Noncommutative Iwasawa Main Conjectures over Totally Real Fields

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Author: John Coates Peter Schneider R. Sujatha Otmar Venjakob

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Springer Proceedings in Mathematics and Statistics Volume 29

Springer Proceedings in Mathematics and Statistics

This book series features volumes composed of select contributions from workshops and conferences in all areas of current research in mathematics and statistics, including OR and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

For further volumes: http://www.springer.com/series/10533

John Coates Peter Schneider Otmar Venjakob

R. Sujatha

Editors

Noncommutative Iwasawa Main Conjectures over Totally Real Fields Münster, April 2011

123

Editors John Coates Department of Pure Mathematics and Mathematical Statistics (DPMMS) University Cambridge Cambridge United Kingdom

Peter Schneider Institute of Mathematics University of Münster Münster Germany

R. Sujatha Department of Mathematics University of British Columbia Vancouver British Columbia Canada

Otmar Venjakob Institute of Mathematics University of Heidelberg Heidelberg Germany

ISSN 2194-1009 ISSN 2194-1017 (electronic) ISBN 978-3-642-32198-6 ISBN 978-3-642-32199-3 (eBook) DOI 10.1007/978-3-642-32199-3 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2012950783 Mathematical Subject Classification (2010): 11R23, 11S40, 14H52, 14K22, 19B28 c Springer-Verlag Berlin Heidelberg 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

Hi no hikari Yabu shi wakaneba IsonokamiFurinishi sato ni Hana mo sakikeri

For the light of the sun Shuns not the wild thickets, Even in IsonokamiThis village grown ancient, The flowers are in bloom. -Furu Imamichi (Kokinshu). The mysterious link between special values of complex zeta and L-functions and purely arithmetic problems was discovered by Dirichlet and Kummer in the nineteenth century, and spectacularly generalized in the twentieth century by Birch and Swinnerton-Dyer with the formulation of their celebrated conjecture on the arithmetic of elliptic curves. We owe to Iwasawa the great discovery that these problems can be attacked by p-adic methods, where p is any prime number, provided one is prepared to work with a class of infinite Galois extensions of the base field F (which is always supposed to be a finite extension of Q). Iwasawa himself only considered those Galois extensions whose Galois group is isomorphic to the additive group of the ring of p-adic integers Zp and the trivial Tate motive. However, it soon became apparent that his methods ought to apply to a much wider class of infinite extensions, namely those whose Galois group over F is a p-adic Lie group of dimension 1, and to a large class of motives defined over F . While it is still not known how to formulate it in complete generality, it is now widely believed that, in this general setting, the link between special values of complex L-functions and arithmetic should be expressed by what is known as a main conjecture. Very roughly speaking, such v

vi

Preface

a main conjecture should assert that an appropriate p-adic L-function, interpolating special values of the relevant complex L-functions, should coincide with a certain algebraically defined invariant, usually called a characteristic element, which arises naturally from the arithmetic of the motive over the p-adic Lie extension. This book arose from a workshop held at the University of Münster from April 25–30, 2011. The principal aim of this Workshop was to present the proof of the first key example of these general ideas, namely, the case when the motive is the trivial Tate motive and the p-adic Lie extension F1 of F is totally real (in addition, we always assume that F1 contains the cyclotomic Zp -extension of Q). The first important progress on this problem goes back to Iwasawa himself, although we owe to Mazur and Wiles the first complete proof of the most classical case of this main conjecture (when F1 is a the compositum of a real abelian base field F with the cyclotomic Zp -extension of Q). Subsequently, Wiles discovered a deep new method, relying heavily on the theory of automorphic forms, for attacking these problems. In this way, he succeeded in proving the main conjecture when the base field F is any totally real number field, and the Galois group of F1 over F is abelian. This book is concerned with the problem of how one can extend Wiles’ work to establish the general non-abelian totally real main conjecture for the trivial Tate motive. Two approaches for doing this were discovered independently and simultaneously, by Kakde on the one hand, generalizing ideas of Kato, and by Ritter and Weiss on the other. Both methods do in fact require one to assume a standard conjecture of Iwasawa about the vanishing of his cyclotomic -invariant, and so far this has only been proven when the base field F is an abelian extension of Q and the Galois group of F1 over F is assumed to be pro-p. Both approaches are discussed in this book, but, following the lectures at the Workshop, it is largely Kakde’s method which is treated in detail here. One reason for doing this is that the remarkable set of congruences established by Kakde to describe the K1 - group of the Iwasawa algebra of any compact p-adic Lie group should also apply to attacking the noncommutative main conjecture for other motives. Finally, for reasons of space, the book only contains a written version of the lectures at the workshop which were closely related to the proof of the main conjecture. The Scientific Committee for the Münster Workshop consisted of J. Coates, P. Schneider (Chairman), R. Sujatha, and O. Venjakob. It was made possible by the generous financial support of Project A2 within the DFG Collaborative Research Center 878 “Groups, Geometry, and Actions” at Münster and by some additional funding from the ERC Starting Grant IWASAWA at Heidelberg. Cambridge, UK Münster, Germany Vancouver, Canada Heidelberg, Germany

John Coates Peter Schneider R. Sujatha Otmar Venjakob

Contents

Introduction to the Work of M. Kakde on the Non-commutative Main Conjectures for Totally Real Fields . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . John Coates and Dohyeong Kim

1

Reductions of the Main Conjecture . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . R. Sujatha

23

The Group Logarithm Past and Present . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Ted Chinburg, Georgios Pappas, and M.J. Taylor

51

K1 of Certain Iwasawa Algebras, After Kakde . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Peter Schneider and Otmar Venjakob

79

Congruences Between Abelian p-Adic Zeta Functions . .. . . . . . . . . . . . . . . . . . . . 125 Mahesh Kakde On the Work of Ritter and Weiss in Comparison with Kakde’s Approach .. . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 159 Otmar Venjakob Noncommutative Main Conjectures of Geometric Iwasawa Theory .. . . . . . 183 Malte Witte

vii

•

Contributors

Ted Chinburg Department of Mathematics, University of Pennsylvania, Philadelphia, USA John Coates Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, UK Mahesh Kakde Department of Mathematics, King’s College London, London, UK Dohyeong Kim Department of Mathematics, POSTECH, Pohang, Gyeongbuk, Republic of Korea Georgios Pappas Department of Mathematics, Michigan State University, East Lansing, USA Peter Schneider Universität Münster, Mathematisches Institut, Münster, Germany R. Sujatha School of Mathematics, Tata Institute of Fundamental Research, Mumbai, India Department of Mathematics, University of British Columbia, Vancouver, Canada Martin J. Taylor Merton College, Oxford University, Oxford, UK Otmar Venjakob Universität Heidelberg, Mathematisches Institut, Heidelberg, Germany Malte Witte Universität Heidelberg, Mathematisches Institut, Heidelberg, Germany

ix

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List of Participants

Participants of the Workshop “Noncommutative Main Conjectures”, Münster 2011. Published with kind permissions of M. Witte and D. Izychev, 2011

K. Ardakov C. Aribam J.-R. Belliard T. Bouganis K. Büyükboduk J. Coates A. Dabrowski R. de Jeu K. Fischer T. Hara D. Izychev S. Jha M. Kakde D. Kim

M. Kolster M. Kümmerer J. Lamplugh C.-Y. Lee G. Mantilla Soler T. Nguyen Quang Do V. Nicolas F. Nuccio A. Pal A. Panciskin J. Park V. Patankar G. Perbet B. Perrin-Riou

A. Riedel C. Rüschoff U. Schmitt S. Shekhar R. Sujatha M. Taylor M. Valentino F. Varescon O. Venjakob S. Viguie M. Witte Y. Zähringer

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Introduction to the Work of M. Kakde on the Non-commutative Main Conjectures for Totally Real Fields John Coates and Dohyeong Kim

Abstract These notes are aimed at providing a not too technical introduction to both the background from classical Iwasawa theory for, as well as a detailed discussion of, the principal result (see Theorem 5.1) of Mahesh Kakde’s fundamental paper [K1] proving, subject to the Iwasawa conjecture, the non-commutative main conjecture for totally real p-adic Lie extensions of a number field. Kakde’s work is the beautiful development of ideas initiated by Kazuya Kato in his important paper [KA]. The material covered roughly corresponds to the oral lectures given by one of us at the Workshop. We have not attempted here to discuss the detailed methods of proof used either by Kakde in his paper, or by Ritter and Weiss in their important related work [RW], leaving all of this to the written material of the subsequent lecturers at the Workshop. We would also like to particularly thank R. Greenberg and K. Ardakov for some very helpful comments which have been included in the present manuscript. In particular, we are very grateful to Greenberg for providing us with a detailed explanation of his observation (Theorem 4.5) that Wiles’ work (Theorems 4.3 and 4.4) on the abelian main conjecture for totally real number fields, can be extended to include the case of abelian characters, whose order is divisible by p.

MSCs: 11G05, 11R23, 16D70, 16E65, 16W70 Keywords Iwasawa algebras • main conjecture • motive

J. Coates () Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Street Wilberforce Road, CB3 0WB, Cambridge, UK e-mail: [email protected] D. Kim Department of Mathematics, POSTECH, San 31 Hyoja-dong, Nam-gu, Pohang, Gyeongbuk, 790-784, Republic of Korea e-mail: [email protected] J. Coates et al. (eds.), Noncommutative Iwasawa Main Conjectures over Totally Real Fields, Springer Proceedings in Mathematics & Statistics 29, DOI 10.1007/978-3-642-32199-3 1, © Springer-Verlag Berlin Heidelberg 2013

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J. Coates and D. Kim

1 Notation Throughout, F will denote a totally real finite extension of Q, and p an odd prime. As always pn , with 1 n 1, is the group of all p n -th roots of unity. Write F cyc for the unique Zp -extension of F contained in F .p1 /, and put D Gal.F cyc =F / so that ' Zp : Let ˙ be a fixed finite set of finite primes of F which contains all the primes dividing p, and write F˙ for the maximal extension of F , which is unramified outside the primes in ˙ and the infinite primes of F . If L is any extension of F contained in F˙ , put G˙ .L/ D Gal.F˙ =L/. Also, define M.L/ to be the maximal abelian p-extension of L contained in F˙ , and put X.L/ D Gal.M.L/=L/: Assume now that L is Galois over F , so that M.L/ is also Galois over F . There is a natural left action of Gal.L=F / on X.L/ defined by g x D gx Q gQ 1 , where gQ denotes any lifting of g in Gal.L=F / to Gal.M.L/=F /: As usual, this left action extends to a left action of the Iwasawa algebra .Gal.L=F //, which is defined by .Gal.L=F // D lim Zp ŒGal.L=F /=U ; U

where U runs over the open normal subgroups of Gal.L=F /. Also, if W is any abelian group, W .p/ will denote the p-primary subgroup of W . A Galois extension F1 of F is defined to be an admissible p-adic Lie extension of F if (1) F1 is totally real, (2) the Galois group of F1 over F is a p-adic Lie group, (3) F1 =F is unramified outside a finite set of primes of F , and (iv) F1 contains F cyc . Given such an admissible p-adic Lie extension, we shall always put G D Gal.F1 =F /; H D Gal.F1 =F cyc /; D Gal.F cyc =F /; and take ˙ to be a finite set of primes of F containing all primes which are ramified in F1 =F . If I denotes the ring of integers of some finite extension of Qp , it will also be convenient to write I ŒŒ for the Iwasawa algebra of with coefficients in I . Fixing a topological generator of , we can, as usual, identify I ŒŒ with the ring I ŒŒT of formal power series in an indeterminate T with coefficients in I , by mapping to 1CT . Finally, we shall write A.G/ for the set of Artin representations of G, and L˙ .; s/ for the complex Artin L-function, with the Euler factors for the primes in ˙ removed, of each in A.G/.

Introduction to the Work of M. Kakde on the Non-commutative Main Conjectures

3

2 Iwasawa’s Work on the Cyclotomic Theory We use the above notation, and we stress that the base field F is always assumed to be totally real. In his fundamental paper [IW], Iwasawa proved the following basic result which is the starting point for the whole theory. Theorem 2.1. For all totally real number fields F , X.F cyc / is a finitely generated and torsion . /-module, which has no non-zero finite . /-submodule. Moreover, we have H 2 .G˙ .F cyc /; Qp =Zp / D 0: (1) Recall that one form of Leopoldt’s conjecture, which remains unproven, is the assertion that F cyc is the unique Zp -extension of F . The above theorem is established by noting that X.F cyc / being . /-torsion is seen, by using the full force of global class field theory, to be equivalent to the assertion that the defect in the Leopoldt conjecture (i.e. the difference between the Z-rank of the unit group and the Zp -rank of its closure, in the p-adic topology, in the product of the local unit groups at the primes above p) is bounded as one mounts the finite layers of the Zp -extension F cyc =F . This boundedness of the defect of Leopoldt is then, in turn, shown to be implied by the boundedness of capitulation of ideal classes in the extension F cyc =F . Finally, Iwasawa gives an ingenious proof of the boundedness of this capitulation. The vanishing statement (1) is then a consequence of an Euler characteristic argument which shows that, in the case of a totally real base field F , the Pontrjagin duals of the two modules H i .G˙ .F cyc /; Qp =Zp /.i D 1; 2/ have the same . /-rank. In addition, a celebrated conjecture of Iwasawa will play an important role in the non-abelian theory developed later. By the structure theory, a finitely generated . /-module W is . /-torsion if and only if W=W .p/ is a finitely generated Zp -module. Moreover, W .p/ is finite if and only if its Iwasawa -invariant is zero. Conjecture A. For totally real F , X.F cyc / is a finitely generated Zp -module. Note that, if Conjecture A is true, Theorem 2.1 shows that X.F cyc / is in fact a free Zp -module of finite rank. The classical Iwasawa D 0 conjecture is the assertion that, for every finite extension K of Q, the Galois group of the maximal unramified abelian p-extension of K cyc is a finitely generated Zp -module. It is well known that, by using an argument from Kummer theory, this classical Iwasawa conjecture for the totally imaginary field K D F .p / implies Conjecture A for the totally real field F . So far, Conjecture A has only been proven when F is an abelian extension of Q, where it is a consequence of the Ferrero-Washington for the cyclotomic Zp extension of the field F .p /, which is again an abelian extension of Q.

4


3 Admissible p-Adic Lie Extensions of F The later material in this book will be concerned with an arbitrary admissible p-adic Lie extension F1 =F , and the .G/ module X.F1 /. We stress that this means, in particular, that F1 must also be totally real. The first thing we should point out is that non-trivial examples of such admissible p-adic Lie extensions are not easy to come by. If Conjecture A is valid for F , we can always take F1 to be the field M.F cyc /. Moreover, assuming that (1) Conjecture A is valid, (2) that G is pro-p with no element of order p, and that (3) G has dimension at least 2 as a p-adic Lie group, it follows from Theorem 3.1 below and Theorem 5.2 of the Appendix that X.F1 / ¤ 0 if and only if the Zp -rank of X.F cyc / is at least 2. Perhaps the most down to earth example of such an admissible p-adic Lie extension F1 with X.F1 / ¤ 0 is to take F to be the maximal real subfield of the field generated over Q by the p-th roots of unity, where p is any odd prime such that at least two of the rational numbers .Q; 1/; .Q; 3/; : : : ; .Q; 4 p/ have their numerators divisible by p (the smallest such prime is p D 157); here we take ˙ to consist of the unique prime of F above p, and .Q; s/ denotes the Riemann zeta function. It is the classical main conjecture for X.F cyc / which guarantees that the Zp -rank of X.F cyc / is at least 2 for such primes p. A much more esoteric example is given by Ramakrishna [RK], who proves the existence of infinitely many Galois extensions L1 of Q, which are totally real, whose Galois group J over Q, is either SL2 .Z7 / or the quotient of SL2 .Z7 / by the subgroup generated by I (where I is the unit matrix), and which are unramified outside a finite set T of primes of Q. Thus we can take F1 to be the compositum of L1 and the cyclotomic Z7 -extension of Q. Note that if we define F to be the fixed field of the image in J of the group of matrices congruent to the identity modulo 7 in SL2 .Z7 /, then the Galois group of F1 =F will be pro-7, and have no element of order 7. Defining ˙ to be the set of primes of F lying above either 7 or the primes in T , it follows from the above remarks that, assuming that Conjecture A is valid for F with p D 7, then the Z7 -rank of X.F cyc / is at least 2, and X.F1 / ¤ 0. The full analogue of Theorem 2.1 for any admissible p-adic Lie extension is proven in the two papers [OV, V, V1]. We say that a left .G/-module W is .G/torsion if every element of W is annihilated by a non-zero divisor in .G/. Theorem 3.1. For every admissible p-adic Lie extension F1 =F , X.F1 / is a finitely generated torsion .G/-module. Moreover, if G has no element of order p, then X.F1 / has no non-zero pseudo-null submodule. Assuming that G is both pro-p and has no element of order p, it follows from the final assertion of Theorem 3.1 and the results of [V] that there is an exact sequence of .G/-modules


0 ! X.F1 /.p/ !

j Dt M

.G/=p nj .G/ ! D ! 0;

5

(2)

j D1

where D is a pseudo-null .G/-module. One then defines G .X.F1 // D n1 C C nt . In particular, we have X.F1 /.p/ D 0 if and only if G .X.F1 // D 0. We shall see below that a suitable form of Conjecture A implies a strong statement about the module X.F1 /, which shows, in particular, that G .X.F1 // D 0. In our present state of knowledge, we do not know how to even formulate the main conjecture using the result of this theorem alone (we cannot define a characteristic element for X.F1 / assuming only that it is finitely generated and torsion over .G/, even if we impose the additional hypothesis that G .X.F1 // D 0). In order to overcome this difficulty, we follow [CFKSV] and introduce the category MH .G/ consisting of all finitely generated .G/-modules W such that W=W .p/ is finitely generated over .H /, where we recall that H D Gal.F1 =F cyc /. While it seems very reasonable to conjecture that X.F1 / always belongs to the category MH .G/, we unfortunately cannot prove this unconditionally at present. Nevertheless, assuming this conjecture, the following result is proven in the Appendix. Theorem 3.2. Assume that the p-adic Lie extension F1 =F is such that (i) G is pro-p and has no element of order p, (ii) G has dimension at least 2 as a p-adic Lie group, and (iii) X.F1 / belongs to the category MH .G/. Then G .X.F1 // D .X.F cyc //, and X.F1 /=X.F1 /.p/ has .H /-rank equal to r 1, where r is the Zp -rank of X.F cyc /=X.F cyc /.p/. Our present inability to prove that X.F1 / lies in the category MH .G/ leads us to work with a stronger conjecture in the subsequent analytic and algebraic arguments. Iwasawa Conjecture: The admissible p-adic Lie extension F1 =F will be said to satisfy the Iwasawa conjecture if there exists a finite extension F 0 of F in F1 such that (1) the Galois group of F1 over F 0 is pro-p, and (2) X.F 0cyc / is a finitely generated Zp -module. We remark that, by the theorem of Ferrero-Washington, this Iwasawa conjecture is true for all p-adic Lie extensions F1 =F such that F is an abelian extension of Q and the Galois group G is pro-p. In particular, when F is an abelian extension of Q and F1 D M.F cyc /, the Iwasawa conjecture is valid. Theorem 3.3. Assume that the p-adic Lie extension F1 =F satisfies the Iwasawa Conjecture. Then X.F1 / is finitely generated over .H /, and X.F1 /.p/ D 0. Proof. Put H 0 D Gal.F1 =F 0cyc /. Then we have the exact sequence of inflation restriction 0 ! H 1 .H 0 ; Qp =Zp / ! Hom.X.F 0cyc /; Qp =Zp / ! Hom.X.F1 /H 0 ; Qp =Zp / ! H 2 .H 0 ; Qp =Zp /:

6


Now H i .H 0 ; Qp =Zp / is a cofinitely generated Zp -module for all i 0. Hence, assuming that X.F 0cyc / is a finitely generated Zp -module, it follows that X.F1 /H 0 is also a finitely generated Zp -module. But, as H 0 is pro-p, .H 0 / is a local ring, and so it follows from Nakayama’s lemma that X.F1 / is finitely generated over .H 0 /, and so all the more so over .H /. To prove the final assertion of the theorem, we note that we can find an open subgroup H 00 of H 0 such that H 00 is pro-p and has no element of order p (possibly H 00 D 0). Since X.F1 / is also finitely generated over .H 00 /, a theorem of Venjakob asserts that every .G/ submodule of X.F1 /, which is .H 00 /-torsion, is pseudo-null as a .G/-module, and so must be zero by Theorem 3.1. In particular, this shows that X.F1 /.p/ D 0.

