proving Theorem 7.4.2. Remark 7.4.8. The analogy between D(x) and log x comes out quite clearly. For one thing, viewing P1C = GL2 (C)/B, D ◦ (cross ratio) can be viewed as a homogeneous 3cocycle just as log  det  is a 1cocycle. Also log cross ratio can be interpreted as the length of a geodesic. It would be very interesting to have a diﬀerentialgeometric interpretation of the regulator on an elliptic curve discussed in the next section.
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https://doi.org/10.1090/crmm/011/09
LECTURE 8
The Regulator Map and Elliptic Curves. I 8.1. Let E be an elliptic curve deﬁned over C. Our objective in the next few lectures will be to write down a regulator map R : K2 (E) → C.
(8.1.1)
When E has complex multiplication over an imaginary quadratic ﬁeld of class number 1, we will relate the zeta function, ζ(E/k, 2) (k = P ray class ﬁeld of the conductor of the Gr¨ossencharakter of E) to the value of R on certain elements in K2 (Ek ). We ﬁx q = e2πiτ ∈ C∗ with q < 1 such that E ∼ = C∗ /q Z . Then R = Rq = Jq + iDq ,
(8.1.2)
where Jq and Dq are real valued. As the notation indicates, Rq , Jq , Dq depend a priori on q. I have not yet worked out the modular behavior of these objects. Lemma 8.1.1. The expression Dq (x) = D(xq n ) n∈Z
deﬁnes a continuous function Dq : E → R. Proof. The fact that the expression in question converges uniformly as n → ∞ follows from the description of D(x) (Corollary 6.1.2). For n → −∞ we use Lemma 6.2.5 to write D(xq n ) = −D(x−1 q −n ).
We will ultimately show Theorem 8.1.2. Dq is a Steinberg function (cf. Deﬁnition 5.3.1) and hence induces a map Dq : K2 C(E) → R. The imaginary part of our regulator is obtained by composing Dq K2 (E) → K2 C(E) −→ R. To explain the real part Jq , we set Let log   · log   :
J(x) = log x log 1 − x. x∈C ∗
C∗ → R denote the map f x → log f  log x. 61
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62
8. THE REGULATOR MAP AND ELLIPTIC CURVES. I
Lemma 8.1.3. The diagram (notation as in Lecture 5) below commutes. tame /
∗ (1 + I)∗ ⊗Z C(P1 )∗ x∈C ∗ C log  ·log  
(5.3.1)
/R
J
x∈C ∗ Z
d Proof. Let f = (t − αj )dj ∈ (1 + I), so dj = 0, αj j = 1, and let g = c (t − βk )ek ∈ C(P1 )∗ . Starting with f ⊗ g in the upper left corner and going clockwise around the square, we obtain (with the natural convention that log 0 log 1 = 0) e d −dj log αj  · logc (αj − β ) + ek log βk  · log (βk − αm ) m j,k
=
m
dj ek log βk αj−1  log 1 − βk αj−1  + e
j
j,k
where e = −
k
dj (log αj )2
ek = ord∞ g. Since
(log α)2 = log α log 1 − α + log α−1  log 1 − α−1  the above expression coincides with J (f )− ∗ (g) .
It follows from the lemma that J is a relative Steinberg function, so we get (8.1.3)
J : K2 (R, I) → R.
Notice that J factors through the tame symbol, so J K3 (C) = 0. Deﬁne ∞ ∞ n (8.1.4) Jq (x) = J(xq ) − J(x−1 q n ). n=0
n=1 ∗
Jq (x) is clearly continuous on C , and we have Jq (qx) − Jq (x) = −J(x−1 ) − J(x) = −(log x)2 . Lemma 8.1.4. Let F = dj (αj ), G = e (β ) be divisors on C∗ , dj ekk k and assume dj = ek = 0, αj = βk = 1. Then F and G project to divisors of elliptic functions f , g on E = C∗ /q Z , and the expression Jq (F − ∗ G) = dj ek Jq (αj−1 βk ) (8.1.5)
j,k
depends only on the divisors of f and g.
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8. THE REGULATOR MAP AND ELLIPTIC CURVES. I
63
Proof. The ﬁrst assertion is wellknown. For the second, it suﬃces to consider − Jq (F ∗ G), F = dj (αj q nj ), dj nj = 0. Write Jq,G (x) =
ek Jq (xβk ).
k
We have from (8.1.5) ek (log βk x)2 = ek (log βk )2 Jq,G (qx) − Jq,G (x) = Jq,G (q n x) − Jq,G (x) = nek (log βk )2 . − Jq (F ∗ G) − Jq (F − ∗ G) = dj Jq,G (αj−1 q −nj ) − Jq,G (αj−1 ) j
=
−dj nj ek (log βk )2 = 0.
j,k
It follows immediately from (8.1.4) that the prescription f ⊗ g → Jq (F − ∗ G), where F lifts (f ) and G lifts (g), deﬁnes a homomorphism Jq
C(E)∗ ⊗Z C(E)∗ −→ R.
(8.1.6)
We will eventually show Theorem 8.1.5. Jq f ⊗ (1 − f ) = 0 for f ≡ 1, so Jq induces a map Jq : K2 (E) → K2 C(E) → R. 8.2. The proof of Theorems 8.1.2 and 8.1.5 require rather detailed knowledge of the behavior of functions like (8.2.1)
F (w) =
∞
(1 − αj q n w−1 )dj
n=0 j
where
∞
(1 − αj−1 q n w)dj
n=1 j
d
αj j = 1. Note F (q −1 w)F (w)−1 = (1 − αj−1 w)dj · (1 − αj w−1 )−dj dj = 0,
j
= (−αj w)−dj = 1,
j
j
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64
8. THE REGULATOR MAP AND ELLIPTIC CURVES. I
so F (w) is the pullback to C∗ of an elliptic function with divisor dj (αj )(modq Z ). We ﬁx a constant K = 0, 1, and for N > 0 we let AN denote the annulus qN ≤ z ≤ q−N .
(8.2.2)
Also for N ≥ 0 an integer we write (w − αj )dj , F0 (w) = j
(8.2.3)
TN (w) =
FN (w) =
F0 (q r w)
r≤N
(1 − αj q n w−1 )dj (1 − αj−1 q n w)dj
n≥N +1 j
so FN (w)TN (w) = F (w). Lemma 8.2.1. There exists an R > 0 independent of N such that FN (w) − K has no zeros oﬀ the annulus AN +R . Proof. Replacing F (w) by F (w−1 ) it will suﬃce to show FN (w) − K has no zero inside a circle of radius qN +R . Then replacing w by q N w it suﬃces to show FN (q N w) − K has no zero inside a circle of radius δ for some ﬁxed δ. Consider the power series expansion F0 (w) = 1 + a1 w + a2 w2 + · · · and choose a constant s > 0 such that an  ≤ sn , all n. We have 1 1 F0 (q w) FN (q w) = r (1 − q sw) (1 − qr sw) r=0 r=0 r=0 N
2N
r
2N
∞
where means term by term domination of the coeﬃcients of the power series expansions. Since the coeﬃcients on the right are all positive real, we ﬁnd ∞ 1 N >0 FN (q w) − K ≥ K − 1 − 1 − r (1 − q sw) 0 for w ≤ δ, some δ.
In what follows, N and are parameters with N large and small. The notation ψ(N, ) = O(N ) + O(1) will mean there exist constants C, C independent of N , such that ψ(N, ) ≤ CN + C . The notation Z(g, R) will denote the number of zeros of a function g(w) on AR .
