Higher Regulators, Algebraic K-Theory, and Zeta Functions of Elliptic Curves
 0821821148

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

Citation preview

Volume 11

C R M

CRM MONOGRAPH SERIES Centre de Recherches Mathématiques Université de Montréal

Higher Regulators, Algebraic K-Theory, 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 K-theory, 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 measure-valued 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 K-Theory, 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 η-function, 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 Hasse-Weil 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 Cataloging-in-Publication Data Bloch, Spencer. Higher regulators, algebraic K-theory, and zeta functions for elliptic curves / Spencer J. Bloch. p. cm. — (CRM monograph series, ISSN 1065-8599 ; v. 11) Includes bibliographical references and index. ISBN 0-8218-2114-8 (alk. paper) 1. Curves, Elliptic. 2. K-theory. 3. Functions, Zeta. I. Title. II. Series. QA567.2.E44.B56 516.352—dc21

2000 00-029969

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 02904-2294 USA. Requests can also be made by e-mail 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 acid-free and falls within the guidelines 

established to ensure permanence and durability. This volume was typeset using AMS-TeX, 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  |D|w

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 non-negative integer m, where K∗ (OK ) denotes the K-theory 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) |D|1/2 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 adelicTamagawa-Weil 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 K-groups 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 Hasse-Weil 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 Swinnerton-Dyer has impressive numerical evidence behind it. The idea is to describe the behavior of the Hasse-Weil 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 Hasse-Weil 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 Hasse-Weil 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 Hasse-Weil 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 set-theoretic 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 K-group 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 semi-local 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 single-valued, vanished at 0, 1, ∞, and represented an R-valued continuous 3-cohomology 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 well-defined 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 non-zero 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 L-series 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 non-trivial 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 non-Archimedean, Oν ⊂ kν ring of integers.



 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 n-form 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 (ν-adic) 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 non-vanishing n-form 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 S-integers, together with a non-vanishing n-form 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 p-adic 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, non-vanishing 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 non-vanishing n-form ω. Let V (Ak ) denote the adele-valued 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 non-vanishing n-form 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 k-scheme T RK/k V (T ) = V (T × K). k

In particular (RK/k V ) × k¯ = k





σ

where σ runs through embeddings K → k¯ over k, and V σ denotes the ¯ k-variety 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 non-vanishing n-form ω, it is natural to look  for a non-vanishing nd-form RK/k ω on RK/k V , and to expect λν = w|ν 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 well-defined 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 non-vanishing left-invariant 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 left-invariant 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)| |z|n )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 one-to-one 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 non-zero 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 translation-invariant 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 X|s−n p (dX)p Mn (Op )∗

where Mn (Op )∗ denotes the set of matrices with non-zero 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 n-tuples 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 A|p | 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 A|sp

A





| det A|sp

(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 G-module, it is natural to conceive of a cohomology theory H ∗ (G, A) analogous to the Eilenberg-MacLane 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 cross-ratio 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 q-forms 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 G-modules. 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 left-invariant 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 R-vector 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 3-forms is a translation-invariant function, i.e., constant, so 3 dim Hcont (G, R) ≤ 1. But the volume form on G/K is G-invariant, 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 semi-simple 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 R-Lie 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 Q-structures. 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 5-term 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 torsion-free 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 right-invariant 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 right-invariant forms which are bi-invariant 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 torsion-free, 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 torsion-free, 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 Q-structure 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 torsion-free 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 torsion-free. 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 non-trivial, 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 non-trivial.  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 + 2j-form ω  which is non-zero, translation-invariant, 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 K-theory. 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 |D|1/2 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) |D|1/2 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) |D|1/2 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) |D|1/2 ζk (m + 1)Lm (Γn , Q).  Let Γn = SLn (Ok ), Γ = SLn (Ok ). Recall that Kp (Ok ) is the pth + homotopy group of an H-space 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) |D|1/2 ζk (m + 1). Remark 4.3.3. The functional equation for ζk shows lim ζk (s)(s + m)−dm ∈ π −d(m+1)+dm |D|1/2 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 3-cocycle 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 K-theory 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 locally-free 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 semi-local 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 K-theory 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 semi-local 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 well-known 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 well-known 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 semi-local 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 well-defined 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 non-principal 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 well-defined 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 well-defined 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 A-module of holomorphic 1-forms 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) it is 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 function-theoretic 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 K-theory, 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 L-series. 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 L-series 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 cross-ratio. 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 under the diagonal action of PGL2 (C) and satisfy the cocycle condition (−1)i f (z1 , . . . , zˆi , . . . , z5 ) = 0 forms a one dimensional R-vector space generated by D({z1 , . . . z4 }). For the proof, let PGL2 (C) and define a complex of G-modules  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 G-modules, 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 G-invariant 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 non-trivial, 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 cross-ratio 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 3-cocycle 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 differential-geometric 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 well-known. 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 |q|N ≤ |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 |q|N +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 − |q|r 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| s|w|) 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(|q|N  ) 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 − |q|n · const) n

which are bounded independent of N . The bound |TN (w) − 1| = O(|q|N  ) 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 |K||TN (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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70

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 |q|N  ) 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 |q|n−N0 log(M |q|n−N0 ) . Since the series ∞ n=0

|q|n

log(M |q|n ) 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 |q|N  ). (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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72

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(|q|r 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| ≤ 2|x − x |. Then |x | ≤ |x − x | + |x| ≤ 3|x − 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 | > 2|x − 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 |q|N  ) + 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 |q|N  ) + 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 |q|N  ) 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 |q|N  ). (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| |q||r|

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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 |q|N  ) + 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 well-defined 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 well-known 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 E given 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 π|μκ | 2 4 iy C

x∈(O/CO)∗ /μ

χ(¯ x)Rq (Sx/C ). κ

The j-invariant 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 π|μκ | 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 π|μκ | 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 non-degenerate 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 non-zero. 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 K-theory 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 York-London, 1966. [BSD65] B. Birch and H. P. F. Swinnerton-Dyer, 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 York-London, 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, Addison-Wesley, Reading, MA-London-Amsterdam, 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 zeta-functions, ´etale cohomology, and algebraic K-theory, Algebraic K-Theory. II. “Classical” Algebraic K-Theory and Connections with Arithmetic (Seattle, WA, 1972) (H. Bass, ed.), Lecture Notes in Math., vol. 342, Springer-Verlag, Berlin-New 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 K-theory. I, Algebraic K-Theory. I. Higher K-Theories (Seattle, WA, 1972) (H. Bass, ed.), Lecture Notes in Math., vol. 341, Springer-Verlag, Berlin-New 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, Gauthier-Villars, 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, Springer-Verlag, New York, 1967.

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Index K-theory, 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 Swinnerton-Dyer, 3 Tamagawa measure, 11–13, 18 Tamagawa number, 9, 12, 16

Deuring, 92 dilogarithm function, ii, 43 Dirichlet L-series, 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 Hasse-Weil 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 K-Theory, and Zeta Functions of Elliptic Curves Spencer J. Bloch This book is the long-awaited 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 intimately-connected new branches of arithmetic algebraic geometry, such as regulators and special values of L-functions 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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