4 The Classical Abelian Main Conjecture In this section, we discuss the classical abelian main conjecture for an arbitrary admissible p-adic Lie extension F1 =F which will be assumed throughout this section to satisfy the: Abelian Hypothesis. G D Gal.F1 =F / is an abelian p-adic Lie group of dimension 1. As before, ˙ will denote the set of primes of F which ramify in F1 . We fix a lifting of D Gal.F cyc =F / to G, which we denote by the same symbol . Thus, since G is abelian, this means that we have G D H . We define K to be the fixed field of the subgroup of G, so that K \ F cyc D F , and F1 is the compositum of K and F cyc . Let HO be the group of 1-dimensional characters of H . Write F W Gal.F .p1 /=F / ! Z p for the cyclotomic character. As we can view as a subgroup of Gal.F .p1 /=F /, it makes sense to consider the restriction of F to . In what follows, we then consider F as a character of G by defining it always to be trivial on H . While complex L-functions can be defined in great generality via Euler products, nothing like this seems to be true in the p-adic world, and, at present, our only way to define p-adic L-function is via p-adic interpolation of essentially algebraic special values of complex L-functions. Viewing an element in HO as being complex valued, let L˙ . ; s/ be the imprimitive complex L-function attached to , with the Euler factors corresponding to the primes in ˙ omitted from its Euler product. The following basic result is due to Siegel. Theorem 4.1. For each in HO , and each even integer n > 0, L˙ . ; 1n/ belongs to the field Q. /, which is generated over Q by the values of . In fact, Siegel’s proof shows that L˙ . ; 1 n/ D L˙ . ; 1 n/

(3)


7

for all in the absolute Galois group of Q, and all even integers n > 0, and even allows us to define this value intrinsically when the character is no longer assumed to have complex values. In fact, we shall assume from now on that the in HO all have values in the algebraic closure of Qp . Let O be the ring of integers of the field obtained by adjoining the values of all

in HO to Qp , and let O .G/ be the Iwasawa algebra of G with coefficients in O. Write QO .G/ for the ring of fractions of O .G/ (i.e. the localization of this ring with respect to its set of non-zero divisors). An element of QO .G/ is defined to be a pseudo-measure on G if . 1/ is in O .G/ for all in G. If W G ! O is any continuous homomorphism, which is distinct from the trivial homomorphism of G, which we denote by 1, it is easily seen that one can define the integral of against , which we denote by Z d; G

and which is a well defined element of the fraction field of O. The following theorem, which generalizes many earlier results starting with Kummer, is due to Cassou-Nogues and Deligne-Ribet. Theorem 4.2. Assume that F1 =F satisfies the abelian hypothesis. Then there exists a unique pseudo-measure F1 =F on G D H such that, for all in HO , we have Z

F n d F1 =F D L˙ . ; 1 n/; (4) G

for all integers n > 0 with n 0 mod ı, where ı D ŒF .p / W F : This theorem is easily seen to imply the following assertion. For each character

in HO , let O be the ring of integers of the field obtained by adjoining the values of

to Qp , and let O ŒŒT be the ring of formal power series in an indeterminate T with coefficients in O . Fix, for the remainder of this section, a topological generator of . Then, if ¤ 1, there exists a unique formal power series W .T / in O ŒŒT such that W . F . /n 1/ D L˙ . ; 1 n/; for all integers n > 0 with n 0 mod ı. In addition, if D 1, there exists a unique power series W1 .T / in Zp ŒŒT such that W1 . F . /n 1/=. F . /n 1/ D ˙ .F; 1 n/; where ˙ .F; s/ denotes the complex zeta function of F , with the Euler factors removed at the primes in ˙. Let be any fixed local parameter for the ring O . We plainly can write

W .T / D V .T /;

(5)

8


where is a non-negative integer, and V .T / is a power series in O ŒŒT, with at least one of its coefficients a unit in O . It is conjectured that we always have D 0 for every F1 =F and every in HO , but this has only been proven in the case F D Q by Ferrero-Washington, and it is unknown for every other totally real base field other than Q. The aim of the abelian main conjecture is to give a precise relation between the analytic pseudo-measure F1 =F on the one hand, and the algebraic structure of the arithmetic O .G/-module X.F1 / on the other hand. However, the exact formulation of this relationship is not straightforward from a classical point of view, because there is no known structure theory for finitely generated torsion O .G/modules when p divides the order of H . For each in HO , let X

.h/h1 e D #.H /1 h2H

be the orthogonal idempotent of in the group ring of H with coefficients in the field of fractions L of O . The simplest thing to do is to simply consider Z.F1 / D X.F1 / ˝Zp L; Z.F1 / D e Z.F1 /;

(6)

which are both finite dimensional vector spaces over L by Theorem 2.1. We then define R .T / to be the characteristic polynomial of 1 acting on Z.F1 / . We omit the proof of the following technical lemma, which is due to Greenberg (see [G1], Proposition 1). Lemma 4.1. Let be any element of HO , and let K 0 be any intermediate field between F and K such that is trivial on Gal.K=K 0 /. Write 0 for , when viewed 0 as a character of Gal.K 0 =F /, and let F1 be the compositum of K 0 and F cyc . Then 0 0 Z.F1 / is isomorphic to Z.F1 / as representations of . In particular, this lemma shows that the polynomial R .T / depends only the character of H , and not on the particular finite extension of F such that factors through the Galois group over F of this extension. The first fundamental result of Wiles (see Theorem 1.3 of [W1]) in the direction of the main conjecture for all totally real number fields F is the following. Theorem 4.3. Assume that F1 =F satisfies the abelian hypothesis. Then, for all characters of H , we have V .T /O ŒŒT D R .T /O ŒŒT:

(7)

The problem with this result is that it does not tell us anything about the -invariants on either the analytic or the algebraic sides. Of course, the analytic -invariant is the integer appearing in (5), and is valid for all characters of H , irrespective of whether the order of is divisible by p or not. The definition of the algebraic -invariant is much more delicate. We first explain what to do in the easy case, when the order of is prime to p. Assuming this to be the case, we may also suppose that


9

K is exactly the fixed field of the kernel of , as this does not change the polynomial R .T / by Lemma 4.1. Define X.F1 / D e .X.F1 / ˝Zp O /: Now, by Theorem 2.1, X.F1 / is a finitely generated torsion O ŒŒT-module, and thus, by the well known structure theory for such modules and the Weierstrass preparation theorem, it has a characteristic ideal of the form C .T /O ŒŒT, where C .T/ is a polynomial in O ŒT of such that

C .T / D

R .T /;

(8)

for some integer 0; here R .T / is, as above, the characteristic polynomial of 1 acting on Z.F1 / . The second fundamental result of Wiles (see Theorem 1.4 of [W1]) is the following. Theorem 4.4. Assume that F1 =F satisfies the abelian hypothesis. If 2 HO has order prime to p, then we have D : (9) In particular, when combined with Theorem 4.3, this result proves the main conjecture asserting that W .T /O ŒŒT D C .T /O ŒŒT;

(10)

for all characters of H of order prime to p. We are very grateful to R. Greenberg (private communication) for the following explanation of how one can define the analogue of the algebraic -invariant

appearing in (8) even for characters of H whose order is divisible by p, and then show that the main conjecture (10) still remains valid for such characters. As we shall need to vary the base field F in this argument, for the remainder of this section we shall write WF; .T /, F; ; : : : to indicate the dependence of the above quantities on the base field F . Fix a character of H , whose order is divisible by p. We shall assume that K is the fixed field of the kernel of . Now we can write in the form

D , where is a character of H of order prime to p, and has p-power order. Define 0 D p , and write L0 ; L for the fixed fields of Ker(0 ), Ker(), respectively. We can now take the restriction L (resp. L0 ) of to the absolute Galois group of L (resp. the absolute Galois group of L0 ). Then K is the fixed field of Ker( L ), and we define K 0 to be the fixed field of Ker( L0 ). Thus we have the tower of fields F L0 L K 0 K:

(11)

0 Write F1 for the compositum of K 0 with F cyc , and, as before, let F1 be the compositum of K with F cyc . To lighten our notation, put

J D O ; I D O ; E D O ;

(12)

10


so that E is the ring generated over J by the values of . We first observe that, up 0 to a pseudo-isomorphism of -modules, we can identify X.F1 / with a quotient 0 of X.F1 /. Indeed, let P (resp. P ) be the Sylow p-subgroup of Gal.K=F / (resp. Gal.K 0 =F /), and put ˝ D Ker.P ! P 0 /; (13) 0 /, which is the so that ˝ has order p. Then the natural map from X.F1 /˝ to X.F1 dual of the restriction map on Galois cohomology, has finite kernel and cokernel. Indeed, by the usual inflation restriction sequence, the cokernel is finite because it 0 is dual to H 1 .Gal.F1 =F1 /; Qp =Zp /, and the kernel is finite because it is dual 2 0 to a submodule of H .Gal.F1 =F1 /; Qp =Zp /; both these cohomology groups 0 are obviously finite because Gal.F1 =F1 / is a finite cyclic group. In particular, 0 it follows that X.F1 /˝ and X.F1 / have the same characteristic power series as -modules. We then define

˘.F1 / D Ker.X.F1 / ! X.F1 /˝ /:

(14)

Explicitly, we have ˘.F1 / D . 1/X.F1 /, where is any generator of ˝. Now the group ring Zp ŒP acts on ˘.F1 /, and this action factors through an action of the ring B D Zp ŒP =.1 C C : : : C p1 /Zp ŒP : But evaluation at the character defines an isomorphism from B onto the ring I . Thus we see that ˘.F1 / has a natural structure as an I ŒŒ -module. Now is a faithful character of Gal.K=L/ of order prime to p, and thus, for any Zp ŒGal.K=L/-module A, we may define A D e .A ˝Zp J /: In particular, we have ˘.F1 / D Ker.X.F1 / ! .X.F1 / /˝ /:

(15)

It is clear that ˘.F1 / has a structure as an EŒŒ -module, because I acts on ˘.F1 /. Moreover, since X.F1 / is a finitely generated torsion Zp ŒŒ -module, it follows that ˘.F1 / is a finitely generated torsion EŒŒ -module. As before, let be any local parameter of the ring E D O . Then, by the structure theory for finitely generated torsion EŒŒ -modules, ˘.F1 / will have a characteristic ideal of the form C .T /EŒŒT, where C .T / is a polynomial such that

C .T / D

R .T/;

(16)

where is some integer 0, and R .T/ is a monic polynomial in EŒT . It is this integer which we define to be the algebraic -invariant of when p divides the order of . On the other hand, since .X.F1 / /˝ is pseudo-isomorphic as a


11

0 J ŒŒ -module to X.F1 / , it follows from (15) that the -invariant of ˘.F1 / as a J ŒŒ -module must be equal to ;L ;L0 , where the subscripts L and L0 indicate that these invariants are now taken with respect to these respective base fields. But, as E is a totally ramified extension of J of degree p m1 .p 1/, where p m is the exact order of the character , it is clear that the -invariant of ˘.F1 / as an EŒŒ -module is equal to its -invariant as a J ŒŒ -module. This is because the residue fields of J and E have the same order. Hence we obtain

D

;L

;L0 :

(17)

Recall that he analytic invariant is defined by the Eq. (5). As Greenberg has remarked, we can now easily establish the following generalization of Theorem 4.4. Theorem 4.5. Assume that F1 =F satisfies the abelian hypothesis, where H is an arbitrary finite abelian group. Then, for all characters of H , we have D :

(18)

Proof. Since we can view , which has order prime to p, as a character of both Gal.K=L/ and Gal.K 0 =L0 /, we can apply Theorem 4.4 to both of the extensions 0 F1 =L and F1 =L0 . We conclude that

;L

D

;L ;

;L0

D

;L0 ;

(19)

where again the subscripts L and L0 mean the invariants are taken relative to the respective base fields. Thus it follows from (17) that

D

;L

;L0 :

(20)

To conclude the proof, we need an analytic argument. Let D be the set of characters of Gal.L=F / which do not factor through Gal.L0 =F /. Plainly D consists of all the characters D a , where a runs over the integers mod p m which are prime to p. It then follows easily from the Artin formalism for complex L-functions that W

;L .T /

DW

;L0 .T /

˘W

;F .T /;

(21)

where now runs over all elements of D. Write for the Galois group of the fraction field of E over the fraction field of J , and recall that this extension is totally ramified of degree p .m1/ .p 1/. We conclude easily from (3) that all of the power series W ;F .T / with 2 D are conjugate under the action of . Hence, since our extension is totally ramified, it follows that all of these power series must have the same -invariant, which must be equal to because is one of the characters in D. Noting that the invariants ;L and ;L0 are defined using a local parameter of the unramified extension J of Zp , it now follows from (21) that

;L

;L0

D :

Combining (21) and (22), the proof of the theorem is now complete.

(22)

12


In particular, when combined with Theorem 4.3, Theorem 4.5 proves the abelian main conjecture in general, asserting that, for every character of H , we have

W .T /O ŒŒT D

R .T /O ŒŒT:

(23)

5 The Non-abelian Main Conjecture Throughout this section, F1 =F will be an arbitrary admissible p-adic Lie extension, which we will always assume for simplicity satisfies: Hypothesis B. The group G D Gal.F1 =F / has no element of order p. While the non-abelian main conjecture can be formulated for all admissible p-adic Lie extensions, the point of imposing this hypothesis is that it makes the whole discussion of the homological properties of the G-module X.F1 / much simpler. In particular, Hypothesis B implies that the Iwasawa algebra .G/ has finite global dimension, and that G has finite p-homological dimension, which is equal to the dimension of G as a p-adic Lie group. Note that Iwasawa [IW] has proven that, for every finite extension F of Q, there is no non-zero finite . /-submodule of X.F cyc /. Hence, whenever Conjecture A is valid for F (for example, when F is an abelian extension of Q), it follows that Hypothesis B is valid in the important example given by taking F1 D M.F cyc /. Note also that Leopoldt’s conjecture implies that such a G must be non-commutative whenever F1 ¤ F cyc : We now rapidly recall the statement of the main conjecture. As in [CFKSV], let S be the subset of .G/ defined by S D ff 2 .G/ W .G/=.G/f is a finitely generated .H / module:g: (24) Further, let S be the subset of .G/ consisting of all elements p n s, where n is some integer 0, and s is in S . Then the following results are proven by rather elementary arguments in [CFKSV]. Firstly, S , and so also S , are multiplicatively closed sets of non-zero divisors in .G/, which satisfy the left and right Ore condition. Thus we can define the localized rings .G/S and .G/S , which are, of course, non-commutative once G is not abelian. Secondly, let NH .G/ (resp. MH .G/) be the category of all finitely generated left .G/-modules W such that W (resp. W=W .p/) is finitely generated over .H /. Then NH .G/ (resp. MH .G/) is precisely the category of finitely generated left .G/-modules which are S -torsion (resp. S -torsion). Write K0 .NH .G// (resp. K0 .MH .G//) for the Grothendieck groups of these two categories. For any ring R with unit, denote by K1 .R/ the K1 group of R. By classical algebraic K-theory, we have boundary maps @ W K1 ..G/S / ! K0 .NH .G//; @ W K1 ..G/S / ! K0 .MH .G//;

(25)


13

whose kernel in both cases is the relevant image of K1 ..G//. Moreover, it is shown in [CFKSV] that both of these maps are surjective. Thus, for each module W in the category NH .G/ (resp. MH .G/), we define a characteristic element of W to be any W in K1 ..G/S / (resp. K1 ..G/S / such that @.W / (resp. @ .W /) is equal to the class of W in the relevant K0 . Granted that our .G/-module X.F1 / lies in the appropriate category, the main conjecture, will, as always, assert that its characteristic element can be taken to be a p-adic L-function, in a sense that we now make precise. Firstly, as is explained in [CFKSV], we can always evaluate an element of K1 ..G/S / (resp. K1 ..G/S /) at any continuous homomorphism from G into GLn .O/, where O is the ring of integers of some finite extension of Qp , obtaining a well-defined value ./ in the fraction field of O, or the value 1. In particular, let A.G/ be the set of all Artin representations of G. Thus an element of A.G/ will be a homomorphism W G ! GLn .O/;