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8. THE REGULATOR MAP AND ELLIPTIC CURVES. I
65
Lemma 8.2.2. Given 0 < < 1 irrational, R > 0, and N0 > 0, there exists N ≥ N0 such that Z(FN (w) − K, N (1 − ) + R) = Z(F (w) − K, N (1 − ) + R). In particular, the number of zeros of FN (w) − K oﬀ AN (1−)+R is O(N ) + O(1) for some sequence of N ’s → ∞, where the sequence of N ’s may depend on . Proof. The second statement follows from the ﬁrst because if F0 (w) has (say) d poles, then F (w) − K will have dN (1 − ) + O(1) poles on AN (1−)+R and (up to O(1)) the same number of zeros. Since FN (w) − K has N d poles and N d zeros, it will have (by the ﬁrst part of the lemma) N d + O(1) zeros oﬀ AN (1−)+R . To prove the lemma, it suﬃces to ﬁnd N ≥ N0 such that FN (w) − F (w) = FN (w)TN (w) − 1 < F (w) − K at the boundary δAN (1−)+R . In fact, this will insure that FN (w) − K F (w) − K − 1 < 1 on δAN (1−)+R whence
FN (w) − K dlog F (w) − K
= 0.
δAN (1−)+R
Since the numbers of poles of FN (w) − K and F (w) − K on AN (1−)+R coincide, we must have Z(F (w) − K, N (1 − ) + R) = Z(FN (w) − K, N (1 − ) + R) as claimed. Write F (w) − K = μ (8.2.4)
∞
−1 ek
(1 − βk q w )
n=0 k
ek = 0,
n
∞
(1 − βk−1 q n w)ek
n=1 k
βkek = 1,
μ ∈ C.
Fix some compact region V ⊂ R/Z such that V = −V = ∅ and such that V does not contain any of the points log βk  log αj  , (mod 1). log q log q Take N ≥ N0 such that N + R (mod 1) ∈ V . (This is possible because is irrational.) Then I claim (i) F (w) − K ≥ δ = δ(V ) > 0 on δAN (1−)+R (ii) FN (w) = O(1) on δAN (1−)+R
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66
8. THE REGULATOR MAP AND ELLIPTIC CURVES. I
(iii) TN (w) − 1 = O(qN ) on AN (1−)+R where δ in (i) and the “big O” in (ii), (iii) are independent of N . Indeed, (i) follows because N ≡ V implies βk q n w−1 = 1 on δAN (1−)+R so F (w) − K has no zeros on δAN (1−)+R . The fact that δ can be chosen independent of N comes from the fact that F (qw) = F (w). For (ii), we may replace FN (w) by FN (w−1 ) so it suﬃces to show boundedness on the inner contour Γ of δAN (1−)+R . Consider the set S of all integers n such that there exists a j with αj q n w−1  ≤ 2 and a j with αj q n w−1  ≥ 12 on Γ. The number of elements in S can be bounded independently of N . Since 1 − αj q n w−1  is bounded below on Γ, (1 − αj q n w−1 )dj (1 − αj−1 q n w)dj n∈S j
is bounded on Γ. Suppose n ∈ / S and (say) αj q n w−1  ≥ 2 for all j. Dividing through by (αj q n w−1 )dj = 1 we ﬁnd (1 − αj q n w−1 )dj = (αj−1 q −n w − 1)dj . j
j
In this way terms for n ∈ / S can be majorized by products (1 − qn · const) n
which are bounded independent of N . The bound TN (w) − 1 = O(qN ) on AN (1−)+R is immediate. The lemma is proved by combining (i), (ii), and (iii). Lemma 8.2.3. Let F (w) − K be given by (8.2.4), and let be sufﬁciently small so the circle γk of radius about βk does not contain 0 or any other βk . Assume also (for simplicity) that all ek = ±1. Then for N 0, any zero or pole of FN (w) − K or AN (1−)+R lies inside exactly one translate γk · q r . No two roots lie in the same translate. Proof. Take N 0, so that KTN (β) − 1 < inf K − F (w) w∈γk
for all β ∈ AN (1−)+R . Notice that K − FN (w), FN (w), F (w), and K − F (w) all have the same poles on AN (1−)+R so we need only be concerned with zeros. If β ∈ AN (1−)+R is a zero of FN (w) − K 0 = (FN (β) − K)TN (β) = 1 − TN (β) K + (F (β) − K) so F (β) − K < inf F (w) − K = inf r F (w) − K. w∈γj
w∈γj q
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8. THE REGULATOR MAP AND ELLIPTIC CURVES. I
67
Since all the zeros of F (w) − K lie inside the γk q r and since ek = ±1 it follows from Rouch´e’s Theorem that β lies in a unique γj q r . Given small and irrational, we ﬁx now N 0 such that Lemmas 8.2.2 and 8.2.3 hold, where R > 0 is ﬁxed such that γk q r ⊂ AN (1−)+R for r ≤ N (1 − ). Let βk (N, r) denote the unique sin gularity of FN (w) − K inside γk q r , r ≤ N (1 − ). Let {βN } denote the other zeros and poles of FN (w) − K. Let eβN = multiplicity of βN . Lemma 8.2.4. eβN = 0 and eβN log βN  = O(N ) + O(1) as N → ∞. Proof. We know ek = 0 because F (w) − K is elliptic, and also eβN = 0 where βN runs through all singularities of FN (w) − K (because K = 1 =⇒ 0 and ∞ are not singularities of FN (w) − K.) It follows that eβN = 0 as well. Also eβN log βN  = log FN (0) − K = log 1 − K. Since
log βk (N, r) − log βk q r  = O() ek log βk q r  = 0, k
and there are O(N ) βk (N, r)’s, we may drop from the sum the βN ’s corresponding to βk (N, r)’s and get  = O(N ) + O(1). eβN log βN
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https://doi.org/10.1090/crmm/011/10
LECTURE 9
The Regulator Map and Elliptic Curves. II 9.1. Our ﬁrst objective is For any elliptic function f and any constant K, Theorem 9.1.1. Jq f ⊗ (K − f ) = 0 (notation as in (8.1.6)). Proof. From the description of Jq , it is clear that Jq (L⊗g) = 0 for any constant L and any elliptic function g. We may therefore replace f by Lf and assume the induced function f on C∗ looks like ∞ ∞ −1 n dj n −1 dj (1 − αj q w) (1 − αj q w ) . F (w) = j
n=0
n=1
Further, since the expression Jq f ⊗(K −f ) is continuous as a function of the divisors (f ) and (K − f ), we may assume K = 0, 1, and f is general, so ∞ ∞ −1 n ek n −1 ek F (w) − K = μ (1 − βk q w) (1 − βk q w ) k
n=0
n=1
with ek = ±1. Let the map (1 + I)∗ ⊗ C(P1 )∗ → R in Lemma 8.1.3 be denoted by f ⊗ g → J(f ⊗ g). in It is an easy exercise from∗ the diagram ∗ Lemma 8.1.3 to show J f ⊗(K−f ) = 0 for f ∈ (1+I) , K ∈ C . Write dj (αj ). (This notation Jf (x) = dj J(αj−1 x), where (f ) = is slightly diﬀerent from that in the proof of (8.1.4).) If (K − f ) = ek (βk ) and K − f has no zero or pole at 0 or ∞, we ﬁnd ek Jf (βk ) = 0. Apply this with f = FN = r≤N F0 (q r w), F0 = j (w − αj )dj . With notation as in Lecture 8, we ﬁnd ek JFN βk (N, r) + eβN JFN (βN ) = 0. (9.1.1) r≤N (1−) k
Lemma 9.1.2.