(26)

which factors through a finite quotient of G, where again O is the ring of integers of some finite extension of Qp . For each 2 A.G/, write L˙ .; s/ for the complex L-function of , with the Euler factors for the primes in ˙ omitted from its Euler product. Now, combining Theorem 4.1 with Brauer’s theorem on finite groups, it follows that, for each even integer n > 0 and each in A.G/, the value L˙ .; 1n/ is an algebraic number satisfying L˙ . ; 1 n/ D L˙ .; 1 n/ ;

(27)

for all in the absolute Galois group of Q. We then have the following conjectural analogue of Theorem 4.2. Conjecture C. For every admissible p-adic Lie extension F1 =F , there exists F1 =F in K1 ..G/S / such that F1 =F . Fn / D L˙ .; 1 n/;

(28)

for all 2 A.G/ and all integers n > 0 with n 0 mod ŒF .p / W F ; here F is the character giving the action of the absolute Galois group of F on the group of all p-power roots of unity. Granted the existence of this p-adic L-function, the general main conjecture can now be stated as follows. If W is any module in MH .G/, write ŒW for the class of W in the Grothendieck group of this category. Conjecture D. Let F1 =F be an admissible p-adic Lie extension satisfying Hypothesis B. Assume that X.F1 / belongs to the category MH .G/, and that there exists F1 =F in K1 ..G/S / satisfying (28). Then @ .F1 =F / D ŒX.F1 / ŒZp :

(29)

14


In his remarkable paper [K1], Kakde has proven this main conjecture, provided the admissible p-adic Lie extension satisfies the generalized Iwasawa conjecture. Theorem 5.1. Assume that F1 =F is an admissible p-adic Lie extension satisfying Hypothesis B. If the generalized Iwasawa conjecture holds for F1 =F , then there exists F1 =F in K1 ..G/S /, satisfying (28) for all 2 A.G/ and all integers n > 0 with n 0 mod ŒF .p / W F , and such that @.F1 =F / D ŒX.F1 / ŒZp : The lectures in this conference are aimed primarily at an account of his proof. The most important unconditional case of this theorem is as follows. Corollary 5.1. Let F be a real abelian field, and take F1 D M.F cyc /. Then there exists F1 =F in K1 ..G/S /, satisfying (28) for all 2 A.G/ and all integers n > 0 with n 0 mod ŒF .p / W F , and such that @.F1 =F / D ŒX.F1 / ŒZp : For example, if we take F to be the maximal real subfield of Q.157 /, take ˙ to consist of the unique prime of F above p, and let F1 D M.F cyc /, then H D Zp 2 , and Theorem 5.2 shows that X.F1 / is non-zero, being isomorphic as a .H /module to a submodule of finite index of .H /. We end by briefly discussing possible applications of Theorem 5.1. Write d C 1, where d is an integer 0, for the dimension of G. We continue to assume that G has no element of order p, so that G has p-homological dimension equal to d C 1. At present, all known applications of Theorem 5.1 are to the computation of the G-Euler characteristics of certain twists of the Iwasawa module X.F1 /. If W is any finitely generated .G/-module, we recall that W is said to have finite G-Euler characteristic if all of the homology groups Hi .G; W / are finite for all i D 0; : : : ; d C 1, and then this Euler characteristic is defined to be

.G; W / D

dY C1

i

.#.Hi .G; W ///.1/ :

(30)

i D0

Similarly, if V is any finitely generated . /-module, we have the analogous notion of it having a finite -Euler characteristic, defined by

.; V / D #.H0 .; V //=#.H1 .; V //: Moreover, the Hochschild-Serre spectral sequence shows that W will have finite G-Euler characteristic if and only if the -modules Hi .H; W / have finite -Euler characteristic for all i D 0; : : : ; d , and when this is the case, we will have d Y i

.; Hi .H; W //.1/ :

.G; W / D i D0

(31)


15

Also, if m is any integer, W .m/ will denote as usual the m-fold Tate twist of W (i.e. the tensor product over Zp of W with the free Zp -module of rank 1 on which Galois acts via the m-th power of the cyclotomic character). Note also that when m is any integer with m 0 mod ŒF .p / W F , G acts on Zp .m/. Our aim now is to see what can be proven about the G-Euler charactersitic of X.F1 /.n/ when n is any integer > 0 with n 0 mod ŒF .p / W F . In the special case when F1 D F cyc , it is a classical consequence of the abelian main conjecture that, for such integers n > 0, X.F cyc /.n/ has finite -Euler characteristic, given by

.; X.F cyc /.n// D jwn .F /˙ .F; 1 n/j1 p ;

(32)

where wn .F / denotes the largest integer r such that the Galois group of F .r / over F has exponent n. In the general case, the following proposition holds: Proposition 5.1. Let n be any integer > 0 with n 0 mod ŒF .p / W F . Then X.F1 /.n/ has finite G-Euler characteristic if and only if Zp .n/ has finite GEuler characteristic. Moreover, assuming that these Euler characteristics are finite, we have

.G; X.F1 /.n//= .G; Zp .n// D j˙ .F; 1 n/j1 p :

(33)

Proof. We first note that H acts trivially on Zp .n/ when n 0 mod ŒF .p / W F . Thus Proposition (5.2) gives immediately an isomorphism of -modules Hi .H; X.F1 /.n// ' Hi C2 .H; Zp .n//;

(34)

for all i 1. Similarly, Proposition (43) gives the exact sequence of -modules 0

/ H2 .H; Zp .n//

/ H0 .H; X.F1 /.n//

/ H1 .H; Zp .n//

/ 0:

/ X.F cyc /.n/ (35)

Now X.F1 /.n/ has finite G-Euler characteristic if and only if the Hi .H; X.F1 /.n// .i D 0; : : : ; d /

(36)

all have finite -Euler characteristics. As X.F cyc /.n/ has finite -Euler characteristic, we conclude from the exact sequences (33) and (35) that the -modules (36) will have finite Euler characteristics if and only if the -modules Hi .H; Zp .n//, for i D 2; : : : ; d , all have finite Euler characteristics. Moreover, H1 .H; Zp .n// has finite Euler characteristic since it is a quotient of X.F cyc /.n/, and

.; H0 .H; Zp .n/// D jwn .F /j1 p :

(37)

16


Combining these results, it follows that X.F1 /.n/ has finite G-Euler characteristic if and only if Zp .n/ has finite G-Euler characteristic. Moreover, assuming these Euler characteristics are finite, and using (31) and (32), the assertion (33) follows immediately, completing the proof. In the above discussion, we have only considered the special values wn .F /.F; 1 n/ for those positive even integers n with n 0 mod d . If one wishes to obtain similar results for all positive even integers n, one should consider the larger admissible p-adic Lie extension F1 D F1 .p /C of F , and study, in an entirely similar manner to the above, the G1 -Euler characteristics of the modules X.F1 /.n/; here G1 denotes the Galois group of F1 over F . It is perhaps worth remarking that, assuming the finiteness of the G-Euler characteristics of X.F1 /.n/ and Zp .n/, and also that X.F1 / belongs to the category MH .G/, one can also deduce (33) from the non-commutative main conjecture (Conjecture D). Moreover, in the special case when F1 D M.F cyc / and H D Zp , then it is well known (see the remark at the end of the Appendix) that X.F1 / D 0, so that, in this case, .G; Zp .n// is indeed finite, and, by virtue of (33), is given by

.G; Zp .n// D j˙ .F; 1 n/jp : (38) When G has dimension at least 3 as a p-adic Lie group, it seems highly likely that Zp .n/ should have finite G-Euler characteristic for all integers n > 0 with n 0 mod ŒF .p / W F . However, it does not seem easy to formulate a general conjecture for the value of this Euler characteristic, assuming that it is finite. We briefly mention two very different specific cases. Lemma 5.1. Let F1 =F be an admissible p-adic Lie extension such that H is isomorphic to an open subgroup of SL2 .Zp /. Then .G; Zp .n// D 1 for all integers n > 0 with n 0 mod ŒF .p / W F . Proof. We have H0 .H; Zp / D Zp , and we are grateful to K. Ardakov for pointing out to us that standard arguments with Lie algebra cohomology show that Hi .H; Zp / is finite for i D 1; 2, and has Zp -rank 1 for i D 3, and that G acts trivially on all of these homology groups modulo their torsion subgroups. Now, as H acts trivially on Zp .n/, we can move this Tate twist inside the H -homology groups. Since the -homology of a finite module is 1, it now follows immediately from (31) and the multiplicativity of the -Euler characteristics in exact sequences, that Zp .n/ has finite G-Euler characteristic equal to 1. Secondly, consider the case when H is abelian, say H D Zp d , with d 2. We then have H1 .H; Zp / D H: (39) and it is well known that Hi .H; Zp / is the i -th exterior power of H as a Zp -module. Now the abelian main conjecture shows that H.n/ has finite -Euler characteristic for all integers n > 0 with n 0 mod ŒF .p / W F . However, we do not know enough about the roots of the p-adic zeta function of F at present to be able to prove that the i -th exterior power of H has finite -Euler characteristic for any of


17

i D 2; : : : ; d , for all such integers n. Thus we cannot even establish the finiteness of the G-Euler characteristic of Zp .n/, let alone compute its exact value. However, if we make the additional hypothesis that n is divisible by a sufficiently large power of p, one easily sees that (37) is valid, and d Y

i

.; Hi .H; Zp .n///.1/ D 1:

(40)

i D1

Thus for all positive integers n, with n 0 mod ŒF .p / W F and n divisible by a sufficiently large power of p, we conclude that Zp .n/ has finite G-Euler characteristic given by

.G; Zp .n// D jwn .F /j1 p :

Appendix Throughout, we assume that both F1 =F is an admissible p-adic Lie extension of totally real fields, and we shall establish some results relating X.F1 / to X.F cyc /. We recall that if G is a p-adic Lie group, and A is a discrete p-primary G-module, then it is well known that, for all i 0, the Pontryagin dual of H i .G; A/ is canonically isomorphic to the homology group Hi .G; B/, where B D Hom.A; Qp =Zp / is the compact Pontryagin dual of A. Proposition 5.2. Let F1 =F be any admissible p-adic Lie extension. Then, for all i 1, we have Hi .H; X.F1 // ' Hi C2 .H; Zp /: Proof. Since (1) is valid not only for F itself, but also for every finite extension of F contained in F1 , we conclude that H i .G˙ .F1 /; Qp =Zp / D 0 .i 2/:

(41)

Applying the Hochschild-Serre spectral sequence to G˙ .F cyc / and its closed normal subgroup G˙ .F1 /, we conclude from (41) (see Theorem 3 of [HS]) that, for all j 1, there is a long exact sequence H j .H; Qp =Zp /

/ H j .G˙ .F cyc /; Qp =Zp /

/ H j C1 .H; Qp =Zp /

/ H j 1 .H; X.F1 //

/ H j C1 .G˙ .F cyc /; Qp =Zp /:

(42)

18


Taking j D i C 1 with i 1, and again using (1), the proof of the proposition is complete after taking Pontrjagin duals. Proposition 5.3. For any admissible p-adic Lie extension F1 =F , we have the exact sequence / H2 .H; Zp /

/ H0 .H; X.F1 //

/ H1 .H; Zp /

/ 0:

0

/ X.F cyc / (43)

Proof. Taking j D 1 in (42), we obtain result on noting that the first arrow on the left in (42) is injective, and the group on the right in (42) is still zero. We assume now that G is pro-p of dimension d C 1 with d 1, and that G no element of order p. We recall that G .X.F1 // is defined via the exact sequence (2). If V is any finitely generated torsion . /-module, we write .V / for its classical -invariant. Since G has finite homological dimension, these -invariants, for p-primary modules, are well known to be related to Euler characteristics as follows (see Corollary 1.7 of [H]). If A is a finitely generated p-primary .G/-module, and B is a finitely generated p-primary . /-module, define the Euler characteristics

.G; A/ D

dY C1

i

.#.Hi .G; A///.1/ ; .; B/ D #.H0 .; B//=#.H1 .; B//:

i D0

Then we have p G .A/ D .G; A/; p .B/ D .; B/:

(44)

Lemma 5.2. Assume that G is pro-p of dimension d C 1 with d 1, and that G has no element of order p. Then Hi .H; X.F1 /.p// is a finitely generated torsion . /-module for all i 0, and G .X.F1 // D

d X

.1/i .Hi .H; X.F1 /.p///:

(45)

i D0

Proof. The first assertion is obvious. To prove the second, we note that, for i 1, the Hochschild-Serre spectral sequence gives the exact sequence 0

/ H0 .; Hi .H; X.F1 /.p/// / H1 .; Hi 1 .H; X.F1 /.p///

/ Hi .G; X.F1 /.p// / 0 ;


19

whence we obtain

.G; X.F1 /.p// D

d Y i

.; Hi .H; X.F1 /.p///.1/ : i D0

The conclusion of the lemma is now clear from (44). Define the .G/-module Y1 by the exact sequence 0 ! X.F1 /.p/ ! X.F1 / ! Y1 ! 0:

(46)

Since X.F1 / is annihilated by p n for all sufficiently large n, we can view Y1 as a .G/-submodule of X.F1 /, and so we also have a short exact sequence 0 ! Y1 ! X.F1 / ! W1 ! 0;

(47)

for some .G/-module W1 . Proposition 5.4. Assume that G is pro-p of dimension d C 1 with d 1, and that G has no element of order p. Then Hi .H; Y1 / is a finitely generated torsion . /-module for all i 0. Moreover, Hd .H; Y1 / D 0. Proof. We first prove that all the groups Hi .H; Y1 / are torsion . /-modules. Taking H -homology of (46), we obtain, for each i 1, the exact sequence Hi .H; X.F1 // ! Hi .H; Y1 / ! Hi 1 .H; X.F1 /.p//: The first term is finitely generated over Zp by Proposition 5.2, and the third term is annihilated by a power of p. Therefore Hi .H; Y1 / is . /-torsion. The assertion for i D 0 follows from (43), since X.F cyc / is . /-torsion and Hj .H; Zp / is a finitely generated over Zp -module for all j 0. To prove the second assertion, we note that H has homological dimension d . Thus, taking H -homology exact of (47), we get the exact sequence 0 ! Hd .H; Y1 / ! Hd .H; X.F1 //:

(48)

But, as d 1, we have Hd .H; X.F1 // ' Hd C2 .H; Zp /, and the latter group is zero since the homological dimension of H is d . Proposition 5.5. Assume that G is pro-p of dimension d C 1 with d 1, and that G has no element of order p. Then we have

G .X.F1 // D .X.F

cyc

//

d 1 X i D0

.1/i .Hi .H; Y1 //:

20


Proof. From (43), we know that .X.F cyc // D .H0 .H; X.F1 /// since the first and last terms in (43) are finitely generated over Zp . Taking H -homology of (46), we obtain the exact sequence H1 .H; X.F1 //

/ H1 .H; Y1 / / .Y1 /H

/ X.F1 /.p/H

/ X.F1 /H

/ 0 :

(49)

By Proposition 5.2, H1 .H; X.F1 // D H3 .H; Zp / is finitely generated over .H /. Hence we obtain .X.F1 // D ..X.F1 //H / ..Y1 /H / C .H1 .H; Y1 //: Moreover, for i 1, homology exact sequence derived from (46) yields .Hi .H; X.F1 /.p/// D .Hi C1 .H; Y1 // since for all i 1, again by Proposition 5.2, Hi .H; X.F1 // D Hi C2 .H; Zp / is finitely generated over Zp . The assertion of the proposition now follows immediately (45). Recall that when H is pro-p and has no elements of order p, we define the rank of Y1 as a .H /-module by rk.H / Y1 D dimQ.H / Y1 ˝.H / Q.H /; where Q.H / is the skew field of fractions of .H /. Theorem 5.2. Assume that G is pro-p of dimension d C 1 with d 1, and that G has no element of order p. Suppose further that X.F1 / is in the category MH .G/. Let r be the Zp -rank of X.F cyc /=X.F cyc /.p/. Then G .X.F1 // D .X.F cyc // and the .H /-rank of X.F1 /=X.F1 /.p/ is r 1. Proof. Our hypothesis that X.F1 / is in MH .G/ implies that Y1 is finitely generated as a .H /-module. Hence the Hi .H; Y1 / are finitely generated as Zp modules for all i 0, and hence have -invariant equal to 0. Thus Proposition 5.5 gives immediately that G .X.F1 // D .X.F cyc //. Now we compute the .H /rank of Y1 . Since H is pro-p and has no elements of order p, it is well known that (see [H]) X rk.H / V D .1/i rkZp Hi .H; V /; (50) i 0


21

for any finitely generated .H /-module V . Now using the long exact sequence of (46), we see immediately that rkZp Hi .H; Y1 / D rkZp Hi .H; X.F1 // for all i 0. On the other hand, since the .H /-module Zp clearly as .H /-rank 0, we have 0D

X

.1/i rkZp Hi .H; Zp /

i 0

D

2 X

.1/i rkZp Hi .H; Zp / C

i D0

X

.1/i rkZp Hi .H; X.F1 //

i 1

X D 1 C rkZp X.F1 /H r C .1/i rkZp Hi .H; X.F1 // i 1

X D 1r C .1/i rkZp Hi .H; Y1 / i 0

D 1 r C rk.H / Y1 : The first and last equalities are (50), the second is Proposition 5.2, and the third follows from the inflation-restriction sequence 0 ! H2 .H; Zp / ! H1 .G˙ .F1 /; Zp /H ! H1 .G˙ .F cyc /; Zp / ! H1 .H; Zp / ! 0:

Remark 5.1. Notice that the hypotheses of the above theorem imply, in particular, that necessarily r 1. We are grateful to Ralph Greenberg for pointing out to us that this fact is a consequence of Proposition 3.9.1 in [NSW], which is also known as Burnside Basis Theorem. Indeed, as he remarked to us, one can easily deduce from this Burnside Basis Theorem that if Conjecture A holds for F and X.F cyc / has Zp -rank at most 1, then M.F cyc / is the maximal pro-p extension of F which is unramified outside ˙. The simplest unconditional example of Theorem 5.2 is as follows. Take F to be the maximal real subfield of Q.p / for an odd prime p, and ˙ to consist of the unique prime of F above p. Since Conjecture A is valid for F by the theorem of Ferrero-Washington, we can take F1 D M.F cyc /. By Theorem 2.1, the Galois group G is then pro-p and has no element of order p, and the subgroup H is a free Zp -module of rank r 0. Moreover, X.F1 / is a finitely generated .H /-module, which has no .H /-torsion by Theorem 3.1. The classical theory of cyclotomic fields tells us that r 1 if and only if p is an irregular prime, and X.F1 / is zero when r D 1. When r 2, Theorem 5.2 and the classical commutative structure theory of finitely generated .H /-modules shows that there is an exact sequence 0 ! X.F1 / ! .H /r1 ! D ! 0; where D is a pseudo-null .H /-module. The smallest prime p for which r 2 is p D 157, and the largest value of r for p < 12;000;000 is r D 7.

22


References [IW] K. Iwasawa, On Zp -extensions of algebraic number fields. Ann. Math. 98, 246–326 (1973) [RK] R. Ramakrishna, Deforming Galois representations and the conjectures of Serre and Fontaine-Mazur. Ann. Math. 156, 115–154 (2002) [OV] Y. Ochi, O. Venjakob, On ranks of Iwasawa modules over p-adic Lie extensions. Math. Proc. Camb. Philos. Soc. 135, 25–43 (2003) [V1] O. Venjakob, On the Iwasawa theory of p-adic Lie extensions. Compos. Math. 138, 1–54 (2003) [CFKSV] J. Coates, T. Fukaya, K. Kato, O. Venjakob, R. Sujatha, The GL2 -main conjecture for elliptic curves without complex multiplication. Publ. Math. IHES 101, 163–208 (2005) [HS] G. Hochschild, J-P. Serre, Cohomology of group extensions. Trans. Am. Math. Soc. 74, 110–134 (1953). Serre’s Collected Papers, v. I [V] O. Venjakob, On the structure theory of the Iwasawa algebra of a p-adic Lie group. J. Eur. Math. Soc. 4(3), 271–311 (2002) [H] S. Howson, Euler characteristics as invariants of Iwasawa modules. Proc. Lond. Math. Soc. 85(3), 634–658 (2002) [NSW] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of number fields (Springer, Berlin, Heidelberg, 2000) [G1] R. Greenberg, On p-adic Artin L-functions. Nagoya Math. J. 89, 77–87 (1983) [W1] A. Wiles, The Iwasawa conjecture for totally real fields. Ann. Math. 131, 493 –540 (1990) [K1] M. Kakde, The main conjectures of Iwasawa theory for totally real fields, to appear [RW] J. Ritter, A. Weiss, On the “main conjecture” of equivariant Iwasawa theory. J. am. Math. Soc. 24, 1015–1050 (2011) [KA] K. Kato, K1 of some non-commutative completed group rings. K-Theory 34, 99–140 (2005)

Reductions of the Main Conjecture R. Sujatha

Abstract The main goal of this article is to discuss the relevant background needed to state the noncommutative main conjecture for certain totally real p-adic Lie extensions, and to make the important reduction to the case when the Galois group of the p-adic Lie extension is of dimension one and pro-p. Keywords K-theory • determinant • p-adic Lie groups • representations

1 Introduction Let F be a totally real number field and p an odd prime. Fix a finite set of primes † of F such that † contains the primes above p and let F† be the maximal extension of F unramified outside †. Suppose that F1 =F is an abelian p-adic Lie extension of F which is totally real, contained in F† , and containing the cyclotomic Zp extension F cyc , such that G D Gal.F1 =F / is an abelian p-adic Lie group of dimension one. Then G ' H where ' Zp is the Galois group of the cyclotomic Zp -extension F cyc =F; and H is a finite abelian group. Put HO for the group of 1-dimensional characters of H . Let O be the ring of integers of the field obtained by attaching the values of elements in HO to Qp . Let ƒO .G/ be the Iwasawa algebra of G with coefficients in O (see Sect. 2) and write QO .G/ for the total ring

R. Sujatha () School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400 005, India Department of Mathematics, University of British Columbia, Mathematics Road 1984, V6T 1Z2, Vancouver, BC, Canada e-mail: [email protected] J. Coates et al. (eds.), Noncommutative Iwasawa Main Conjectures over Totally Real Fields, Springer Proceedings in Mathematics & Statistics 29, DOI 10.1007/978-3-642-32199-3 2, © Springer-Verlag Berlin Heidelberg 2013

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R. Sujatha

of fractions of ƒO .G/. An element of QO .G/ is called a pseudomeasure on G if . 1/ is in ƒO .G/ for all in G. Given a pseudomeasure , the integral Z d; G

of any nontrivial continuous homomorphism W G ! O against gives a well defined element of the fraction field of O: Let F denote the character which vanishes on H and restricts to the cyclotomic character on . A result due to Cassou-Noguès and independently, Deligne-Ribet, asserts the existence of a unique pseudomeasure F1 =F on G such that for all in HO , we have Z G

Fn d F1 =F D L† . ; 1 n/;

(1)

for all integers n > 0 with n 0 mod k, where k D ŒF .p / W F , p denotes the group of p-th roots of unity, and L† . ; s/ is the imprimitive complex L-function with the Euler factors corresponding to the primes in † being omitted in the Euler product. For any field extension L of F , let M.L/ be the maximal abelian p-extension of L contained in F† and let X.L/ D Gal.M.L/=L/: Now let K be any finite extension of Q (K need not be totally real). The classical Iwasawa D 0 conjecture for K and the prime p is the assertion that, the Galois group over K cyc of the maximal unramified abelian p-extension of K cyc is a finitely generated Zp -module. This is known to be true for all abelian extensions K of Q and all primes p, thanks to work of Ferrero-Washington. Moreover, a well-known argument in Kummer theory shows that if F is any totally real field and K D F .p /, then the validity of the classical Iwasawa conjecture (see [CK, Sect. 3]) for K implies that X.F cyc / is a finitely generated Zp -module. The aim of the abelian main conjecture is to give a precise relation between the analytic pseudomeasure F1 =F and the algebraic structure of the ƒO .G/-module X.F1 /. The precise assertion is discussed in the article by Coates and Kim [CK] in this volume. As is explained there, stating the assertion of the main conjecture is delicate even in this classical case, when H has order divisible by p. The reader is referred to loc.cit. for a careful discussion leading to the assertion of the main conjecture, which is a fundamental theorem due to Wiles [Wi]. The goal then is to first formulate an analogue of the main conjecture for field extensions F1 =F that are admissible, (see [CK, Sect. 1] for the precise definition) totally real p-adic Lie extensions of F , and prove the conjecture in some special cases. As in [CK, Sect. 3], the extension F1 =F is said to satisfy Iwasawa Conjecture if there exists a finite extension F 0 of F in F1 such that Gal.F1 =F 0 / is pro-p, and X.F 0cyc / is a finitely generated Zp -module. In our present state of knowledge, we can prove this main conjecture only when the Iwasawa conjecture holds (see [CK, Sect. 3]) for F1 . Let G D Gal.F1 =F / and ƒ.G/ be the corresponding Iwasawa algebra, which is a Noetherian noncommutative ring. Then

Reductions of the Main Conjecture

25

X.F1 / is a finitely generated module over ƒ.G/. The philosophy of the main conjecture is to relate an algebraic invariant (the ‘characteristic element’ of X.F1 /) to an analytic invariant, (namely the ‘p-adic L-function’) that interpolates values of the complex L-function. These should be viewed as the counterparts of the characteristic element and the pseudomeasure discussed earlier in the abelian case. However, in the nonabelian case, these invariants lie in the K1 group of a certain noncommutative localization of the Iwasawa algebra ƒ.G/: A precise formulation of the noncommutative main conjecture, and its proof due to M. Kakde was the goal of the series of lectures given at the workshop. The reader is referred to [CK, Sect. 5] and Sect. 2 below for a fuller discussion of this theme. In this introduction, we only point out that even for formulating the main conjecture in this context, it is necessary to assume the analogue of Iwasawa’s D 0 conjecture (see [CK, Sect. 1]), which is equivalent here to assuming that X.F1 / is finitely generated as a ƒ.H /-module where H D Gal.F1 =F cyc /. Further, in this nonabelian context, the analogue of (1) involves a formulation in terms of noncommutative determinants using K-groups and Euler characteristics. The main aim of this article is to formulate the noncommutative main conjecture, and to discuss the reduction of its proof for totally real fields to the case when the p-adic Lie extension F1 =F has dimension one, with pro-p Galois group G. An intermediate important step is to consider the case when G is isomorphic to Gp , where Gp is a compact pro-p, p-adic Lie group of dimension one and is a finite group of order prime to p. We stress that in the general nonabelian case, the hypothesis that X.F1 / satisfies the analogue of Iwasawa’s D 0 conjecture is necessary both for the formulation of the main conjecture and in its proof. That such a reduction to the one dimensional case is possible, was first proved by Burns [Bu]. When is abelian, this reduces to the classical abelian case where the main conjecture is known, thanks to the result of Wiles, and is dealt with in [CK]. In the general case, Kakde’s proof then proceeds by a delicate study of the K1 -group of the relevant localization of the Iwasawa algebra ƒ.G/ and forms the technical core of the proof. This is the subject of the article by Schneider and Venjakob [SV] in this volume. These algebraic results serve as a crucial step in reducing the proof to verifying certain congruence relations between the Deligne-Ribet, Cassou-Nogués p-adic zeta functions for abelian one dimensional extensions, and this forms the basis of constructing the desired p-adic L-function. This step uses the analogue of 0 the D 0 hypothesis, i.e. that X.F1 =F / is a finitely generated Zp -module, where 0 F1 is the extension of F corresponding to the fixed field of F1 by the subgroup Gp , and rests on a strategy of Burns and Kato. The final step of verifying the required congruences is executed using the Deligne-Ribet q-expansion principle. Here is the plan of the article. As the general formulation of the noncommutative main conjecture needs the localization sequence in algebraic K-theory, we will begin by providing a quick review of the K-groups K0 and K1 and also the relevant localization exact sequence in Sect. 1. In Sect. 2, we discuss the necessary background that is needed and then state the noncommutative conjecture. For the proofs of the results stated, the reader is referred to [B] or [Sw]. Finally, in Sect. 3, we show how the proof of the main conjecture can be reduced to the

26

R. Sujatha

case of a special one dimensional p-adic Lie extension, following the arguments of Kakde. The reader is strongly encouraged to consult the papers of Ritter and Weiss (especially [RW]) for another approach of the proof of the main conjecture. We mention that Venjakob’s article [V1] in this volume compares Kakde’s proof with that of Ritter and Weiss. Throughout these notes, all rings considered will be (not necessarily commutative) unital, associative, left and right Noetherian rings, and modules shall mean left modules. Finally, the author is grateful to John Coates and the referee for their comments that helped improve the exposition of the article.

2 Algebraic K-Theory Let C be an abelian category. Recall that a nonempty full subcategory D of C is a Serre subcategory if given an exact sequence f

g

A0 ! A ! A00 in C, we have that A0 and A00 are objects in D if and only if A is an object in D: The group K0 .C/ is the abelian group generated by classes ŒA, where A runs over the objects of C, and subject to the relation ŒA D ŒA0 C ŒA00 ; for each short exact sequence 0 ! A0 ! A ! A00 ! 0 in C: The group K0 .C/ is universal with respect to this property in the following sense. Suppose we are given an abelian group G and a map f W Ob.C/ ! G; such that for each short exact sequence 0 ! A0 ! A ! A00 ! 0 in C, we have f .A/ D f .A0 / C f .A00 / in G. Then there exists a unique homomorphism W K0 .C/ ! G making the diagram Ob.C/ ! K0 .C/ &f

. G

commutative. More generally, the group K0 can be defined analogously for any exact category, which we will not define rigorously (see [B]); intuitively, it is an additive category in which exact sequences are well-defined. Then the map C ! K0 .C/


27

is a covariant functor from the category of abelian categories and exact functors to Ab, the category of abelian groups. The following proposition is well-known. Proposition 2.1. Let A; B be objects of C: If ŒA D ŒB in K0 .C/, then there is an object C 2 C such that A ˚ C ' B ˚ C: Examples: (i) Let k be a field and C be the category of finite dimensional vector spaces. Then K0 .C/ ' Z; the isomorphism being given by the dimension homomorphism. (ii) Let C be the category of finite abelian groups. Then K0 .C/ is free abelian on the basis ŒZ=pZ; where p runs over all primes. This is seen to be true via an induction argument on the cardinality of the finite group, combined with the structure theorem for finite abelian groups. The statement also follows from the Jordan-Hölder theorem along with the fact that a simple abelian group has the form Z=pZ for some prime p. (iii) Let R be a ring and CR be the category of finitely generated (left) R-modules and let PR be the full subcategory of CR consisting of finitely generated projective left R-modules. This is an exact category, and the natural inclusion PR CR induces a homomorphism K0 .PR / ! K0 .CR /: Definition 2.1. Let R be a ring. The group K0 .R/ is defined to be the group K0 .PR /: Proposition 2.2 (see [Sw, Part II, Chap. 1]). Let R be a ring. Then the following assertions hold: (i) The group K0 .PR / is generated by ŒP , where P is a finitely generated projective R-module with the relations ŒP D ŒQ if P ' Q and ŒP ˚ Q D ŒP C ŒQ: (ii) Any element of K0 .R/ can be written as ŒP ŒQ for finitely generated projective R-modules P and Q. Further, ŒP ŒQ D ŒP 0 ŒQ0 in K0 .R/ if and only if there is a finitely generated projective R-module M such that P ˚ Q0 ˚ M D P 0 ˚ Q ˚ M: Given a ring homomorphism W R ! R0 , the base change map P 7! R0 ˝R P induces a group homomorphism K0 .R/ ! K0 .R0 / as induces an exact functor from the category PR to PR0 : We shall also need the following result (see [Sw, Part II, Chap. 2, Proposition 2.29]). Proposition 2.3. If R is complete with respect to a two sided ideal I, the map K0 .R/ ! K0 .R=I / induced by the projection R ! R=I is an isomorphism. The following result due to A. Heller, is important in relating K0 .R/ and K0 .CR /. Recall that a subcategory A is closed under subobjects (resp. quotient objects) in C if subobjects (resp. quotient objects) in C of an object in A are again in A. Theorem 2.1 ([Sw, Part II, Chap. 3, Theorem 3.1]). Let C be an abelian category, A and B be full subcategories with A B, and such that A is closed in C under

28

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subobjects and quotient objects. Assume further that every object of B has a finite filtration with all quotients in A. Then the canonical map K0 .A/ ! K0 .B/ is an isomorphism. Definition 2.2. For a ring R, the group G0 .R/ is defined to be the abelian group K0 .CR /. Note that the inclusion PR CR induces a group homomorphism (see loc.cit. Corollary 4.6) i W K0 .R/ ! G0 .R/; (2) called the Cartan homomorphism. Theorem 2.2 ([Sw, Part II, Chap. 3, Proposition 3.4]). Let I be a nilpotent two sided ideal of a ring R. Then the map G0 .R=I / ! G0 .R/ given by ŒA 7! ŒA is an isomorphism. Suppose that h W R ! R0 is a ring homomorphism such that R0 is flat as an Rmodule. Then the natural functor CR ! CR0 obtained from the base change map induces a homomorphism G0 .R/ ! G0 .R0 /: To make precise the relationship between G0 .R/ and K0 .R/, we need some more groundwork. Definition 2.3. Let C be an abelian category. Recall that an object P 2 C is projective if the functor Hom .P; / W C ! Set preserves epimorphisms. The projective dimension of A 2 Ob.C/ denoted pd.A/ is the smallest n such that there exists an exact sequence 0 ! Pn ! Pn1 ! P0 ! A ! 0 with all the Pi ’s projective. If the projective dimension of A is undefined we write pd.A/ D 1: In the particular case that C D CR , we say that R has finite global dimension n if every object of CR has finite projective dimension at most n. Commutative regular local rings have finite global dimension n where n is the Krull dimension of R. This is a deep result of Auslander-Buchsbaum in commutative ring theory. In noncommutative ring theory, Noetherian Auslander regular local rings [Bj] are known to have finite global dimension. For such rings, there is a well developed dimension theory for modules which plays a central role. Theorem 2.3 ([Sw, Theorem 4.4]). Let C be an abelian category and P be the full subcategory of projective objects. If pd.A/ < 1 for every A in C, then the natural map K0 .P/ ! K0 .A/ is an isomorphism. In particular, if a ring R has finite global dimension, then the natural homomorphism i W K0 .R/ ! G0 .R/ is an isomorphism. Corollary 2.1 (cf. [Sw, Corollary 4.6]). Let R be a commutative Noetherian regular local ring or a noncommutative Noetherian Auslander regular local ring. Then the natural homomorphism i W K0 .R/ ! G0 .R/ is an isomorphism.