) = O(N ) + O(1). eβN JFN (βN
69
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9. THE REGULATOR MAP AND ELLIPTIC CURVES. II
Proof. Let C = CF = j dj (log αj )2 . By (8.2.4) it suﬃces to show  log βN −N = O(N ) + O(1). eβN JFN (βN ) + C log q βN
Dropping the sum, this will follow if we know for β = βN log β − N C = O(1). (9.1.2) JFN (β) + log q Write F0− (w) = j (w − αj−1 )dj , and note
JF0 (x) + JF0− (x−1 ) = C. It follows from Lemma 8.2.1 that max −N, − log β + log β = O(1), log q log q so we get
JFN (β) =
r≤N
=
JF0 (βq r ) N
max(−N,− log β/ log q)
JF0 (βq ) − r
r=max(−N,− log β/ log q)
JF0− (β −1 q −r )
r=−N
log β C + O(1). + N− log q
Note that in the ranges indicated for the summations, βq r and β −1 q −r n < 1. For βq N0  < 1, however, one easily checks that ∞ n=N0 J(βq ) = O(1) uniformly in β. Thus the two sums on the right are O(1), and (9.1.2) follows. We write JF,q (x) =
dj Jq (αj−1 x)
(again note that αj−1 ; a change from the notation in the proof of Lemma 8.1.4). Lemma 9.1.3. JF,q (x) − JFN (x) + N CF  = O(N qN ) as N → ∞, uniformly in x for x ∈ AN (1−)+R .
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9. THE REGULATOR MAP AND ELLIPTIC CURVES. II
71
Proof. We have (because J(x) + J(x−1 ) = (log x)2 ): dj J(αj−1 q r x) JFN (x) = r≤N j
=
N
dj J(αj−1 q r x)−
r=0 j
N N r −1 dj J(αj q x )+ dj (log αj−1 q −r x)2 , r=1 j
r=1 j
so
∞ ∞ −1 r r −1 dj J(αj q x)+ dj J(αj q x ). JF,Q (x)−JFN (x)+N CF  ≤ r=N +1 j
r=N +1 j
Lemma 9.1.3 now follows from
∞ n Sublemma 9.1.4. If xq N0  ≤ M , then n=N0 J(xq ) = O(M log M ) as M → 0. Proof. J(xq n ) = O M qn−N0 log(M qn−N0 ) . Since the series ∞ n=0
qn
log(M qn ) log M
can be bounded uniformly in M for M 1, we get J(xq n ) = O(M log M ). This proves Sublemma 9.1.4 and also Lemma 9.1.3.
Returning now to the proof of Theorem 9.1.1, we can rewrite (9.1.1) (since ek = 0): ek JF,q β(N, r) = O(N ) + O(1) + O(N 2 qN ). (9.1.3) r≤N (1−) k
Lemma 9.1.5. For x, x ∈ C∗ , x − x  ≤ x , we have JF,q (x) − JF,q (x ) = O( log ),
→0
uniformly in x, x . Proof. We have JF,q (qx) − JF,q (x) = −CF , so JF,q (qx) − JF,q (qx ) = JF,q (x) − JF,q (x ). We may assume, therefore, that x ∈ A1 . Note J(xq r ) − J(x q r ) = log xq r  log 1 − xq r  − log x q r  log 1 − x q r  1 − xq r r + log x log 1 − x q r . = log xq  log x 1 − x q r
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9. THE REGULATOR MAP AND ELLIPTIC CURVES. II
Since
1 − xq r q r (x − x) = 1 + 1 − x q r 1 − x q r and 1 − x/x  ≤ , we see J(xq r ) − J(x q r ) = O(qr r) as r → ∞ uniformly for x ∈ A1 . It suﬃces to show, therefore, that J(x) − J(x ) = O(x − x  log x − x ) on some ﬁxed disk Ba (0) of radius a about 0. Since J(x) = J(1 − x) we may assume x, x ∈ / B1/3 (1), and since the inequality is more or less trivial on any compact region where J has continuous partial derivatives, we may assume a small, say a ≤ e−1 /4. Suppose ﬁrst (say) x ≤ 2x − x . Then x  ≤ x − x  + x ≤ 3x − x . Since log 1 − x = O(x) on Ba (0), and since the function y log y is monotone increasing for y ≤ e−1 , we get J(x), J(x ) individually O(x − x  log x − x ). The remaining case to consider is when x, x  > 2x − x , so 1 > 1 − x/x  and log x/x  = O(1 − x/x ). We have 2 log x log 1 − x − log x  log 1 − x  x ≤ log x log 1 − x − log 1 − x  + log log 1 − x  x x = O(x − x  log x − x ) + O 1 − x  x = O(x − x  log x − x ). We can now ﬁnish oﬀ the proof of Theorem 9.1.1. Equation (9.1.3) can be rewritten ek JF,q (βk q r ) = O(N ) + O(1) + O(N 2 qN ) + O(N log ) r≤N (1−) k
Since JF,q (qx) − JF,q (x) = −CF and 1 + 2N (1 − ) ek JF,q (βk ) k
ek = 0, this becomes
= O(N ) + O(1) + O(N 2 qN ) + O(N log ).
Dividing by 2N (1 − ) + 1 and letting N → ∞ gives ek JF,q (βk ) = O( log ). k
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9. THE REGULATOR MAP AND ELLIPTIC CURVES. II
Now let → 0.
73
9.2. The proof of the corresponding result for Dq is similar but slightly less complicated. Theorem 9.2.1. Dq is a Steinberg function (Deﬁnition 5.3.1) on E. Proof. We keep the same notations (F , FN , F −K, βk (N, r), etc.) as above. We have, since D is a relative Steinberg function (9.2.1) 0= ek DFN βk (N, r) + eβN DFN (β ). r≤N (1−) k
Note n∈Z D(xq n ) converges, so DFN (x) = O(1) uniformly in x and N . Since the number of βN is O(N ) + O(1), we get ek DFN βk (N, r) = O(N ) + O(1). (9.2.2) r≤N (1−) k
Lemma 9.2.2. DFN (x) − DF,q (x) = O(N qN ) as N → ∞ uniformly in x for x ∈ AN (1−)+R . Proof. It suﬃces to show ∞ n N D(xq ) = O(N q ) n=N +1
on AN (1−)+R . This is an easy consequence (cf. the proof of Sublemma 9.1.4) of the obvious bound D(y) = O(y log y),
y → 0.
Substituting in (9.2.2), ek DF,q βk (N, r) = O(N ) + O(1) + O(N 2 qN ). (9.2.3) r≤N (1−) k
Lemma 9.2.3. For x, x ∈ C∗ , x − x  ≤ x  we have DF,q (x) − DF,q (x ) = O( log ),
→0
uniformly in x, x . Proof. It suﬃces to show D(xq r ) − D(x q r ) = O x − x  log x − x  + r qr
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74
9. THE REGULATOR MAP AND ELLIPTIC CURVES. II
uniformly for x in an annulus A. For this note x−x  = O(1/x−1/x ) on A, and D(x−1 ) = −D(x) so we can replace xq r by x−1 q −r and assume r ≥ 0. It thus suﬃces to show D(x) − D(x ) = O(x − x  log x − x ) for x, x ∈ Ba (0), the disk of radius a about 0. Since D(x) = −D(1−x), we may assume x, x ∈ / B1/3 (1), and since further the estimate is easy on compact sets on which D has continuous partial derivatives, we may assume a ≤ e−1 /4. The argument now is identical with the proof of Lemma 9.1.5. We can now rewrite (9.2.3) ek DF,q (βk q r ) = O(N ) + O(1) + O(N 2 qN ) + O(N log ). r≤N (1−) k
Divide by 2N (1 − ) + 1 and let N → ∞, getting ek DF,q (βk ) = O( log ). k
Now let → 0.