29

If C is an abelian category, P is the full subcategory of projective objects, and H the full subcategory of all A 2 C with pdA < 1 (note that H need not be an abelian category, but it is an exact category), then the natural map K0 .P/ ! K0 .H/ is again an isomorphism. If B is a full, exact subcategory such that P B H, then the natural map K0 .P/ ! K0 .B/ is a split mono. It is onto if for every exact sequence 0 ! C ! P ! B ! 0; P 2 P and B 2 B implies C 2 B. We next deal with localization and the corresponding maps between G0 (see [Sw, Chap. 5]). Let R be a ring and S a subset such that S contains 1 and is either a multiplicatively closed set contained in the centre of R or a left and right Ore set. For ease of exposition, we shall always assume that the elements of S are not zero divisors in R. In both these cases, the corresponding localizations exist and we denote by RS the localized ring. For a module M over R, the localization is denoted by MS . The natural map f W M ! MS given by f .m/ D m=1 is an Rhomomorphism and a ring homomorphism in the special case when M D R. The kernel Ker .f / consists of the set of elements m 2 M such that there exists an element s 2 S with sm D 0: Recall that RS is a flat R-module. Let CS be the subcategory consisting of finitely generated R-modules M such that MS D 0. Then CS is a Serre subcategory and there is an exact sequence (loc.cit. Corollary 5.14) K0 .CS / ! K0 .R/ ! K0 .RS /:

(3)

The right map is not always surjective, but there is always an exact sequence K0 .CS / ! G0 .R/ ! G0 .RS / ! 0:

(4)

If the ring R (and hence the ring RS ) has finite global dimension, then (3) gives an exact sequence K0 .CS / ! K0 .R/ ! K0 .RS / ! 0: (5) To deal with a broader class of rings which do not necessarily have finite global dimension, and also to extend the above exact sequences on the left with K1 groups, we introduce the notion of the relative K-group. For more details, the reader is referred to Weibel’s book [We]. Definition 2.4. Let f W R ! R0 be a ring homomorphism. The category Cf consists of triplets .P; a; Q/ as objects, where P and Q are finitely generated projective R-modules, and a is an isomorphism a

a W R0 ˝R P ! R0 ˝R Q: Morphisms between two objects .P; a; Q/ and .P 0 ; a0 ; Q0 / consist of a pair of Rmorphisms g W P ! P 0 and h W Q ! Q0 such that a0 ı .1R0 ˝ g/ D .1R0 ˝ h/ ı a:

30

R. Sujatha

It is an isomorphism if both g and h are isomorphisms. A sequence of morphisms in Cf (see [Sw, Part II, Chap. 13]) 0 ! .P 0 ; a0 ; Q0 / ! .P; a; Q/ ! .P 00 ; a00 ; Q00 / ! 0

(6)

is a short exact sequence if the underlying sequences 0 ! P 0 ! P ! P 00 ! 0;

0 ! Q0 ! Q ! Q00 ! 0

are short exact. The group K0 .f / (also called the relative K0 ) is an abelian group defined by generators Œ.P; a; Q/ with .P; a; Q/ an object in the category Cf subject to the following relations: • Œ.P; a; Q/ D Œ.P 0 ; a0 ; Q0 / if the two objects are isomorphic. • Œ.P; a; Q/ D Œ.P 0 ; a0 ; Q0 / C Œ.P 00 ; a00 ; Q00 / for every short exact sequence as in (6). • Œ.P1 ; b ı a; P3 / D Œ.P1 ; a; P2 / C Œ.P2 ; b; P3 /: For the natural map W R ! RS , the relative K-group K0 ./ is usually denoted by K0 .R; RS /: This group can also be identified with the following group. Let CShb denote the category of bounded complexes of finitely generated projective Rmodules whose homology modules are S -torsion. The abelian group K0 .CShb / is defined as the abelian group with generators ŒC ; where C is in CShb , and relations (1) ŒC D 0 if the complex C is acyclic, and (2) ŒC D ŒC 0 C ŒC 00 , for every short exact sequence of complexes 0 ! C 0 ! C ! C 00 ! 0 in CShb : Let HS denote the category of finitely generated R-modules which are S -torsion and which have a finite resolution by finitely generated projective Rmodules. There are isomorphisms K0 .CShb /

K0 .R; RS / ! K0 .HS /:

(7)

To describe the isomorphisms in the special case when S is central, first note that every isomorphism a from RS ˝R P to RS ˝R Q is of the form s 1 ˛, where ˛ is an R-map from P to Q and s 2 S . The first isomorphism is given by ˛

s

Œ.P; a; Q/ 7! ŒP ! Q ŒQ ! Q; while the second is given by Œ.P; a; Q/ 7! ŒQ=˛.P / ŒQ=sQ: For more general localizing sets S , the isomorphisms are a little more subtle and proceed via the Euler characteristic.


31

Definition 2.5. Given a bounded chain complex C W 0 ! Cm ! ! C0 ! 0 of objects in an abelian category C., the Euler characteristic .C / of C is defined to be the element †.1/i ŒCi of K0 .C/: If C is a bounded complex of objects in C, the element .C / depends only upon the homology of C and .C / D †.1/i ŒHi .C /: In particular, if C is acyclic, then

.C / D 0: In the general case, the isomorphisms in (7) are fleshed out using the Euler characteristic map. The first isomorphism takes Œ.P; a; Q/ to the complex a Œ0 ! P ! Q ! 0 while the second maps Œ.P; a; Q/ to ŒCoker .a/ ŒKer .a/: We now discuss the group K1 .R/ for a ring R. There are various (equivalent) definitions though checking the equivalence involves several technicalities and is hence beyond the scope of these lectures. We shall largely consider the algebraic definitions which only use linear algebra and elementary group theory. Let GLn .R/ denote the group of .n n/ matrices over R that are invertible. Each such matrix g g0 is naturally viewed as an .n C 1/ .n C 1/ matrix given by and thus gives 01 an embedding of GLn .R/ into GLnC1 .R/: The group GL.R/ is defined as the union [

GLn .R/;

n

and is called the infinite linear group over R. Recall the commutator subgroup ŒG; G of a group G, which is always a normal subgroup of G and is universal with respect to the quotient group being abelian. The first definition of K1 .R/ (also called the Whitehead group of R) is as the quotient of GL.R/ by its commutator subgroup. Definition 2.6. K1 .R/ is the abelian group GL.R/=ŒGL.R/; GL.R/: The abelian group K1 .R/ thus has the universal property that every homomorphism from GL.R/ to an abelian group must factor through K1 .R/: Given a ring homomorphism R ! R0 , it induces a homomorphism from K1 .R/ to K1 .R0 / and K1 is thus a covariant functor from the category of rings to Ab: Example: If R happens to be commutative, then the determinant homomorphism is a group homomorphism from GL.R/ onto the (abelian) group R of units in R, and we denote the induced surjective map from K1 .R/ by det, det W K1 .R/ ! R ; and the kernel of this surjection is defined to be the abelian group SK1 .R/: There is a direct sum decomposition K1 .R/ ' R ˚ SK1 .R/: Recall that for an element r in R, the elementary matrix eij .r/ in GL.R/ is the matrix which has 1 across the diagonal, r at the .i; j /-spot, and zero elsewhere.

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R. Sujatha

If En .R/ is the subgroup of GLn .R/ generated by all elementary matrices eij .r/ with 1 i; j n; i ¤ j , then the group E.R/ is the union [ En .R/: The classical n

Whitehead’s lemma asserts that the group E.R/ is the commutator subgroup and therefore K1 .R/ ' GL.R/=E.R/: The group K1 .R/ measures the obstruction to taking an arbitrary invertible matrix over R and reducing it to the identity via a series of elementary operations. Recall that a ring R is said to be semilocal if R=radR is a left artinian ring, or equivalently R=radR is a semisimple ring; here radR is the radical of the ring R. If R has finitely many maximal left ideals, then R is semilocal. The following result of Vaserstein is very useful. Proposition 2.4. Let R be a semilocal ring. Then the natural inclusion of R D GL1 .R/ into GL.R/ induces an isomorphism K1 .R/ ' R =ŒR ; R : Example: (i) If R is the group ring ZŒG of a group G, the (first) Whitehead group Wh.G/ is the quotient of K1 .ZŒG/ by the subgroup generated by ˙1 and the elements of G, considered as elements of R and hence in GL1 .R/: If G is abelian, then ZŒG is commutative and we have Wh.G/ D ZŒG = ˙ G ˚ SK1 .ZŒG/: If G is finite, then Wh.G/ is known to be a finitely generated group. In fact, it is known that rank K1 .ZŒG/= rank Wh.G/, and hence SK1 .ZŒG/ is the full torsion subgroup of K1 .ZŒG/ and is in fact equal to the kernel of the natural map K1 .ZŒG/ ! K1 .QŒG/: (ii) More generally if A is a Dedekind domain with quotient field K, and G is a finite group, the group SK1 .AŒG/ is defined by SK1 .AŒG/ D Ker .K1 .AŒG/ ! K1 .KŒG//:

(8)

The important results proved by Wall, Oliver and others will be used subsequently and [O] is a good reference for this. The group SK1 .ZŒG/ has interesting topological applications. Let us now turn to an alternative definition of K1 .R/ using projective modules. Let P be a finitely generated projective R-module. Choosing an isomorphism P ˚ Q ' Rn gives a group homomorphism from Aut.P / to GLn .R/, by sending ˛ to ˛ ˚ 1Q : This homomorphism is well-defined when viewed as a homomorphism into GLn .R/, up to inner automorphisms of GLn .R/; and thus there is a well-defined homomorphism Aut.P / ! K1 .R/: Definition 2.7. For a ring R, the group K1 .R/ is an abelian group which consists of generators ŒP; a where P is a finitely generated projective R-module and a is in Aut.P /, subject to the relations


33

• ŒP; a D ŒQ; b if there is an isomorphism f W P ! Q such that f ı a D b ı f; • ŒP; a ı b D ŒP; aŒP; b • ŒP ˚ Q; a ˚ b D ŒP; aŒQ; b: The equivalence of this definition with the earlier one is seen using the welldefined homorphism from Aut.P / to GL.R/ as discussed above. There is a homological interpretation of K1 .R/ given by K1 .R/ D H1 .GL.R/; Z/; the first group homology of GL.R/ with Z-coefficients, (see [Mi]). A further topological interpretation elucidates K1 .R/ as 1 .K.R// where K.R/ is a certain topological space built using the category PR : Let R be a ring and S a (left and right) localising Ore set as before. Consider the relative K-group K0 .R; RS /: There is a connecting homomorphism @ W K1 .RS / ! K0 .R; RS /

(9)

that maps an element ŒPS ; ˛S in K1 .RS / (PS is a finitely generated projective module over RS and ˛S is an automorphism of PS ) to ŒRn ; ˛; Q Rn . Here P is a projective module such that RS ˝R P D PS and Q is a projective R-module such that P ˚Q ' Rn , and ˛Q is an endomorphism of Rn such that 1RS ˝R .˛˚1Q / ' ˛S : We can also use the isomorphism of K1 .R/ with GL.R/=E.R/ to define this homomorphism. Map an element g in GL.n; R/ to Œ.Rn ; g; Rn / in K0 .R; RS /: This commutes with the map GL.n; R/ ! GL.n C 1; R/ and thus gives a well-defined homomorphism from GL.R/ to K0 .R; RS /. But the latter group being abelian, this homomorphism factors to give the desired map from K1 .R/: There is also a homomorphism W K0 .R; RS / ! K0 .R/ (10) that maps Œ.P; a; Q/ to ŒP ŒQ in K0 .R/: These maps are used to construct a long exact sequence of K-groups as in the theorem stated below. A proof of this theorem when S is a central multiplicatively closed set can be found in [Sw, Part II, Chap. 15]. Theorem 2.4. Let R be a ring and S a left and right Ore set of R whose elements are nonzero divisors. Then there is an exact localization sequence @

K1 .R/ ! K1 .RS / ! K0 .R; RS / ! K0 .R/ ! K0 .RS /:

(11)

The sequence is surjective on the right if K0 .RS / is replaced by G0 .RS /, and the same holds if the ring R has finite global dimension. Another important property of the groups K0 and K1 is Morita invariance. If R is a ring, then this property states that there are natural isomorphisms Ki .R/ ' Ki .Mn .R// for i D 0; 1: We shall need the norm or transfer maps defined in Ktheory, which we recall for the K1 -groups.

34

R. Sujatha

Definition 2.8. Let R be a ring and R0 a ring containing R such that R0 is finitely generated and projective as an R-module. Let PR0 and PR denote the categories of finitely generated projective modules over R0 and R respectively. Then the natural forgetful functor PR0 ! PR induces a homomorphism K1 .R0 / ! K1 .R/ which is called the norm or transfer homomorphism. A particular case of this will be that of group rings, with G a subgroup of finite index in another group G 0 , R0 D ZŒG 0 and R D ZŒG: We shall close this section with the following definition which will be extended later to Iwasawa algebras. Definition 2.9. Let G be a finite group and A be a Dedekind domain. Consider the group ring R D AŒG: The group K10 .A/ is defined by K10 .R/ D K1 .R/=SK1 .R/, where SK1 .R/ is defined as in (8). Finally, we remark that the integral logarithm which will be discussed in the lectures of [SV] gives a nice description of K10 .R/.

3 The Noncommutative Main Conjecture In this section, we shall formulate the Noncommutative Main Conjecture for the trivial motive over totally real fields. We shall follow notation as in the lectures of J. Coates [CK]. Thus F will denote a totally real number field and F1 will be an admissible, totally real p-adic Lie extension of F . An extension F1 =F as above is said to be admissible if (1) the Galois group Gal.F1 =F / is a p-adic Lie group, (2) F1 =F is unramified outside a finite set of primes † of F containing the primes lying above p, and (3) F1 contains the cyclotomic Zp -extension F cyc : We shall further assume that the Iwasawa conjecture holds for the admissible extension F1 : Put G D Gal.F1 =F /, H D Gal.F1 =F cyc /; and let ƒ.G/ denote the Iwasawa algebra ƒ.G/ WD lim Zp ŒG=U

where U varies over open normal subgroups of G and the inverse limit is taken with respect to the natural surjections. If O is the ring of integers in a finite extension of Qp , then ƒO .G/ is the Iwasawa algebra of G over O, defined as above with O replacing Zp . Fix a finite set † of primes of F that contain the primes above p, and denote the maximal extension of F that is unramified outside † by F† . Recall that for any extension L of F contained in F† , X.L/ is the Galois group of the maximal abelian p-extension contained in F† . As discussed in the lectures by Coates, X.F cyc / is a torsion module over the Iwasawa algebra ƒ./, where ' Zp D Gal.F cyc =F /: The ring ƒ.G/ is known to be a left and right noetherian ring. If G is pro-p and has no elements of order p, it is an Auslander regular, local domain (see [V]). In general however, it is a semilocal ring. The statement of the main conjecture needs


35

the localization sequence, which is contingent on a choice of a left and right Ore set. Recall that H D Gal.F1 =F cyc /: Definition 3.1. The set S is defined as S D fx 2 ƒ.G/ j ƒ.G/=sƒ.G/ is a finitely generated ƒ.H / moduleg: It is proven in [CFKSV] that the set S consists of nonzero and is a left and S divisors right Ore set of ƒ.G/: The set S is defined by S D p n S: n0

In this set-up, we have the localization exact sequence

@

K1 .ƒ.G// ! K1 .ƒ.G/S / ! K0 .ƒ.G/; ƒ.G/S / ! K0 .ƒ.G// ! K0 .ƒ.G/S /: (12)

It is proven in [FK] that if G is a profinite group, then K1 .ƒO .G// ' lim K1 .OŒG=U /;

(13)

where the groups G=U vary over the finite quotients of the profinite group G. An additional important result (see for example [K]) is that the connecting homomorphism @ is surjective. This is easily seen to be true when ƒ.G/ is local and G has no element of order p, as then the map is zero since the last two groups on the right are isomorphic to Z, the isomorphism being given by the rank. The general case needs a little more work, along with the fact that G always contains an open normal pro-p subgroup P . The hypothesis that D 0 for the Iwasawa module X.F cyc / guarantees that the module X.F1 / is in fact an S -torsion module. However, as we are not assuming that G has no elements of order p, X.F1 / might not have a finite resolution by projective modules. We shall therefore associate a complex which we denote by C.F1 =F /, (see [FK]) with the following properties: • Hi .C.F1 =F // D 0 for i ¤ 0; 1 • H0 .C.F1 =F // D Zp and H1 .C.F1 =F // D X.F1 /. Indeed the complex C.F1 =F / is defined by RHom.R.Gal.F† =F1 /; Qp =Zp //. Moreover, it was proved by Fukaya and Kato [FK] that the complex C.F1 =F / is quasi-isomorphic to a bounded complex of finitely generated projective ƒ.G/modules. It is clear by the remarks above that ƒ.G/S ˝L ƒ.G/ C.F1 =F / is acyclic. Further, there are complexes C.K=F / for any extension F K F1 with derived isomorphisms ƒ.Gal.K=F // ˝L ƒ.G/ C.F1 =F / ' C.K=F /: Thus there is an object ŒC.F1 /=F in the category CShb which we then view as being in the relative K-group K0 .ƒ.G/; ƒ.G/S / via the isomorphism (7). This object is

36

R. Sujatha

closely linked to X.F1 /: In fact, when G has no elements of order p, the module X.F1 / can be viewed as an object in the category HS and under the isomorphisms in (7), the complex C.F1 =F / maps to the module X.F1 /. Definition 3.2. Let O be the ring of integers in a finite extension L of Qp . If G is a profinite group, then SK1 .ƒO .G// is defined as the inverse limit SK1 .ƒO .G// WD lim SK1 .OŒG=U /

3

where U varies over open normal subgroups of G and SK1 .OŒG=U / is defined for the group ring as in (8), noting that G=U is a finite group. The ring ƒO .G/S is the completion (in the p-adic topology) of the localisation of the Iwasawa algebra ƒO .G/S : We define SK1 .ƒO .G/S / and SK1 .ƒO .G/S / to be the image of SK1 .ƒO .G// in K1 .ƒO .G/S / and in K1 .ƒO .G/S / respectively under the corresponding natural maps

2 3

3

K1 .ƒO .G// ! K1 .ƒO .G/S / and K1 .ƒO .G// ! K1 .ƒO .G/S /:

The following definition extends that of 2.9.

3

Definition 3.3. Let R denote any of the rings ƒO .G/ or ƒO .G/S or ƒO .G/S : Then we define K10 .R/ D K1 .R/=SK1 .R/: It is easily seen that there is a localization exact sequence @

K1 .ƒ.G// ! K10 .ƒ.G/S / ! K0 .ƒ.G/; ƒ.G/S / ! 0:

(14)

The formulation of the main conjecture uses this exact sequence along with the fact noted earlier, namely that ŒX.F1 / lies in the relative K-group K0 .ƒ.G/; ƒ.G/S /: It is convenient to work with K10 rather than the K1 groups in the localization sequence as this gives a uniqueness statement in the main conjecture. To make a precise formulation, we need to discuss noncommutative determinants. Let be a finite dimensional Artin representation of Gal.FN =F / factoring through G. Then we have a continuous representation W G ! GLn O where O is the ring of integers of a finite extension of Qp . Let QO ./ be the fraction field of the Iwasawa algebra ƒO ./: There is an induced homomorphism (see [CFKSV]) ˆ W ƒ.G/S ! Mn .QO .// which by functoriality induces a group homomorphism


37

ˆ0 W K10 .ƒ.G/S / ! K10 .Mn .QO .// D QO ./ ; where the last equality holds by Morita invariance. On the other hand, the augmentation map W ƒO ./ ! O extends to a homomorphism W ƒO ./p ! L; where p is the kernel of and ƒO ./p is the corresponding localisation. Extend this map to 0 W QO ./ ! L [ 1 by mapping any x 2 QO ./ n ƒO ./p to 1: Composing ˆ0 with 0 , we get a map K10 .ƒ.G/S / ! L [ 1 x 7! x./: Thus elements of K1 .ƒ.G/S / can be evaluated on Artin characters to get values. The main conjecture predicts a precise interpolation of these values as below. For any Artin representation of the Galois group of F , let L† .; s/ denote the imprimitive L-function associated to with the Euler factors at the primes in † being removed. Theorem 3.1 (MAIN CONJECTURE). Let F1 =F be an admissible p-adic Lie extension satisfying the hypothesis D 0: Then there is a unique element .F1 =F / in K10 .ƒ.G/S / such that @..F1 =F // D ŒC.F1 =F / where C.F1 =F / is the complex defined above, and such that for any Artin character of G and any positive integer r divisible by the extension degree ŒF1 .p / W F1 , we have .F1 =F /. Fr / D L† .; 1 r/; where is the cyclotomic character. The element .F1 =F / is called a p-adic zeta function for the extension F1 =F . It depends on the finite set † but we shall suppress this in the notation. It can be shown that the validity of the Main Conjecture is independent of † as long as it contains all the primes of F that ramify in F1 :

4 Reductions In this section, we show how the proof of the main conjecture can be reduced to the case when the Galois group G has dimension one, with G ' H Gp , where Gp is a pro-p, p-adic Lie group and H is a finite group. In other words, the validity of the

38

R. Sujatha

main conjecture for one dimensional p-adic Lie groups implies the main conjecture for p-adic Lie groups of arbitrary dimension. This was first proven by Burns [Bu] using slightly different methods. We briefly outline the key reduction steps. Needless to say, each stage of the reduction is accompanied by a meticulous book-keeping on the analytic side for the p-adic L-function and a corresponding reduction for the K-groups, on the algebraic side, but we shall suppress a discussion of these aspects in this outline. The reader can check the corresponding precise assertions in the statements of the theorems where the reduction steps are executed. Of course, the “ D 0” hypothesis, and the admissibility hypothesis for the reduction steps need to be ensured as well, but this is easily checked. The first reduction is to the case when G is replaced by a quotient G=U where U is an open subgroup of H (here H D Gal.F1 =F cyc /), and normal in G (see Theorem 4.1). We next reduce to the case where G may be replaced by the inverse image in G of a Qp -elementary subgroup (see Definition 4.2) in a suitable finite quotient Gn under the natural quotient map G ! Gn ; this is accomplished in Theorem 4.3. The next reduction (cf. Theorem 4.4) is to the case when G may be assumed to be of the form G ' E 0 E , where E 0 is a finite group of order prime to p and E is a finite abelian group. This paves the way via Theorem 4.5 to reduce to the case when G is in fact an l Qp -elementary group for a prime l ¤ p (cf. Definition 4.3). Theorem 4.6 proves that the main conjecture is true also when G is p Qp -elementary. Combining all these results, one then shows that it suffices to prove the main conjecture when G ' Gp , where is a finite cyclic group of order prime to p, and Gp is a compact p-adic Lie group of dimension one. This is the final assertion which will eventually be proven. Let us start with some preliminary observations. By a result of Higman (see [O]), it is known that the groups SK1 .OŒ/ are finite for all finite groups : Combining this with (13), for a compact p-adic Lie group P , we have K10 .ƒO .P // ' lim K10 .OŒ/;

(15)

where varies over the finite quotients of P . Let G be the Galois group F1 =F as before. The following set of quotient groups of G plays an important role. Definition 4.1. The set Q1 .G/ D fG=U W U is an open subgroup of H and is normal in Gg. Proposition 4.1. The natural map K10 .ƒ.G// !

lim

G 0 2Q

is an isomorphism.

1 .G/

K10 .ƒ.G 0 //


39

Proof. We have lim

G 0 2Q

1 .G/

K10 .ƒ.G 0 // ' lim lim K10 .Zp ŒG 0 / 0 G

G 0

' lim K10 .Zp Œ/

' K10 .ƒ.G//; where G 0 runs through finite quotients of G 0 and runs through all finite quotients of G: t u The first reduction is the following: Theorem 4.1. Assume F1 =F is an admissible p-adic Lie extension satisfying the D 0 hypothesis. Then the main conjecture is true for F1 =F if it is true for each U of the extensions F1 =F for all open subgroups U H such that U is normal in G, and if for each group GU WD G=U; the group K10 .ƒ.GU // injects into K1 .ƒ.GU /S /: U =F is an admissible p-adic Lie extension which Proof (sketch). Note first that F1 again satisfies the D 0 Hypothesis. (This is true because the D 0 hypothesis is equivalent to X.F1 / being finitely generated as a ƒ.H /-module, and observing that ƒ.U / is a subalgebra of ƒ.H / with the property that ƒ.H / is finite as a ƒ.U /module). Consider now the following commutative diagram: K10 .ƒ.G//

!

K10 .ƒ.G/S /

@

!

K0 .ƒ.G/; ƒ.G/S /

! 0

? ? y

? ? y

? ? oy

1 ! lim K10 .ƒ.GU // ! lim K10 .ƒ.GU /S / ! lim K0 .ƒ.GU /; ƒ.GU /S /: GU

GU

GU

Using the hypothesis on the validity of the main conjecture for the quotients G=U , we see that there exist elements U in K10 .ƒ.GU /S / satisfying the main conjecture for each GU . Let f 2 K10 .ƒ.G/S / be a characteristic element of C.F1 =F / and let .fU /U 2 lim K10 .ƒ.GU /S / be the image of f under the second vertical map. Put GU

wU D U fU1 2 K10 .ƒ.GU /S /: Then .wU /U 2Q1 .G/ 2 lim K10 .ƒ.GU //. There is an GU

element w 2 K10 .ƒ.G// that maps to .wU /U under the first vertical isomorphism. Let us check that WD wf is the required p-adic zeta function. Uniqueness is clear, and an easy diagram chase gives @./ D ŒC.F1 =F /: To check the interpolation property, let be an Artin representation of G. Note first that there is a G 0 in Q1 .G/ such that factors through G 0 : Since maps to G 0 in K1 .ƒ.G 0 /S /; for any positive integer r divisible by ŒF1 .p / W F1 ; we have . Fr / D G . Fr / D L† .; 1 r/; as required.

t u

40

R. Sujatha

The next reduction steps help us to reduce further to special kinds of admissible p-adic extensions of dimension one. Let ' Zp be the Galois group of the cyclotomic extension F cyc =F . Pick and fix a lift of in G so that we get an e isomorphism G ' H Ì . Fix also an open subgroup p of that acts trivially on e H and put Ge D G= p : We need the following definitions. Definition 4.2. Let l be prime integer. A finite group P is called l-hyperelementary if P is of the form Cn Ì P1 ; with Cn a cyclic group of order n, l a prime such that l − n, and P1 is a finite l-group. Let K be a field. An l-hyperelementary group Cn Ì P1 is called l-K-elementary if ImŒP1 ! Aut.Cn / ' .Z=nZ/ Gal.K.n /=K: A hyperelementary group is one which is l-hyperelementary for some prime l. A K-elementary group is one which is l-K-elementary for some prime l. The above definition is extended to p-adic Lie groups as follows. Definition 4.3. Let l be a prime. A p-adic Lie group is called l-K-elementary if it is of the form P Ì for a finite group P and such that there is a central open subgroup r r p of P Ì such that .P Ì /= p is an l-K-elementary finite group. A p-adic Lie group is K-elementary if it is l-K-elementary for some prime l. The next tool we use is the induction theory of A. Dress which reduces the computation of K10 for finite group rings OŒP to l-K-elementary subgroups of the finite group P , where K is the fraction field of O: In particular, we need the following theorem (see [O, Theorem 11.2] and also [W]). Theorem 4.2. Let P be a finite group. Then there is an isomorphism K10 .OŒP / ' lim K10 .OŒ /; where runs through all K-elementary subgroups of P . The inverse limit is taken with respect Q to norm maps and the maps induced by conjugation. In other words, .x / 2 K10 .OŒ / lies in lim K10 .OŒ / if and only if

• For all g 2 P; gx g

1

D xg g1 ; and

• For 0 P; the norm homomorphism K10 .OŒ 0 / ! K10 .OŒ / maps x 0 ! x : Hence in the above isomorphism, we may restrict the inverse limit to only maximal K-elementary subgroups of P . Lemma 4.1. Let Gn D G= p

nCe

for all n 0: For each n, we have an isomorphism


41

K10 .ƒ.G// ' lim K10 .ƒ.UP //; P

where the inverse limit ranges over all Qp -elementary subgroups of Gn , and it is with respect to the maps induced by conjugation and the norm maps. The group UP is the inverse image of P in G under the surjection G ! ! Gn : Proof. Note that Gn is a finite group as we have reduced to the case when G has dimension one. Letting P run through all Qp -elementary subgroups of Gn and taking all inverse limits with respect to the maps induced by conjugation and the norm maps, by Theorem 4.2 we have an isomorphism K10 .Zp ŒGn / ' lim K10 .Zp ŒP / P

Claim:

K10 .Zp ŒGnCi / ' lim K10 .Zp ŒUP = p

nCeCi

/:

P

nCeCi

We remark that UP = p is the inverse image of P in the group GnCi for all i 0, under the surjection GnCi ! Gn : Further, any Qp -elementary subgroup of nCeC1 GnCi is contained in UP = p for some P . Hence lim K10 .Zp ŒUP = p P

nCeC1

/ ' lim lim K10 .Zp ŒCP / P

CP

'

lim K10 .Zp ŒQ/

'

K10 .Zp ŒGnCi /I

Q

nCeCi

where CP runs through all Qp -elementary subgroups of UP = p and Q runs through all Qp -elementary subgroups of GnCi : Hence the claim is proved and the lemma follows by passing to the inverse limit and using (15). t u The following theorem reduces proving the main conjecture to the case of Qp elementary extensions. Theorem 4.3. Assume that the main conjecture is valid for all one dimensional, admissible p-adic Lie extensions satisfying D 0 hypothesis and whose Galois group is Qp -elementary. Also assume that for all Qp -elementary p-adic Lie groups U , the group K10 .ƒ.U // injects into K10 .ƒ.U /S /: Then the main conjecture is valid for all admissible p-adic Lie extensions F1 =F satisfying D 0 hypothesis. Moreover, if G D Gal.F1 =F /, then K10 .ƒ.G// injects into K10 .ƒ.G/S /: Proof. We sketch the essential steps. First note that if the D 0 hypothesis is valid for an admissible p-adic Lie extension F1 =F of dimension one, then it is valid for all its admissible p-adic Lie subextensions as well. This is a consequence of the fact that D 0 is equivalent to X.F1 / being a finitely generated module over Zp ŒH where H D Gal.F1 =F cyc /: Assume that the main conjecture is

42

R. Sujatha

valid for all admissible p-adic Lie subextensions of F1 =F whose Galois group nCe is Qp -elementary, and let Gn D G= p , for n 0: Consider the following commutative diagram where P runs through all Qp -elementary subgroups of Gn and UP denotes the inverse image of P in G. K10 .ƒ.G//

K10 .ƒ.G/S /

!

@

!

K0 .ƒ.G/; ƒ.G/S /

! 0

? ? y

? ? y

? ? oy

1 ! lim K10 .ƒ.UP // ! lim K10 .ƒ.UP /S / ! lim K0 .ƒ.UP /; ƒ.UP /S /: P

P

P

Pick f in K10 .ƒ.G/S / such that @.f / D ŒC.F1 /=F and let .fP / be its image in lim K10 .ƒ.UP /S /: From our hypothesis, we get a p-adic zeta function P for each P

P , which gives, by uniqueness, an element .P / 2 lim K10 .ƒ.UP /S /: Put uP D P

P fP1 2 K10 .ƒ.UP // and note that there is a u 2 K10 .ƒ.G// mapping to .uP /: Then D uf is the sought after p-adic L-function. Being the only element in K10 .ƒ.G/S / such that @./ D ŒC.F1 =F / and whose image in K10 .ƒ.UP // is P , it is independent of the choice of n. The interpolation property is seen to hold as follows. Let be an Artin character of G, then factors through a quotient Gn for some n. By the Brauer Induction theorem (see [CR, (15.9)]), we can write D † np IndG UP P ; P

remembering that P runs through Qp -elementary subgroups of Gn , and also that P ’s need not be one dimensional. Let FP denote the field extension of F that is the fixed field of UP and suppose that FP is the corresponding cyclotomic character. For any positive integer r divisible by ŒF1 .p / W F1 , we then have . Fr / D D D

Q P Q P Q

r n .IndG UP P F /

P .P Fr P /nP L† .P ; 1 r/nP ;

P

thereby establishing the interpolation property. The remaining assertion follows easily from a diagram chase and the theorem is proved. t u We have thus reduced the proof of the main conjecture to Qp -elementary extensions. Within this reduction, we proceed to analyse the case under the assumption that G is l Qp -elementary for some prime l ¤ p: The first observation is the following


43

Lemma 4.2. If a p-adic Lie group G is l-Qp -elementary for some prime l ¤ p, e then it is isomorphic to p E for some finite l-Qp -elementary group E. r

r

Proof. Write G D H Ì: Let p be a central open subgroup of G such that G= p e is l Qp -elementary. Choose e 0 to be the smallest integer such that p is an 0 pe open central subgroup of G. Then G WD G= is an l Qp -elementary finite e group. We prove that G D p G 0 : Write G 0 D C Ì P; where C is a cyclic subgroup of G of order prime to l and P is an l-group. We have C Ì P D G 0 D e H Ì = p : The order of P being prime to p, it is a subgroup of H and is hence e also a subgroup of G. Let UC be the inverse image of C in G. As p is central and C is cyclic, UC is an abelian group. We write UC D Q D, where Q is a pro-p, pro-cyclic subgroup of UC and D is a torsion subgroup. Then G D .Q D/ Ì P: The action of P on Q is necessarily trivial, as it is trivial on an open subgroup of Q. e Hence Q is a central pro-cyclic subgroup of G whence Q D p and D D C: This completes the proof. t u This lemma enables us to take the further step of reducing to the case when G is of the form E, with E an l-Qp -elementary finite group. We may write E D C Ì U where C is a cyclic group of order prime to l and U is a finite l-group. If p − the order of C , then the main conjecture is valid for F1 =F by the following classical theorem (see [CK]). Theorem 4.4. Let F1 =F be a p-adic Lie extension with Galois group G D E 0 E where E 0 is a finite group whose order is prime to p and E is a finite abelian group. Then the main conjecture is true for the extension F1 =F: t u The main idea in the proof of the above theorem is to use the images C of the complex C.F1 =F / for characters in the set R. / of irreducible characters of E 0 , where C WD .ƒO .E //S ˝L ƒ.G/S C.F1 =F /: Then @.Lp . // D ŒC 2 K0 .ƒO .E /; ƒO .E /S /: One then uses the isomorphism K0 .ƒ.G/; ƒ.G/S / !

˚

2R.E 0 /

K0 .ƒO .E /; ƒO .E /S /:

Thus if p divides the order of the cyclic group C , working with the Sylow subgroups of C , we may split the semidirect product and assume that our group G is an l-Qp -elementary group of the form G D H with H D C Ì P;

(16)

with C a cyclic group of p-power order and P a finite group of order prime to p. By a result of Oliver (see [O, Proposition 12.7]), if G 0 is a finite group such that the p-Sylow subgroup of G 0 has an abelian normal subgroup with cyclic quotient,

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then SK1 .Zp ŒG 0 / D 1: By passing to the inverse limit, one then concludes that SK1 .Zp ŒG/ D 1 for G as reduced above in (16). Definition 4.4. Let G be as in (16). Since P acts on C , the homology groups Hi .P; C / are defined. Put CN D H0 .P; C /; HN D CN P and GN D HN : Define P1 D Ker .P ! Aut.C //; H1 D C P1 ; and G1 D H1 : We have the natural maps norm W K1 .ƒ.G// ! K1 .ƒ.G1 //: N K1 .ƒ.G1 //; W K1 .ƒ.G// ! K1 .ƒ.G// N S / K1 .ƒ.G1 /S /; S W K1 .ƒ.G/S / ! K1 .ƒ.G/

(17)

where the map in the first component is the one induced by the natural surjection and the map in the second component is the norm map. This is used to define the following set which will play a key role in the description of K1 .ƒ.G//: Definition 4.5. Consider the maps can W K1 .ƒ.G1 // ! K1 .ƒ. CN //;

N ! K1 .ƒ. CN //: norm W K1 .ƒ.G//

Let ˆ (resp. ˆS ) be the group whose underlying set consists of all pairs .x0 ; x1 / 2 N K1 .ƒ.G1 // (resp. K1 .ƒ.G/ N S / K1 .ƒ.G1 /S /// such that norm.x0 / D K1 .ƒ.G/ can.x1 /, and x1 is fixed under the conjugation action by every element of P . Proposition 4.2. The map induces an isomorphism between K10 .ƒ.G// and ˆ. The image of S is contained in ˆS . In particular N K1 .ƒ.G1 /// D Im./: Im.S / \ .K1 .ƒ.G// Thus there is a commutative diagram

K1 .ƒ.G// ! ? ? y

N K1 .ƒ.G1 // ˆ K1 .ƒ.G// ? ? y

S N S / K1 .ƒ.G1 /S /: K1 .ƒ.G/S / ! ˆS K1 .ƒ..G//

Proof. Let us verify that the image of is contained in ˆ; the corresponding verification for S is similar. Let B be the set of right coset representatives of C in H . Then B is a basis of the ƒ.G1 /-module ƒ.G/. Let x 2 K1 .ƒ.G//: The image of x in K1 .ƒ.G1 // under the norm map is described as follows. Let xQ be a lift of x in ƒ.G/ : Multiplication on the right by xQ gives a ƒ.G1 /-linear map on ƒ.G/. Let A.B; x/ Q be the matrix of this map with respect to the basis B. Then the norm of x is the class of this matrix in K1 .ƒ.G1 //, and is independent of xQ and the basis B.