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https://doi.org/10.1090/crmm/011/11
LECTURE 10
Elements in K2 (E) of an Elliptic Curve E 10.1.
We now have a regulator map Rq : K2 C(E) → C
which induces by restriction a regulator on the global K2 (E). Abusing notation, we will also write Rq = Jq +iDq for the function on C∗ . Thus, dj ek if f , g are elliptic functions, dj = ek = 0, αj = βk = 1, and (f˜) = dj (αj ), (˜ g) = ek (βk ) are liftings of (f ), (g) to C∗ , we have Rq {f, g} = dj ek Rq (αj−1 βk ). The next step will be to construct interesting elements in K2 (E) to which we can apply Rq . Recall the exact sequence (valid for any smooth curve over a ﬁeld k) tame K2 (E) → K2 k(E) −−→ k(x)∗ . x∈E
When k is a number ﬁeld, the left hand arrow has torsion kernel, so it will suﬃce to construct elements in the kernel of the tame symbol. Also, Rq is trivial on symbols {f, c}, c ∈ C∗ , so it will suﬃce for our purposes to specify elements modulo the subgroup of K2 k(E) generated by symbols with one element constant. Proposition 10.1.1. Let f, g be functions on E deﬁned over k. Let C be an integer and assume all points of order C are deﬁned over k and that the divisors of f and g are supported in the group EC of such points. Then there exist fi ∈ k(E)∗ , ci ∈ k ∗ such that {f, g}C · {fi , ci } ∈ Ker(tame symbol). 75
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76
10. ELEMENTS IN K2 (E) OF AN ELLIPTIC CURVE E
Proof. Consider the diagram
div ⊗ Id / x∈E k ∗ k(E)∗ ⊗ k ∗ closed point
/ Pic(E) ⊗ k ∗
_
(10.1.1)
symbol
K2 k(E)
tame
/
x∈E k(x) closed point
∗
/0
.
Since f , g and all points of EC are deﬁned over k, we see
tame{f, g} ∈ k∗. Clearly tame{f, g}C → 0 in Pic(E)⊗k ∗ , so modifying {f, g}C by something in the image of k(E)∗ ⊗ k ∗ we get an element of Ker(tame). Exercise 10.1.2. Prove an analogous result without assuming all points of order C deﬁned over k. Let ρ denote a function on E with poles of order 1 at every nonzero point of EC and a zero of order C 2 − 1 at the origin. Note that such a function exists when EC ∼ = (Z/C)⊕2 . For example, if k → C, E(C) = C/[1, τ ], we have C−1
a + bτ C(C − 1) C(C − 1) = + τ ∈ [1, τ ]. C 2 2 a,b=0 For a ∈ EC , a = 0, let fa denote the function with zero of order C at a and pole of order C at 0. Deﬁne Sa ∈ Ker(tame) by (10.1.2)
Sa ≡ {ρ, fa }
(mod Image k(E)∗ ⊗ k ∗ ).
Sa is actually welldeﬁned modulo torsion and symbols with both entries constant, as one sees by looking at the kernel of div ⊗ Id in (10.1.1). When K2 (k) is torsion (e.g., k a number ﬁeld), Sa is welldeﬁned modulo torsion. 10.2. Our main result can be given a purely analytic formulation as follows: ﬁx E = C/[1, τ ] with y = Im τ > 0, q = e2πiτ , and let C > 0 be an integer. Let f : Z/CZ × Z/CZ → C be an odd function, f (−a, −b) = −f (a, b), and write (10.2.1)
C−1 1 ˆ f (k, ) = 2 f (a, b)e2πi(−ak+b)/C . C a,b=0
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10. ELEMENTS IN K2 (E) OF AN ELLIPTIC CURVE E
77
Theorem 10.2.1. With notation as above, C−1
iy 2 C 3 fˆ(k, )Rq (S k+τ ) = C π k,=0
f (m, n) . (mτ + n)2 (mτ + n)
m,n∈Z (m,n)=(0,0)
The relationship between the right hand side and the zeta function of E in the complex multiplication case will be discussed in the next lecture. The remainder of this lecture is devoted to the proof of Theorem 10.2.1. Lemma 10.2.2. We have 3 2 3 2πi(k+τ )/C 2 2 Rq (S(k+τ )/C ) = C Rq (e , ) + 4π y − + 3C 3 2C 2 6C where Rq on the right denotes the function Rq and Rq on the left is the regulator. Proof. Since Rq is trivial on symbols with one element constant Rq (S(k+τ )/C ) = Rq {ρ, f(k+τ )/C }
= Jq {ρ, f(k+τ )/C } + iDq (ρ) ∗ (f(k+τ )/C ) .
Writing σ : C → E = C/[1, τ ] and writing D q for the function on E, we see k + τ + Dq (−a) −D q −a + σ D q (ρ) ∗ (f(k+τ )/C ) = C C Ca=0 a=0 k + τ 2 − Dq (0) . + C(C − 1) Dq σ C Since Dq (0) = n∈Z D(q n ) = 0, the sum on the right collapses to k + τ , CDq σ C so (10.2.2) Dq (ρ) ∗ (f(k+τ )/C ) = C 3 Dq (e2πi(k+τ )/C ). To evaluate Jq (S(k+τ )/C ) we choose liftings of (ρ) and (f( k +τ )/C) to divisors on C∗ as follows (10.2.3)
(˜ ρ) = −
C−1
(e2πi(a+b)/C ) + (e2πiC(C−1)τ /2 ) + (C 2 − 1)(1)
a,b=0 2πi(k+τ )/C ) − (C − 1)(1) − (e2πiτ ). (f(k+τ )/C ) = C(e
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78
10. ELEMENTS IN K2 (E) OF AN ELLIPTIC CURVE E
We write
' −
A = Jq
C−1
2πi(a+bτ )/C
(e
( ) + C (1) ∗ (f(k+τ )/C ) 2
a,b=0
, B = Jq (eπiC(C−1)τ ) − (1) ∗(f(k+τ )/C ) so
ρ) ∗ (f(k+τ Jq (˜ )/C ) = A + B.