45

Let g 2 P . Then gBg 1 is also a ƒ.G1 /-basis of ƒ.G/, and we have gA.B; x/g Q 1 D A.gBg 1 ; g xg Q 1 /. Since g xg Q 1 is also a lift of x, the class 1 1 of A.gBg ; g xg Q / in K1 .ƒ.G1 // is the same as that of A.B; x/: Q Hence .x/ satisfies the second condition on the definition of ˆ, namely invariance under N conjugation. Choosing the same set B as a basis of the ƒ. CN /-module ƒ.G/; one sees that the following diagram commutes: norm

K1 .ƒ.G// ! ? ? cany

K1 .ƒ.G1 // ? ?can y

N ! K1 .ƒ. CN //: K1 .ƒ.G// norm

Hence .x/ also satisfies the first condition in the definition of ˆ: We show that surjects onto ˆ. Let .x0 ; x1 / 2 ˆ: Since K1 .ƒ.G// surjects onto N we can assume that x0 D 1. Then the first condition in the definition K1 .ƒ.G//, of ˆ implies that x1 2 J WD Ker .can W K1 .ƒ.G1 // ! K1 .ƒ. CN ///: By Oliver [O, Theorem 2.10] and Fukaya and Kato [FK, Proposition 1.5.3], the subgroup J is pro-p. Let J P be the subgroup of elements of J that are fixed pointwise under the conjugation action of P . Note that the above result tells us that J P is pro-p as well. Let x1 2 J P , and suppose that n is the order of P . Let z 2 J P be such that zn D x1 : Denote by z the image of z in K1 .ƒ.G//. Let .z/ D .z0 ; z1 /: By construction, we have z1 D x1 , and we have norm.z0 / D zn0 D 1 in K1 .ƒ. CN //: But z0 lies in a pro-p subgroup and hence z0 D 1, thereby proving the surjectivity of onto ˆ: We now show that is injective. For a finite group G 0 , let C onj.G 0 / denote the set of conjugacy classes of G 0 . Let n be a non-negative integer. Consider the map N p / Qp ŒG1 = p ˇ W Qp ŒConj.G= p / ! Qp ŒConj.G= n

n

n

where the map into the first component is induced by the natural surjection and the map into the second component is defined as follows. Let B be a set of left coset n n representatives of C in G= p and let g 2 G= p : Then the map is the Qp -linear map induced by g 7! † fx 1 gx W x 1 gx 2 G1 = p g: n

x2B

To see this injectivity, we need to use some representation theory. Recall that H is isomorphic to C Ì P with C a cyclic group of order a power of p which is prime to l and P a finite l-group. If CO denotes the set of irreducible characters of C , then P acts on CO by .g: /.h/ D .ghg 1 /; for g 2 P; 2 CO and h 2 C: Under this action by P , the stabilizer of is P1 if ¤ 1: Indeed, let 2 CO and ¤ 1:

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R. Sujatha

If g: D ; then .ghg 1 / D .h/ for all h 2 C and hence the commutator ghg 1 h1 2 Ker for all h 2 C: In particular, if c is a generator of C , then as

¤ 1, Ker is a proper subgroup of C . Thus if gcg1 D c a , then a 1 mod p. But the order of P is prime to p and hence ap1 1 mod j C j : Hence a 1 modulo j C j and g 2 P1 : For a system of representatives fi g of orbits of this action of P on CO , let Pi denote the stabilizer of i . By Serre [Se, Proposition 25] (see also [CR, Theorem 11.11]), we deduce that every irreducible representation of H is obtained by inducing an irreducible representation of C Ì Pi , which in turn is obtained by taking an irreducible representation of Pi , inflating it to C Ì Pi , and twisting it by i : We have just argued that Pi is either P (when the character is trivial) or is equal to P1 . Noting that no nontrivial character of C is fixed by this action of P , the latter case reduces to inflating from P , which does give a smaller set of representations than those inflated from CN P , we obtain the following lemma. Lemma 4.3. Any irreducible representation of H is obtained either by inflating an irreducible representation of CN P or by inducing an irreducible representation of H1 . t u For a finite group K, let R.K/ be the ring of virtual characters of K. The map N p ˝Z R.G1 = pn / ! Q N p ˝Z R.G= pn / N p ˝Z R.G= N pn / Q Q . ; / 7! Inf. / C Ind./; where Inf denotes the inflating homomorphism and Ind is the induced representation, is surjective by Lemma 4.3. Clearly, IdQN p ˝ˇ is dual to the above map described in Lemma 4.3. It follows therefore that ˇ is injective. It induces an injection n N pn lim Qp ŒG1 = pn : ˇ W lim Qp ŒConj.G= p ! lim Qp ŒConj.G= n n n

A final crucial ingredient in proving the injectivity of is the logarithm map on K1 (see [SV]) defined by Oliver and Taylor. For a finite group G 0 , there is a group homomorphism log W K1 .Zp ŒG 0 / ! Qp ŒConj.G/ with kernel the torsion subgroup. By a result of Wall [W, Proposition 6.5], the torsion subgroup of K1 .Zp ŒG 0 / is p1 G ab SK1 .Zp ŒG/: Thus the exact sequence n

n

log

n

1 ! p1 .G= p /ab ! K1 .Zp ŒG= p / ! Qp ŒConj.G= p / induces a homomorphism log

n

1 ! p1 G ab ! K1 .ƒ.G// ! lim Qp ŒConj.G= p /: n


47

There is a commutative diagram (see proof of [O, Theorem 6.8]) K1 .ƒ.G// ? ?

n

!

lim Qp ŒConj.G= p / n ? ?ˇ y

log

y log

N K1 .ƒ.G1 // ! lim Qp ŒConj.G= N p / lim Qp ŒG1 = p : K1 .ƒ.G// n

n

n

n

We complete the proof of the injectivity of : Let x 2 Ker./: As ˇ is injective, x 2 Ker .log/; and hence x 2 p1 G ab : But under the natural surjection N K1 .ƒ.G// ! K1 .ƒ.G//; which is in fact the first component of the map , p1 G ab maps identically on N ab (note that G ab D GN ab ). Hence x D 1 and the proposition is proved. u p1 .G/ t Theorem 4.5. Let F1 =F be an admissible extension satisfying the hypothesis D 0 and let G D Gal.F1 =F /: Assume that G is l Qp -elementary for some prime l ¤ p: Then the main conjecture for F1 =F is valid. Moreover, K1 .ƒ.G// injects into K1 .ƒ.G/S /: Proof. Let f 2 K1 .ƒ.G/S / such that @.f / D ŒC.F1 =F /; and let S .f / D .f0 ; f1 / 2 ˆS : Recall that G1 ' C with C a cyclic group of p-power order, and G1 0 GN ' CN P; with P a finite group of order prime to p. Let L D F1 ; and F1 =F 0 N By the earlier reductions, be the Galois extension of F such that Gal.F1 =F / D G: 0 the main conjecture is valid for subextensions F1 =L and the extension F1 =F . Let 1 and 0 be the corresponding p-adic zeta functions satisfying the main conjecture. By the uniqueness and the interpolation property, one verifies that .0 ; 1 / 2 ˆS : Let ui D i fi1 (for i D 0; 1). Then .u0 ; u1 / 2 ˆ; and we pick u 2 K1 .ƒ.G// be the unique element such that .u/ D .u0 ; u1 /: Now D uf is the required padic -function. That @.f / D ŒC.F1 /=F is clear, and we check the interpolation property. By Lemma 4.3, is either obtained by inflating a representation N of GN or by inducing a representation 1 of G1 : Hence for any positive integer r divisible by ŒF1 .p / W F1 , we have for i D 0 or 1, and 0 D F , 1 D L ; . Fr / D i .i ir / D L† .i ; 1 r/ D L† .; 1 r/: To see the assertion about injectivity, note that if x 2 K1 .ƒ.G// maps to 1 in K1 .ƒ.G/S /; then .x/ D 1 and hence x D 1. Hence the theorem is proved. u t The final reduction is to pro-p extensions. We may assume by the results e e proved so far that G= p is p-Qp -elementary, say G= p D Cn Ì Hp where Hp is a p-group and Cn is a cyclic group of order n prime to p. The Galois group Gal.Qp .n /=Qp / acts on the set of one dimensional characters COn of Cn , and we let C be the orbit set of COn under this action. Then the ring Zp ŒCn decomposes as

48

R. Sujatha

Zp ŒCn ' ˚ O ;

where varies over a representative character from each orbit, and O is the ring of integers in a finite extension L of Qp : The action of Hp on Cn induces an action on each O through Hp ! Gal.L =Qp /: Set UHp to be the inverse image of Hp in G which for ease of notation, we denote by U: Then G D Cn Ì U . Let t denote the composition t W U ! Hp ! Gal.L =Qp /; and let U denote Ker .t /: Proposition 4.3. We have an isomorphism K10 .ƒ.G// ! ˚ K10 .ƒO .Ui //U=U :

nCe

Proof. For every n 0; let Gn D G= p : Define U ;n to be the kernel of the composite map eCn ! Hp ! Gal.L =Qp /: U= p Then by Oliver [O, Theorem 12.3], we have K10 .Zp ŒGn / ' ˚K10 .O ŒU ;n /U=U ;n :

The required result follows on passing to the inverse limit.

t u

Let us recall the set-up; we have now reduced to the case where G D Gal.F1 =F / is a p-adic Lie extension such that it satisfies the D 0 hypothesis and such that G is a p-Qp -elementary subgroup which contains a fixed open subgroup e p of which is central in G. Let be a character as in the paragraph above. Theorem 4.6. Let F1 =F be a p-adic Lie extension satisfying D 0 hypothesis such that G is a p-Qp -elementary group G D Cn Ì U . With notation as above, C ÌU Ker

assume that the main conjecture is true for F1 =F1n for each 2 COn and 0 0 that K1 .ƒO .U // injects into K1 .ƒO .U /S /: Then the main conjecture is true for F1 =F and K10 .ƒ.G// injects into K10 .ƒ.G/S /: Proof. Assume that the main conjecture is valid for each of the extensions C ÌU Ker

F1 =F1n : Let 2 K10 .ƒ..Cn =.Ker / U /S // be the p-adic zeta function in the main conjecture. Let Lp . / be the image of under the natural map K10 .ƒ..Cn =Ker / U /S / ! K10 .ƒO .U /S /;


49

induced by the natural surjection Zp ŒCn =Ker ! O : Consider the following commutative diagram K10 .ƒ.G// ? ? oy

K10 .ƒ.G/S / ? ? y

!

@

!

K0 .ƒ.G/; ƒ.G/S / ? ? y

@

˚K10 .ƒO .U //U=U ! ˚K10 .ƒO .U /S /U=U ! K0 .ƒO .U /; ƒO .U /S /:

We have Lp . / 2 K10 .ƒO .U //U=U . Suppose f 2 K10 .ƒ.G/S / is such that @.f / D ŒC.F1 =F /: Denote the image of f under the middle vertical arrow by .f /; and put u D Lp . /f 1 : Then .u / 2 ˚ K10 .ƒO .U //U=U ; and there is

a unique u in K1 .ƒ.G// mapping to .u / under the left vertical isomorphism. Then WD uf is the sought after p-adic L-function. We only check the interpolation property. Let be an irreducible Artin representation of G. Then by Serre [Se, Proposition 25], there is a and an Artin representation of U such that D IndG Cn ÌU . /: Thus for any positive integer r divisible by ŒF1 .p / W F1 ; we get . Fr / D . r / D L† .; 1 r/; C ÌU

where is the p-adic cyclotomic character of F1n : The uniqueness of the padic zeta function and the statement about K10 -groups follows from an easy diagram chase and the theorem is proved. t u We have thus reduced the proof of the main conjecture to the case where the Galois group G is of the form G D Gp , where is a finite cyclic group of order prime to p and Gp is a pro-p compact p-adic Lie group of dimension 1. This completes the reduction we set out to make and the algebraic aspects of the study of K1 .ƒ.G// for such groups G will be discussed in the article [SV].

References [B] H. Bass, Algebraic K-Theory (W. A. Benjamin, Inc., New York/Amsterdam, 1968) [Bj] J.-E. Björk, Filtered Noetherian rings, in Noetherian Rings and Their Applications (Oberwolfach, 1983), 5997. Mathematical Surveys and Monographs, vol. 24 (American Mathematical Society, Providence, 1987) [Bu] D. Burns, On main conjectures in noncommutative Iwasawa theory and related conjectures, preprint (2010) http://www.mth.kcl.ac.uk/staff/dj burns/newdbpublist.html [CK] J. Coates, D. Kim, Introduction to the work of M. Kakde on the non-commutative main conjectures for totally real fields, This volume

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[CFKSV] J. Coates, T. Fukaya, K. Kato, R. Sujatha, O. Venjakob, The GL2 main conjecture for elliptic curves without complex multiplication. Publ. Math. IHES 101, 163–208 (2005) [CR] C. Curtis, I. Reiner, Methods of Representation Theory with applications to Finite Groups and Orders (Wiley, New York, 1981) [FK] T. Fukaya, K. Kato, A formulation of conjectures on p-adic zeta functions in noncommutative Iwasawa theory, in Proceedings of the St. Petersburg Mathematical Society, vol. 12, ed. by N.N. Uraltseva (American Mathematical Society, Providence, 2006), pp. 1–85 [K] M. Kakde, The Main Conjecture of Iwasawa theory for totally real fields, preprint (2011) [Mi] J. Milnor, Introduction to Algebraic K-Theory. Annals of Mathematics Studies (Princeton University Press, Princeton, 1971) [O] R. Oliver, Whitehead Groups of Finite Groups. London Mathematical Society Lecture Note Series, vol. 132 (Cambridge University Press, Cambridge/New York, 1988) [RW] J. Ritter, A. Weiss, On the ‘main conjecture’ of equivariant Iwasawa theory. J. Am. Math. Soc. 24, 1015–1050 (2011) [SV] P. Schneider, O. Venjakob, K1 of certain Iwasawa algebras, after Kakde, This volume [Se] J.-P. Serre, Linear Representation of Finite groups. Graduate Texts in Mathematics (Springer, New York, 1977) [Sw] R. G. Swan, Algebraic K-Theory. Lecture Notes in Mathematics, vol. 76 (Springer, Berlin/New York, 1968) [V] O. Venjakob, On the structure theory of the Iwasawa algebra of a p-adic Lie group. J. Eur. Math. Soc. 4, 271–311 (2002) [V1] O. Venjakob, On the work of Ritter and Weiss in comparison with Kakde’s approach, This volume [W] C.T.C. Wall, Norms of units in group rings. Proc. Lond. Math. Soc. 29, 593–632 (1974) [We] C. Weibel, The K-book: an introduction to algebraic K-theory, Available at http://www. math.rutgers.edu/ weibel/Kbook.html [Wi] A. Wiles, The Iwasawa conjecture for totally real fields. Ann. Math. 131, 493–540 (1990)

The Group Logarithm Past and Present Ted Chinburg , Georgios Pappas , and M.J. Taylor

Abstract We give an account of the group logarithm and discuss a number of its applications. We begin by rehearsing some basic theory, and then go on to consider its use both in the theory of class groups of group rings and in the Galois structure of rings of algebraic integers. We conclude with a brief insight into its recent use in the construction of an adelic second Chern class for a non-commutative Riemann Roch theorem. The use of the group logarithm in non-commutative Iwasawa theory is discussed elsewherein this volume. Keywords Algebraic K-theory • Group rings • Determinants and logarithms • Galois modules

partially supported by NSF grant DMS-1100355 partially supported by NSF grant DMS-1102208

T. Chinburg () Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, 19104-6395, Philadelphia, PA, USA e-mail: [email protected] G. Pappas Department of Mathematics, Michigan State University, 619 Red Cedar Road, 48824, East Lansing, MI, USA e-mail: [email protected] M.J. Taylor Merton College, Oxford University, Merton Street, OX1 4JD, Oxford, UK e-mail: [email protected] J. Coates et al. (eds.), Noncommutative Iwasawa Main Conjectures over Totally Real Fields, Springer Proceedings in Mathematics & Statistics 29, DOI 10.1007/978-3-642-32199-3 3, © Springer-Verlag Berlin Heidelberg 2013

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1 Introduction The group logarithm is a fundamental tool in the study of K1 of the group ring RŒG of a finite group G and a p-adically complete ring of coefficients R. The goal of this article is to present both an accessible account of the basic theory of the group logarithm and also to give an overview of some of its most important applications. Group logarithmic techniques were first used in the study of class groups of group rings, and in particular they were used to construct easily calculable homomorphic images of various subgroups of such class groups. This produces completely new insights on class groups of group rings (see [T1, T2, T3]). There is a very readable account of this approach in Sect. 54 of [CR2]. R. Oliver used his own version of the group logarithm to study the subgroup SK1 .RŒG/ of p-adic group rings when R is the valuation ring of a finite extension of a p-adic field (see [O1, O2, O3, O4, O5]). There is a good account of the state of knowledge of such SK1 groups prior to Oliver’s work in Sect. 45 of [CR2]: from this one can see the major progress that his work represented. The application of the group logarithm which was the driver for the third named author’s work was the Fröhlich conjecture. This relates the Galois module structure of rings of algebraic integers to the constants occurring in the functional equations of certain Artin L-functions. The resolution of the Fröhlich conjecture involved two distinct uses of the group logarithm: the fixed point theorem for determinants (see Theorem 5.2 in Sect. 5), and the proof that certain Galois Gauss sums can be written as determinants of group ring elements (see Sect. 6.2). In recent times the group logarithm has been extended to various Iwasawa algebras, with a view to tackling the non-commutative Iwasawa conjecture. (See for instance the initial work of Kato in [K] and then [Ka1, Ka2] and [RW1].) This conjecture has now been proved (under various hypotheses) by Ritter and Weiss [RW2], and by Kakde [Ka3]. The main topic of this volume is to present a detailed account of Kakde’s proof. The required extensions of the group logarithm are fully described elsewhere in this volume; see in particular the article of P. Schneider and O. Venjakob [SV]. A further recent use of the group logarithm has been in the construction of a second Chern class for suitable equivariant vector bundles on arithmetic surfaces, which can then be used to give an equivariant Riemann-Roch theorem (see [CPT3]). This requires the use of a group logarithm for group rings with higher dimensional coefficient rings. This development comes from [CPT1, CPT2] and involves a substantial generalization of the original work of both Oliver and the third named author. As explained above, it is our intention to provide a speedy and accessible introduction to the group logarithm, together with an overview of some of its many applications. For this reason we shall often focus on the case when G is a p-group. This is in many ways the heart of the matter; one can then use induction techniques to obtain results for arbitrary finite groups. This article is therefore structured as follows. In Sect. 2 we define our notation and recall some standard results in

The Group Logarithm Past and Present

53

K-theory. In Sect. 3 we describe the group logarithm for p-groups over a fairly general class of coefficient rings and in Sect. 4 we use Oliver’s work to describe SK1 for such group rings. In Sect. 5 we give a very brief insight into the role of induction techniques: this Section is only a very cursory account: the reader is referred to [CPT1, CPT2] and above all Chap. 12 in Oliver’s book [O5] for more details. The article concludes with an account of some of the most important applications of the group logarithm. We begin by describing some algebraic results. These include the proof of Ullom’s Conjecture on Swan groups and Oliver’s description of class groups of group rings for p-groups. We finally go on to consider two major applications of a more arithmetic flavor: the proof of the Fröhlich conjecture and the proof of an adelic Riemann-Roch theorem for an equivariant second Chern class. As indicated previously, a number of articles in this volume deal with applications to non-commutative Iwasawa theory, and so we avoid unnecessary duplication and shall not go into this application here.