Note Jq (qx) − Jq (x) = −(log x)2
C(C−1)/2−1
Jq (q
C(C−1)/2
x) − Jq (x) = −
and if we replace x by a divisor D = dj αj = 1, we get Jq (q C(C−1)/2 ∗ D) − Jq (D) = −
(log xq r )2
r=0
dj (αj ) on C∗ with
dj = 0,
C(C − 1) dj (log αj )2 . 2 j
Take D = (f(k+τ )/C ), so 2 2 2 2 2 − dj (log αj ) = 4π y C (10.2.4) B = 2π 2 y 2 2 (C − 1)2 . We now compute A. A = C 3 Jq (e2πi((k+τ )/C) ) − (1) C−1 2πi[a+k+(b+)τ ]/C 2πi(a+b)/C − CJq (e ) − (e ) a,b=0 C−1 2πiτ 2 2πi[(a+bτ )/C+τ ] 2πi(a+bτ )/C )−(1) +Jq (e )−(e ) − C Jq (e a,b=0
= A1 − A2 − A3 + A4 . Note Jq (1) = 0, so
A1 = C 3 Jq (e2πi((k+τ )/C) ) C−1 −1 2πi[(a+bτ )/C+τ ] 2πi(a+bτ )/C (e ) − (e ) A2 = CJq a=0 b=0
= −4π 2 y 2 C 2
−1 2 b 2 2 ( − 1)(2 − 1) = −2π y 2 C 3 b=0
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10. ELEMENTS IN K2 (E) OF AN ELLIPTIC CURVE E
79
A3 = C 2 Jq (e2πiτ ) = −4π y C 2 2
2
−1
b2 = −2π 2 y 2
b=0
A4 = −4π y C 2 2
−1 C−1 b=0 r=0
=
2 2 C−1
−4π y C
s=0
s2 =
b r+ C
( − 1)(2 − 1) 2 C 3
2
−2π 2 y 2 (C − 1)(2C − 1) . 3
Combining these calculations, Rq (S(k+τ )/C ) = Jq (˜ ρ) ∗ (f(k+τ )/C ) + iD q (ρ) ∗ (f(k+τ )/C ) = A1 − A2 − A3 + A4 − B + iC 3 Dq (e2πi(k+τ )/C ) = C 3 Rq (e2πi(k+τ )/C ) − A2 − A3 + A4 + B ' C 2 (−1)(2−1) 3 2πi(k+τ )/C 2 2 (−1)(2−1) + = C Rq (e )+2π y 3 3 ( (C − 1)(2C − 1) 2 2 + (C − 1) − 3 3 2 3 2πi(k+τ )/C 2 2 . ) + 4π y − + = C Rq (e 3C 3 2C 2 6C C−1 2 2πi(−ak+b)/C Lemma 10.2.3. Recall fˆ(k, ) = /C . f (a, b)e a,b=0 We have 3 C−1 2 iy 2 f (0, n) 2 2 = 4π y − + . fˆ(k, ) 3 2 3 3c 2C 6C π n k,=0 n∈Z n=0
Proof. We use the wellknown Fourier series 3 e2πinx x x2 x 3 = 4π i − + n3 3 2 6 n∈Z n=0
to write 3 2 e2πin/C −y 2 i ˆ 2 2 4π y = − + fˆ(k, ) f (k, ) 3C 3 2C 2 6C π k, n3 n∈Z k, n=0
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80
10. ELEMENTS IN K2 (E) OF AN ELLIPTIC CURVE E
=
e2πin/C −y 2 i 2πi(−ak+b)/C f (a, b)e πC 2 a,b,k, n3 n=0
=
−y 2 i f (0, −n) y 2 i f (0, n) = , π n=0 n3 π n3
* because f is odd.
10.3. Using Lemmas 10.2.2 and 10.2.3, the main result, Theorem 10.2.1 can be rewritten C−1 f (m, n) y2i (10.3.1) . fˆ(k, )Rq (e2πi(k+τ )/C ) = π m,n∈Z (mτ + n)2 (mτ + n) k,=0 m=0
For the proof, it will be convenient to write ∞ ∞ Rq (x) = log xq n  log 1 − xq n  − log x−1 q n  log 1 − x−1 q n  n=0
n=1
dt log xq  arg(1 − xq ) − Im +i log(1 − t) t 0 n=0 x−1 qn ∞ dt −1 n −1 n −i log x q  arg(1 − x q ) − Im log(1 − t) t 0 n=1 ∞ ∞ n n −1 n −1 n log xq  log(1 − xq ) − log x q  log(1 − x q ) = n=0
−i
∞
∞
n
=
−
xq n
n=1
xq n
Im
n=0
Rq (x)
n
0
dt log(1 − t) − Im t n=1 ∞
x−1 q n 0
dt log(1 − t) t
iRq (x).
(In what follows xq n  < 1 for n ≥ 0 and x−1 q n  < 1 for n ≥ 1 so there will be no ambiguity about branches of functions.) Put L= (10.3.2)
C−1
fˆ(k, )Rq (e2πi((k+τ )/C) )
k,=0
M = −i
fˆ(k, )Rq (e2πi(k+τ )/C ).
Proposition 10.3.1. We have −y f (m, n) . L= 2π m∈Z n∈Z m(mτ + n)2 m=0
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10. ELEMENTS IN K2 (E) OF AN ELLIPTIC CURVE E
81
Proof. ∞ log 1 − e2πi[(k+τ )/C+nτ ] L = −2πy n+ fˆ(k, ) C n=0 k,=0 ∞ 2πi[(−k−τ )/C+nτ ] log 1 − e n− − . C n=1 C−1
To consolidate the two series note that (replacing k, by C − k, C − and n by n + 1), C−1
−
fˆ(k, )
k,=0
∞ n=1
n− C
log 1 − e2πi[(−k−τ )/C+nτ ]
log 1 − e2πi[(k+τ )/C+nτ ] = n+ C n=0 k=0 =1 C−1 ∞ ˆ log 1 − e2πi[(k+τ )/C+nτ ] n+ f (k, ) = C n=0 k,=0 C C−1
+
fˆ(k, )
C−1
∞
fˆ(k, C)
−
fˆ(k, 0)
k=0
=
C−1
fˆ(k, )
L = −4πy
C−1
fˆ(k, )
n log 1 − e2πi(k/C+nτ )
n=0
∞ n=0
k,=0
∞
∞ n=0
k,=0
Thus
n log 1 − e2πi(k/C+nτ )
n=1
k=0 C−1
∞
n+ C
n+ C
log 1 − e2πi[(k+τ )/C+nτ ] .
log 1 − e2πi[(k+τ )/C+nτ ]
∞ ∞ e2πim[(k+τ )/C+nτ ] 4πy 2πi[(−ak+b)/C] . n+ f (a, b)e = 2 C k,,a,b C m n=0 m=1 This can be rewritten (10.3.3)
∞ ne2πi[(mτ +b)/C] 4πy L= 2 . f (a, b) C a,b m n=0,m=1 m≡a mod C
Lemma 10.3.2. For Im x > 0,
∞
n=0
ne2πinx = −(4 sin2 πx)−1 .
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82
10. ELEMENTS IN K2 (E) OF AN ELLIPTIC CURVE E
Proof.
2πinx
ne
1 e2πix 1 d = = 2πi dx 1 − e2πix (1 − e2πix )2 1 −1 . = πix = (e − e−πix )2 4 sin2 πx
Returning to the expression for L (10.3.3), we see L=
−πy f (a, b) C 2 a,b
∞
1 . m sin π[(mτ + b)C] 2
m=1 m≡a mod C
Now use π2 1 = 2 sin πx n∈Z (x + n)2 to write L=
−y f (a, b) πC 2 a,b
−y = 2π
∞
m=1 n∈Z m≡a mod C
1 m[(mτ + b)/C + n]2
∞
f (m, n) . m(mτ + n)2 m,n=−∞ m=0
This proves Proposition 10.3.1. Proposition 10.3.3. Let M be as in (10.3.2) Then ∞ 1 −1 . f (m, n) Im M= 2π m,n=−∞ m2 (mτ + n) m=0
For x < 1, the “natural” branch of Proof. ∞ m − m=1 x /m2 . Thus
x 0
log(1 − t) dt/t is
* ∞ ∞ 2πi[(k+τ )/C+nτ ]m 2πi[(−k−τ )/C+nτ ]m e e . − fˆ(k, ) Im M =i m2 m2 n=0 n=1 k,=0 C−1
2
m=1
m=1
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10. ELEMENTS IN K2 (E) OF AN ELLIPTIC CURVE E
83
As with L, it will be convenient to combine the two sums: −
C−1
fˆ(k, )
k,=0
=
∞ e2πim[(k+τ )/C+nτ ] m2 m=n=1
C C−1
fˆ(k, )
k=0 =1 C−1
=
fˆ(k, )
k,=0
so
∞ e2πim[(k+τ )/C+nτ ] m2 m=1 n=0
∞
C−1 ∞ e2πim[(k+τ )/C+nτ ] ˆ e2πimk/C − , f (k, 0) 2 2 m m m=1 m=1 k=0 n=0
* ∞ 2πi[(k+τ )/C+nτ ]m e M = 2i fˆ(k, ) Im m2 n=0 k,=0 C−1 ∞ m=1 2πimk/C e −i fˆ(k, 0) Im m2 m=1 k=0 C−1
2
= M1 + M2 . Lemma 10.3.4. C−1 ∞ ∞ 1 f (m, b) 1 1 . M1 = − f (m, n) Im C b=0 m=1 m2 2π m,n=−∞ m2 (mτ + n) m=0
Proof.
* 2 ∞ e2πi[(k+τ )/C+nτ )]m 2i . f (a, b)e2πi[(−ak+b)/C] Im M1 = 2 2 C a,b,k, m =0 m=1
Note a,b f (a, b) Re(e2πi[(−ak+b)/C] ) = 0 because f (a, b) = −f (−a, −b), and also Im ie2πi[(−ak+b)/C] (∗) = − Im e2πi[(−ak+b)/C] Im(∗) + Re e2πi[(−ak+b)/C] Re(∗). It follows that ∞ e2πi[(m−a)k/C+(nC+)(mτ +b)/C] 2 M1 = 2 f (a, b) Im i C a,b,k, m2 n=0 2 = f (a, b) Im i C a,b
m=1 ∞
∞ e2πin(mτ +b)/C
m=1 n=0 m≡a mod C
m2
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84
10. ELEMENTS IN K2 (E) OF AN ELLIPTIC CURVE E
Using the identity 1 i cot πx + 1 = 2πix 1−e 2 we can rewrite 1 f (a, b) Im M1 = C a,b
∞ m=1 m≡a mod C
− cot π([mτ + b]/C) + i m2
2 ∞ * ∞ f (m, b) 1 1 mτ + b 1 . − f (a, b) Im cot π = 2 2 C m=1 m C a,b m C =1 b
Note
m≡a
N 1 1 cot πx = lim , π N →∞ n=−N x + n
1 1 Im Im(cot πx) = π n∈Z x+n
and
(the series converging without any special sort of summation), so we get ∞ ∞ 1 f (m, b) 1 1 M1 = − f (a, b) Im C m=1 m2 πC a,b m2 ([mτ +b]/C +n) m=1 n∈Z 1 = C
b ∞
m≡a
f (m, b) 1 1 . − f (m, n) Im m2 2π m∈Z n∈Z m2 (mτ + n) m=1 b
m=0
This completes the computation of M1 .
Lemma 10.3.5. M2 =
∞ C−1 −1 f (m, b) . C m=1 b=0 m2
Proof. mk C−1 ∞ e2πi C ˆ f (k, 0) Im M2 = −i m2 m=1 k=0 mk ∞ e2πi C −i −2πiak/C f (a, b)e Im = 2 C k,a,b m2 m=1 k ∞ e2πi C (m−a) −1 f (m, b) −1 = = 2 f (a, b) Im i . 2 2 C k,a,b m C m m=1
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10. ELEMENTS IN K2 (E) OF AN ELLIPTIC CURVE E
85
We return now to the proof of Proposition 10.3.3. −1 1 f (m, n) Im M = M1 + M2 = 2π m=0 n∈Z m2 (mτ + n)
as desired. We can now complete the proof of Theorem 10.2.1: C−1
fˆ(k, )Rq (e2πi(k+τ )/C ) = L + M 2 * k,=0 1 −y 1 1 . = f (m, n) + Im 2 2π m∈Z n∈Z m(mτ + n)2 y m (mτ + n) m=0
But 1 1 1 + Im 2 2 m(mτ + n) y m (mτ + n) 1 1 1 1 − + = m(mτ + n)2 2iy m2 (mτ + n) m2 (mτ + n) 1 1 = − 2 m(m + τ n) m(mτ + n)(mτ + n) 1 = −2iy (mτ + n)2 (mτ + n) so C−1
f (m, n) y2i 2πi(k+τ )/C ˆ . f (k, )Rq (e )= π m,n∈Z (mτ + n)2 (mτ + n) k,=0 m=0
This completes the proof of (10.3.1) and also Theorem 10.2.1.
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https://doi.org/10.1090/crmm/011/12
LECTURE 11
A Regulator Formula 11.1. In this ﬁnal lecture, we will show how Theorem 10.3.2 leads to a regulator formula for the value at s = 2 of the zeta function of an elliptic curve E with complex multiplication by the ring of integers in an imaginary quadratic ﬁeld κ with class number 1. We begin with some general lemmas, valid without hypotheses on E. Notation and hypotheses will be as in Lecture 10. Lemma 11.1.1. Rq (S(a+bτ )/C ) = −Rq (S(−a−bτ )/C ). Proof. The divisor (ρ) (10.1.2) is invariant under the automorphism of Egiven by multiplication by −1, so 2(ρ) can be lifted to a ∗ −1 C divisor dj (αj ) invariant under α → α and such that dj = 0, dj αj = 1. Let ek (βk ) be a lifting of C ([a + bτ ]/C) − (0) with ek (βk−1 ) lifts C ([−a − bτ ]/C) − (0) . ek = 0, βkek = 1. Then We must show dj ek Jq (αj−1 βk ) + Jq (αj βk−1 ) = 0 = dj ek Dq (αj−1 βk ) + Dq (αj βk−1 ) . It follows from Lemma 6.2.5 that Dq (x−1 ) = −Dq (x), so the right hand equality is clear. For the left hand, we know Jq (x)+Jq (x−1 ) = (log x)2 from which the desired result follows. It is convenient to alter slightly the deﬁnition of the Fourier transform employed in Lecture 10. We write f (a + bτ ) in place of f (a, b), and deﬁne (11.1.1)
C−1 1 fˆ(k + τ ) = f (a + bτ )e2πi[(−a+bk)/C] . C a,b=0
(Note the interchange of k and .) Our fundamental result, Theorem 10.2.1, becomes 87
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88
11. A REGULATOR FORMULA
(11.1.2)
C−1
fˆ(k + τ )Rq (S(k+τ )/C )
k,=0
=
iy 2 C 4 π
(m,n)=(0,0)
f (m + nτ ) . (m + nτ )2 (m + nτ )
Lemma 11.1.2. Rq (S(a+bτ )/C ) =
y2C 3 π
(m,n)=(0,0)
sin(2π[an − bm]/C) (m + nτ )2 (m + nτ )
Proof. Let ga+bτ (k + τ ) = e2πi(a−bk)/C /C. Then 1 2πi(a−bk)/C 2πi(−kn+m)/C e e C 2 k, 1 2πi[(a+m)−k(b+n)]/C = 2 e  C k, 0 a = −m or b = −n = 1 a = −m, b = −n
gˆa+bτ (m + nτ ) =
= δ−a−bτ (m + nτ ) with obvious notation. Writing fa+bτ = g−a−bτ − ga+bτ we get 1 ˆ fa+bτ (k + τ )Rq (S(k+τ )/C ) 2 iy 2 C 4 fa+bτ (m + nτ ) = 2π (m + nτ )2 (m + nτ ) y 2 C 3 sin(2π[an − bm]/C) = . π (m + nτ )2 (m + nτ )
Rq (S(a+bτ )/C ) =
The following description of the “modular behavior” of Rq seemed worth including although it will not be used: α γ Lemma 11.1.3. Let τ = (aτ + β)/(γτ ∈ SL2 (Z). + δ) with β δ α γ b 2πiτ b and let ( a ) = β δ a . Then Write q = e Rq (S(a +b τ )/C ) = (γτ + δ)2 (γτ + δ)Rq (S(a+bτ )/C ). 0 1 Proof. Since SL2 (Z) preserves the symplectic form ( −1 0 ), we ﬁnd α γ −a + b k = −a + bk, where ( k ) = β δ k . Thus
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11. A REGULATOR FORMULA
Rq (S(a +b τ )/C ) =
89
yC 3 sin(2π[a − b k ]/C) π (k + τ )2 (k + τ ) k ,
yC sin(2π[a − bk]/C) . = π k , (k + τ )2 (k + τ ) 3
Note k + τ = (γτ + δ)−1 (k δ + β + (k γ + α)τ ) = (γτ + δ)−1 (k + τ ) whence Rq (S(a +b τ )/C ) =
yC 3 (γτ + δ)2 (γτ + δ) sin(2π[a − bk]/C) π (k + τ )2 (k + τ ) k,
= (γτ + δ)2 (γτ + δ)Rq (S(a+bτ )/C ).
It is convenient to denote by , the bilinear form , : O/CO × O/CO → C∗ a + bτ, k + τ = e2πi(−a+bk)/C . We assume henceforth that O = Z + Zτ is an order in an imaginary quadratic ﬁeld κ. Lemma 11.1.4. xy, z = x, y¯z, where y¯ is the complex conjugate. Proof. Write x = x1 +x2 τ , y = y1 +y2 τ , z = z1 +z2 τ , and suppose τ 2 + Aτ + B = 0, with xi , yi , zi , A, B ∈ Z. Then xy = x1 y1 − x2 y2 B + (x1 y2 + x2 y1 − x2 y2 A)τ x2 y2 Bz2 −x1 y1 z2 +x1 y2 z1 +x2 y1 z1 −x2 y2 Az1 xy, z = exp 2πi C y¯z = y1 z1 + y2 z2 B + y2 z1 τ + y1 z2 τ = (y1 z1 + y2 z2 B − y2 z1 A) + (y1 z2 − y2 z1 )τ −x1 y1 z2 +x1 y2 z1 +x2 y1 z1 +x2 y2 z2 B −x2 y2 z1 A x, yz = exp 2πi C Corollary 11.1.5. If ζ ∈ Oκ is a unit (i.e., a root of 1) then Rq (Sζ(a+bτ )/C ) = ζ −1 Rq (S(a+bτ )/C ).
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90
11. A REGULATOR FORMULA
Proof. Rq (Sζ(a+bτ )/C ) = =
y 2 C 3 Imζ(a + bτ ), k + τ π (k + τ )2 (k + τ ) y 2 C 3 Ima + bτ, ζ(k + τ )
π (k + τ )2 (k + τ ) y 2 C 3 Ima + bτ, k + τ = π ζ(k + τ )2 (k + τ ) = ζ −1 Rq (S(a+bτ )/C ).
Assume now for simplicity that O is the ring of integers in κ, and κ has class number 1. Corollary 11.1.6. Suppose C = f g, f , g ∈ Oκ . Then g¯ Rq (S(a+bτ )/C+f μ/C ) = Rq (S(a+bτ )/f ). μ∈O/gO
Proof. We have
N (g) g¯  k + τ μ, f¯(k + τ ) = 0 otherwise. μ∈O/gO
Thus
Rq (S(a+bτ )/C+f μ/C ) μ∈O/gO y2C 3 Ima + bτ, k + τ = μ, f¯(k + τ ) π (k + τ )2 (k + τ ) μ k, = N (g) =
y 2 C 3 Ima + bτ, g¯(r + sτ ) π r,s (r + sτ )2 (r + sτ )¯ g2 g
y 2 C 3 Img(a + bτ ), r + sτ g¯π r,s (r + sτ )2 (r + sτ )
= (¯ g )−1 Rq (Sg(a+bτ )/C ) = (¯ g )−1 Rq (S(a+bτ )/f ). Lemma 11.1.7. Let χ have conductor f  C, C = f g. Then (i) χ (x) = 0 unless C  f¯x. (xy) = χ(¯ x) χ(y). (ii) ) For x ∈ (O/CO)∗, χ ¯ (iii) Let f1 f and suppose C  f1 x. Then χ (x) = 0.
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11. A REGULATOR FORMULA
Proof. (i) We have 1 1 χ (x) = χ(y)x, y = C C y∈O/CO
χ(y )
y mod f
91
x, y x, f y
y mod g
1 χ(y )x, y xf¯, y C y mod f y mod g 0 g¯ x = N (g)( y mod f χ(y )x, y )/C g¯  x, =
where y (resp. y ) run through representatives in O for the congruence classes modulo f (resp. g). (ii) If x ∈ (O/CO)∗, 1 1 χ(z)xy, z = χ(¯ y) χ(z y¯)x, y¯z = χ(¯ y ) χ(x). χ (xy) = C C z z∈O/CO
(iii) Since f1 f = conductor χ, we can ﬁnd y ≡ 1 mod f¯1 , y a unit in O/CO, with χ(¯ y ) = 1. Since C  f¯1 · x χ (x) = χ (xy) = χ(¯ y ) χ(x). Thus χ (x) = 0.
11.2. We now have the tools we need for the regulator formula. Assume as above that O = Z + Zτ is the ring of integers in an imaginary quadratic number ﬁeld κ of class number 1. We ﬁx an embedding κ → C, an integer C, and a character χ of (O/CO)∗ which restricts to the given embedding μκ → C∗ on the roots of 1. Let χGross denote the Gr¨ossencharakter χGross (p) = hχ(h),
p = (h), p C.
Let f generate the conductor ideal of χ and write C = f g. From (11.1.2), Corollary 11.1.5, Corollary 11.1.6, and Lemma 11.1.7 we get π L(2, χGross ) = 2 4 (11.2.1) χ (w)Rq (Sw/C ) iy C w∈O/CO π = 2 4 χ (x¯ g )Rq (Sxf¯−1 ) iy C ¯ ∗ x∈(O/f O)
=
πχ (¯ g) 2 iy C 4
χ(¯ x)Rq (Sxf¯−1 )
x∈(O/f¯O)∗
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92
11. A REGULATOR FORMULA
= =
πχ (¯ g )g 2 iy C 4
χ(¯ x)Rq (Sx/C )
x∈(O/CO)∗
χ(¯ g )g πμκ  2 4 iy C
x∈(O/CO)∗ /μ
χ(¯ x)Rq (Sx/C ). κ
The jinvariant j(E) is known to be real and to generate the Hilbert class ﬁeld of κ. Since κ has class number 1, j(E) ∈ Q so we can choose a model EQ deﬁned over Q. Deuring’s theory associates to EQ a Gr¨ossencharakter χGross of κ with values in κ∗ , χGross (p) = x¯χ(x), x) = where (x) = p C. The character χ takes values in μκ and χ(¯ χ(x) (this follows from [Lan73, Theorem 10, p. 140], which implies χGross (p) = χGross (p),) so we can rewrite (11.2.1) χ(¯ g )g πμκ  Gross (11.2.2) L(2, χ )= Rq S xχ(x) C iy 2 C 4 ∗ (O/CO) /μκ
The ray class group (O/CO)∗ /μκ acts on points of order C by $ y % x−1 χ(x)y = x· C C and conjugation acts on these points in the natural way. The element dfn (11.2.3) U = Sxχ(x)/C (O/CO)∗ /μκ
is invariant under both these actions, and hence lies in K2 (EQ ) ⊗ Q. (This follows from the existence of a norm map K2 (EL ) → K2 (EQ ) for L/Q ﬁnite.) We get Theorem 11.2.1. With notations as above, let χGross be the Gr¨ossencharakter associated to EQ . Let f be a generator for the conductor ideal and let f g = C ∈ Z, g ∈ O. Then there exists U ∈ K2 (EQ ) ⊗ Q (deﬁned as above) such that (11.2.4)
L(2, χGross ) =
χ(¯ g )g πμκ  Rq (U ). iy 2 C 4
Remark 11.2.2. (i) L(s, χGross ) is related to the zeta function of EQ , ζEQ (deﬁned to be the product over all p ∈ Z
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11. A REGULATOR FORMULA
93
such that E has nondegenerate reduction modp of the zeta function of the corresponding curve over Fp ), by ζEQ (s) = ζQ (s)ζQ (s − 1)L(s, χGross )−1 . (ii) It is presumably possible to renormalize the additive character deﬁning χ so as to remove C and g from (11.2.4) and obtain a formula solely in terms of the conductor of χ. If the conductor ideal comes from Z, we can take g = 1 and get such a formula immediately. Since L(s, χGross ) has a product expansion converging for Re s > 32 , we see from (11.2.4) Corollary 11.2.3. The element U ∈ K2 (EQ ) ⊗ Q deﬁned above is nonzero. In view of Borel’s work, the following conjecture seems irresistible: Conjecture 11.2.4. U spans K2 (EQ ) ⊗ Q. More generally, for any elliptic curve over a number ﬁeld one could conjecture that rk K2 (E) = order of zero of L(E, s) at s = 0, assuming of course the existence of a functional equation relating L(E, s) and L(E, 2 − s).
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Bibliography [Bas71] H. Bass, K2 des corps globaux [d’apr`es J. Tate, H. Garland,. . . ], Exp. No. 394, S´eminaire Bourbaki, 23e ann´ee (1970/1971), Lecture Notes in Math., vol. 244, 1971, pp. 233–255. [Blo78] S. Bloch, Applications of the dilogarithm function in algebraic Ktheory and algebraic geometry, Proceedings of the International Symposium on Algebraic Geometry (Kyoto, 1977) (M. Nagata, ed.), Kinokuniya , Tokyo, 1978, pp. 103–114. [Bor53] A. Borel, Sur la cohomologie des espaces ﬁbr´es principaux et des espaces homog`enes de groupes de Lie compacts, Ann. of Math. (2) 57 (1953), 115–207. ´ [Bor74] A. Borel, Stable real cohomology of arithmetic groups, Ann. Sci. Ecole Norm. Sup. (4) 7 (1974), 235–272. [Bor77] A. Borel, Cohomologie de SLn et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 4, 613– 636. [BS66] Z. I. Borevich and I. R. Shafarevich, Number theory, Pure Appl. Math., vol. 20, Academic Press, New YorkLondon, 1966. [BSD65] B. Birch and H. P. F. SwinnertonDyer, Notes on elliptic curves. II, J. Reine Angew. Math. 218 (1965), 79–108. [Cox35] H. S. M. Coxeter, The functions of Schl¨ aﬂi and Lobatschevsky, Quart. J. Math. Oxford Ser. 6 (1935), 13–29. [Hel62] S. Helgason, Diﬀerential geometry and symmetric spaces, Pure Appl. Math., vol. 12, Academic Press, New YorkLondon, 1962. [HM62] G. Hochschild and G. D. Mostow, Cohomology of Lie groups, Illinois J. Math 6 (1962), 367–401. [Keu78] F. Keune, The relativization of K2 , J. Algebra 54 (1978), no. 1, 159–177. [Lan73] S. Lang, Elliptic functions, AddisonWesley, Reading, MALondonAmsterdam, 1973. [Ler51] J. Leray, Sur l’homologie des groupes de Lie, des espaces homog`enes, et des espaces ﬁbr´es principaux, Colloque de topologie (espace ﬁbr´es) (Bruxelles, 1950), G. Thone, Li`ege, 1951, pp. 101–115. [Lic73] S. Lichtenbaum, Values of zetafunctions, ´etale cohomology, and algebraic Ktheory, Algebraic KTheory. II. “Classical” Algebraic KTheory and Connections with Arithmetic (Seattle, WA, 1972) (H. Bass, ed.), Lecture Notes in Math., vol. 342, SpringerVerlag, BerlinNew York, 1973, pp. 489– 501. [Moo76] C. C. Moore, Group extensions and cohomology. III, Trans. Amer. Math. Soc. 221 (1976), no. 1, 1–33.
95
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96
BIBLIOGRAPHY
[Qui73] D. Quillen, Higher algebraic Ktheory. I, Algebraic KTheory. I. Higher KTheories (Seattle, WA, 1972) (H. Bass, ed.), Lecture Notes in Math., vol. 341, SpringerVerlag, BerlinNew York, 1973, pp. 85–147. n 2 [Rog07] L. J. Rogers, On function sums connected with the series x /n , Proc. London Math. Soc. (2) 4 (1907), 169–189. [Tat71] J. Tate, Symbols in arithmetic, Actes du Congr`es International des Math´ematiciens (Nice, 1970), vol. 1, GauthierVillars, Paris, 1971, pp. 201–211. [vE53] W. I. van Est, Group cohomology and Lie algebra cohomology in Lie groups. I, Nederl. Akad. Wetensch. Proc. Ser. A 56 (1953), 484–492, II, 493–504. [Wei61] A. Weil, Adeles and algebraic groups, Institute for Advanced Study, Princeton, NJ, 1961. [Wei67] A. Weil, Basic number theory, Grundlehren Math. Wiss., vol. 144, SpringerVerlag, New York, 1967.
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Index Ktheory, 1, 32, 35, 36, 51
Quillen, 1, 2, 35–37
Bass, 1 Birch, 3 Borel, ii, 2, 3, 12, 25, 29, 33, 34, 54, 58, 93 Borel regulator, ii, 4
regulator, ii, 1, 3, 6, 35, 60, 61, 75, 77 regulator formula, ii, 51, 87, 91 regulator map, 3, 5, 6, 34, 35, 53, 61, 75 restriction of scalars, 12
class number formula, 1, 3 Coates, 3 continuous cohomology, 2, 23, 24, 29, 32 convergence factors, 11–13, 15, 16 Coxeter, 60
Steinberg function, 40, 43, 48, 61, 62, 73 Steinberg symbol, 4 Stienstra, 40 SwinnertonDyer, 3 Tamagawa measure, 11–13, 18 Tamagawa number, 9, 12, 16
Deuring, 92 dilogarithm function, ii, 43 Dirichlet Lseries, 6, 53, 56
Wigner, ix, 4 Wiles, 3
Eilenberg, 23 elliptic curves, ii, ix, 2–4, 35, 39, 60, 61, 87, 93 elliptic functions, 5, 62, 64, 67, 69, 75
zeta function, 1, 6, 17, 18, 20, 51, 61, 92, 93
Fubini, 10 HasseWeil zeta function, ii, 3 Hilbert class ﬁeld, 3, 92 indecomposables, 26, 30 Keune, 4, 39, 40 Kummer, 5, 6, 51 Lichtenbaum, 1, 2 MacLane, 23 Matsumoto, 39, 40 Milnor, 1 Poisson summation, 20, 21 97
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Higher Regulators, Algebraic KTheory, and Zeta Functions of Elliptic Curves Spencer J. Bloch This book is the longawaited publication of the famous Irvine lectures by Spencer Bloch. Delivered in 1978 at the University of California at Irvine, these lectures turned out to be an entry point to several intimatelyconnected new branches of arithmetic algebraic geometry, such as regulators and special values of Lfunctions of algebraic varieties, explicit formulas for them in terms of polylogarithms, the theory of algebraic cycles, and eventually the general theory of mixed motives which unifies and underlies all of the above (and much more). In the 20 years since, the importance of Bloch's lectures has not diminished. A lucky group of people working in the above areas had the good fortune to possess a copy of old typewritten notes of these lectures. Now everyone can have their own copy of this classic.
CRMM/11.S
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