2 Preliminaries 2.1 Notation Throughout this article R denotes an integral domain with field of fractions N which has characteristic zero, and we write N c for a chosen algebraic closure of N . We impose two further conditions: Standing Hypotheses: (i) R is p-adically complete, so that the natural map R ! lim R=p n R is an n isomorphism. (ii) R supports a lift of Frobenius; that is to say there is a ring endomorphism F of R with the property that F .r/ r p mod pR. Examples of such rings R are: the valuation ring of a non-ramified extension of the p-adic field Qp ; the p-adic completion of the polynomial ring over Zp in indeterminates T1 ; : : : ; Tn Zp hhT1 ; : : : ; Tn ii D lim Zp ŒT1 ; : : : ; Tn =.p n /I n

the p-adic completion of the ring of Laurent series over Zp in indeterminates T1 ; : : : ; Tn Zp ffT1 ; : : : ; Tn gg D lim Zp ..T1 ; : : : ; Tn //=.p n /: n

p

In each of the latter two examples we may take F .Ti / D Ti .

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2.2 Some Elementary K -Theory Let S denote a unitary ring (which is not necessarily commutative). In the usual way we define GL.S / to be the direct limit of the groups GLn .S /; in particular we view the group of units S D GL1 .S / as a subgroup of GL.S / by embedding S into GLn .S / as the diagonal matrices which are 1 in the non-leading diagonal positions. En .S / denotes the group of n n elementary matrices with coefficients in S , and we denote the direct limit of the En .S / by E.S /. Then, in the usual way, the group K1 .S / is defined as: GL.S / : K1 .S / D E.S / For each n > 0 we write Mn .S / for the ring of n n matrices with entries in S . If S is commutative, then the determinant induces a homomorphism det W K1 .Mn .S // Š K1 .S / ! S I

(1)

furthermore this map is an isomorphism if S is also a local ring. See [R] for more details. For a finite group G we form the group ring RŒG. The group algebra N c ŒG admits a natural algebra decomposition Y N c ŒG D Mn .N c / (2)

where the product extends over the irreducible N c -characters of G. Hence, by the above, we know that we have Y K1 .N c ŒG/ D .N c / : (3)

We shall write Det for the composite map K1 .RŒG/ ! K1 .N c ŒG/ D

Y

.N c / :

(4)

The group SK1 .RŒG/ is defined to be the kernel of Det, so that we have the exact sequence 1 ! SK1 .RŒG/ ! K1 .RŒG/ ! Det.K1 .RŒG// ! 1:

(5)

This exact sequence is fundamental for our understanding of K1 .RŒG/: note that Det.K1 .RŒG// and SK1 .RŒG/ have rather different properties but the group logarithm gives great insight into both of these terms. We shall write Det.RŒG / for the image of Det on the image of RŒG in K1 .RŒG/. In the case when G is a p-group it follows easily by left and right multiplication by elementary matrices that: Proposition 2.1. Det.RŒG / D Det.K1 .RŒG//.


55

Proof. See Theorem 1.2 in [CPT1].

t u

Remark 2.1. The corresponding result for an arbitrary finite group will be given in Theorem 5.1; but, note that for this more general result, we require that R is also Noetherian and normal. If the group G is abelian, then we have a diagram Q c N c ŒG D

.N / " " Det

RŒG ! Det.RŒG / where the vertical maps are inclusions; hence, in this case, Det is injective on RŒG and so we have shown: Lemma 2.1. If G is abelian, then Det W RŒG ! Det.RŒG / is an isomorphism.

3 The Group Logarithm Throughout this section G is a finite p-group. We write IG .R/ for the augmentation ideal ker." W RŒG ! R/, where the augmentation map " is the R-linear ring homomorphism with ".g/ D 1 for all g 2 G; and we let JG .R/ denote the Jacobson radical of RŒG. When R is clear from the context we shall write IG for IG .R/ and JG for JG .R/. Note that IG .Fp ŒG/ is the Jacobson radical of the Artinian ring Fp ŒG it therefore follows that we can find a positive integer m such that IG .Fp ŒG/m D 0. Note also that JG .R/ IG .R/ D RIG .Zp / mod pRŒGI and so

IG .R/m pRŒG; JG .R/m pRŒG: Therefore, for x 2 JG .R/, the logarithmic series log.1 x/ D

X n>0

xn n

converges to an element of N ŒG. We now use passage to conjugacy classes, together with a pth power map, to produce a logarithm which converges to integral values. Let Conj.G/ denote the set of conjugacy classes of G and let D G W N ŒG ! N ŒConj.G/ be the N -linear map which maps each group element to its conjugacy class. We define the p-th power operation D F to be the F -semilinear endomorphism of RŒConj.G/ which maps the conjugacy class of each group element to the conjugacy class of its p th power; so that for r 2 R, g 2 G .r.g// D F .r/.g p /:

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We then define .x n / X ı .x n / C n>0 n>0 n np X .x n / X .x np / X ı .x n / C D p−n>0 n>0 n>0 n np np n n np X X .x / ı .x / .x / C : D p−n>0 n>0 n np

L.1 x/ D

X

(6) (7)

Theorem 3.1. We have (a) L.1 C JG .R// RŒConj.G/, and (b) L.1 C IG .R// .IG .R//. Although the group logarithm is defined on all of 1 C JG .R/, we shall often restrict it to 1 C IG .R/ as this often gives a cleaner statement of results (see for example Theorem 3.5). Proof (Sketch). The proof of the theorem comes from a straightforward noncommutative version of the binomial theorem (see 6.2 in [T1] and Lemma 5.3 in Chap. 2 in [F1] for details). We need to prove that for each n > 0 ı .x n / .x np / 2 npRŒConj.G/: The key-idea can be seen from the case when the group G is abelian: indeed, in that case P is the identity and we know by the usual binomial theorem that for any xD rg g 2 RŒG xp D .

X

rg g/p

X

rgp g p

X

F .rg /g p D .x/ mod pRŒG

and from this it follows easily that for any n > 0 x np .x/n D .x n / mod npRŒG:

t u

We relate the logarithm to Det via the following lemma which depends on the well known fact that “the logarithm of the determinant is trace of the logarithm”: Lemma 3.1. Let be an N c -valued virtual character of G. Recall that the p-th Adams operation on is defined by the formula p .g/ D .g p / and p is also a virtual character of G. We extend F to RŒG by coefficientwise action; that is to say X X F. rg g/ D F .rg /g: Then, for any x 2 IG .R/ we have p .L.1 x// D log.Detp .1 x// log.Det

p

.1 F .x///:

(8)


57

We see that L is a homomorphism since the characters of G span the dual of N ŒConj.G/. Since Det .1 x/ D 1 implies that Det .1 F .x// D 1, it follows from Theorem 3.1 that ker.Det/ ker.L/; and so there is a unique map W Det.1 C IG / ! .IG / which makes the following diagram commutative: 1 C IG

Det

DD DD D!

L

/

Det.1 C IG / uuu zuuu

(9)

.IG / As .IG / N ŒConj.G/, clearly the elements of Det.1 C IG / of finite order lie in ker . Using the fact that p is nilpotent on virtual characters of degree zero, it easy to establish the reverse inclusion: namely, that ker is contained in the subgroup elements of Det.1 CIG / of finite order. A result of C. T. C. Wall ([W1,W2]) extends readily (see Theorems 3.14 and 3.15 in [CPT1]) to give: Theorem 3.2. We have ker D Det.G/. In Lemma 2.1 we have seen that Det is an isomorphism on RŒG ab ; to get a complete picture of Det on all of RŒG we now consider the ideal: AG .R/ D ker.RŒG ! RŒG ab /: Wall’s theorem tells us immediately that is injective on Det.1 C AG /. The fundamental result in the theory of the group logarithm is the fact that the image of on Det.1 C AG / is a natural R-lattice (see Theorem 3.5 in [CPT1]): Theorem 3.3. We have the equality in N ŒConj.G/

.Det.1 C AG // D .AG /: We can then piece this result together the abelian result to get the important exact sequences: 1 ! .AG / ! Det.RŒG / ! RŒG ab ! 1 1 ! .AG / ! Det.1 C IG .R// ! 1 C IG ab .R/ ! 1:

(10)

Application. As a first illustration of the power of the group logarithm, we use this exact sequence to prove a fixed point theorem for determinants for group rings over p-groups. Let S be a finite Galois extension of R with D Gal.S=R/, and suppose that S also satisfies the Standing Hypotheses in Sect. 2. For the sake of simplicity we shall assume that the lift of Frobenius FS of S is compatible with the lift of Frobenius FR of R in the sense that FS jR D FR : We let act on Det.S ŒG / via coefficients; so that for ı 2 ; x 2 S ŒG we have defn

Det.x/ı D Det.x ı /:

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Clearly Det.RŒG / Det.S ŒG / . Using the group logarithm we can now show: Theorem 3.4. Det.RŒG / D Det.S ŒG / : Remark 3.1. In fact, with a bit more effort, we can avoid the condition that the lifts of Frobenius are compatible; see Sect. 4.1 of [CPT1] for details. Proof. First note that .AG .S // D S ˝R .AG .R//;

.S ŒG ab / D RŒG ab :

By the naturality of the group logarithm (which depends on a compatible choice of lifts of Frobenius, so that LS ŒG jRŒG D LRŒG ), from (10) we have the commutative diagram: ! Det.RŒG / ! RŒG ab ! 1 .AG .R// # # #D 1 ! .S ˝R .AG .R/// ! Det.S ŒG / ! .S ŒG ab / ! 1

1!

with the bottom row exact since S ˝R .AG .R// is RŒ-projective, since S is RŒ-projective. Since the left and right vertical arrows are equalities the central vertical arrow is also an equality. t u We now complete our understanding of the group logarithm by describing the cokernel of the map W Det.1 C IG / ! .IG /. The evaluation of this cokernel was first carried out by R. Oliver in [O2] for the case when R is a non-ramified p-adic ring of integers: Theorem 3.5. Let iG denote the inclusion .Det.1 C IG // ,! .IG /; then there is a natural isomorphism: coker.iG / Š G ab ˝Z

R : .1 F /R

Proof (This is a sketch. For a full proof see 3.d in [CPT1]). First we observe that by Theorem 3.3, coker.iG / Š coker.iG ab / and so without loss of generality we may suppose G to be abelian. We then have the standard differential map for augmented commutative rings: 1 d W RŒG ! ˝RŒG=R D

IG .R/ D G ˝Z R; x 7! x ".x/ mod IG .R/2 : IG .R/2

1 via R. Recall that we also have the R-semilinear Thus F acts naturally on ˝RŒG=R endomorphism of RŒG with the property that for r 2 R, g 2 G, .rg/ D F .r/g p ; and moreover, since G is now abelian, is now a ring homomorphism.


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By using lifts to characteristic zero, it is possible to give a sense to the following equality for each x 2 IG 1 F .dx/ D d .x/I (11) p see 3.d in [CPT1] for details. Then we have the equalities: d.L.1 x// D d.log.1 x// d.log

1 .1 x// p

(12)

D dx C F dx D .F 1/dx and so d. .1 C IG // .1 F /.R ˝Z G/ D .1 F /R ˝Z G. It is then relatively straightforward to show the equality d. .1 C IG // D .1 F /R ˝Z G and to see that d induces an isomorphism R 1 C IG Š ˝Z G:

.1 C IG / .F 1/R

t u

4 SK1 .RŒG / In this section we suppose that G is a p-group and that, in addition to the Standing Hypotheses, pR is now a prime ideal of R; although it is not essential, we shall also suppose, for the sake of presentational simplicity, that SK1 .R/ D f1g. For the treatment of the more general case when SK1 .R/ is non-trivial see [CPT2]. We now follow R. Oliver and use the more general group logarithm to describe SK1 .RŒG/. Recall that in general we have the long exact sequence of K-theory ! K2 .R/ ! K1 .RŒG; IG / ! K1 .RŒG/ ! K1 .R/ ! We write K10 .RŒG; IG / for the image of K1 .RŒG; IG / in K1 .RŒG/. We denote the Whitehead group K10 .RŒG; IG /=Im.G/ by Wh0G .R/, or Wh0G when R is clear from the context. Since SK1 .R/ D f1g and since SK1 .RŒG/ \ ŒG D f1g, we note that SK1 .RŒG/ is naturally a subgroup of Wh0G .R/. Using Theorems 3.2 and 3.5 we get the exact sequence: 1!

R Det.1 C IG / ! .IG / ! G ab ˝Z !1 Det.G/ .1 F /R

and using the exact sequence (5), together with our hypothesis that SK1 .R/ D f1g, we get the further exact sequence 1 ! SK1 .RŒG/ ! Wh0G .R/ !

Det.1 C IG / ! 1: Det.G/

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These two exact sequences may be spliced to together to give the four term exact sequence 1 ! SK1 .RŒG/ ! Wh0G .R/ ! .IG / ! G ab ˝Z

R ! 1: .1 F /R

(13)

Suppose we have an extension of finite p-groups i

˛

e ! G ! 1I 1!H !G

(14)

then we have a commutative diagram: 1 ! SK1 .RŒH / ! Wh0H ! H .IH / Û H ab ˝Z !

R .1F /R

! 1

# # #i ˝1 # i 0 R ab e e Û e 1 ! SK1 .RŒG/ ! Whe G .Ie G / ! G ˝Z .1F /R ! 1 G ! # ˛ # # # ˛ ab ˝ 1 0 R ab ! 1: 1 ! SK1 .RŒG/ ! WhG ! G .IG / ! G ˝Z .1F /R ab

(15)

We then use the dotted arrows, which are just set-theoretic lifts,—in a similar manner to the construction to the boundary map in the snake lemma, but done twice—to produce a map: W ker.iGab / ˝Z

R ! coker.˛ /: .1 F /R

Next we consider the following two subgroups of G: e G e H0 D H \ ŒG; e H1 D hh 2 H j h D Œe g 1;e g 2 for e g 1;e g 2 2 GiI e commutators which lie in H , so that that is to say H1 is generated by G obviously ŒH; H H1 H0 and ker.iGab / D H0 =ŒH; H . We write e ˛ for the composite map e ˛ W H0 ˝Z

R R ! ker.iGab / ˝Z : .1 F /R .1 F /R

As in Proposition 16 in [O2] (see also 3.3 in [CPT2]) we know that Proposition 4.1. The map e ˛ induces an isomorphism a W

R H0 ! coker.˛ /: ˝Z H1 .1 F /R


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Suppose now that the extension (14) is a central extension. From [O2] and [CPT2] we have: Theorem 4.1. (a) (Hopf) There is a natural map ı˛ W H2 .G; Z/ ! H and Im.ı˛ / D H0 ; (b) Define H2ab .G; Z/ to be the subgroup of H2 .G; Z/ generated by the images of the H2 .A; Z/ for all abelian subgroups of G; then ı˛ .H2ab .G; Z// D H1 . We define H 2 .G; Z/ D

H2 .G; Z/ : H2ab .G; Z/

Piecing the above together we get maps ˛1

SK1 .RŒG/ coker.˛ / Š

ı˛ R R H0 : ˝Z H 2 .G; Z/ ˝Z H1 .1 F /R .1 F /R

By considering families of central extensions, as above, Oliver shows that it is possible to find situations where ı˛ is injective (and hence an isomorphism); this then yields a surjective map (which by naturality is independent of the particular such central extension and map ˛ used) W SK1 .RŒG/ ! H 2 .G; Z/ ˝Z

R : .1 F /R

The following result is due to Oliver for p-adic rings of integers (see [O2, O3]). Note that in this case and with our assumptions, the ring of p-adic integers R will be the ring of integers of an non-ramified extension of Qp and so R=.1 F /R identifies with Zp via the trace; thus, in this case, the second tensor factor in the above expression may be omitted. This result was generalized to the rings of integers of the completed maximal non-ramified extension of Qp in [IV]; and in [CPT2] it is proved for general R in which the ideal pR is prime and which satisfy the Standing Hypotheses of this article. Theorem 4.2. The map is an isomorphism which is natural in G and R.

5 Induction Methods We have seen that the group logarithm is a powerful tool for the study of K1 .RŒG/ when G is a p-group. There are a number of induction techniques which allow us to use results for p-groups to obtain results for arbitrary finite groups. In this section we assume that, in addition to satisfying the Standing Hypotheses, R is also a Noetherian and normal ring. Then by Theorem 1.2 in [CPT1] we have: Theorem 5.1. For an arbitrary finite group G and for R as above Det.K1 .RŒG// D Det.RŒG /:

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In Chap. 11 of [O5], Oliver presents an extremely effective set of induction techniques using Green rings and Mackey functors—building on work of Dress in [D]. For instance, under suitable circumstances, this allows one to describe a result for an arbitrary finite group G as a direct limit over the elementary p-groups of G: In [Sn], V. Snaith exhibits a technique of explicit Brauer induction. This has the remarkable property of being natural with respect to Adams operations. Since the p-th Adams operation plays a key-role in the properties the group logarithm (see for instance Lemma 3.1), Snaith’s method is particularly well adapted to inductive techniques involving the group logarithm. In a number of situations, however, it suffices to use the simpler technique of character action due to S. Ullom. We briefly illustrate this method by sketching a proof of the following general version of the fixed point Theorem 3.4 (see Theorem 6.4 in [CPT1] for a full proof): Theorem 5.2. Let G be an arbitrary finite group and we keep the notation of Theorem 3.4. Suppose that, in addition to the Standing Hypotheses, S is both Noetherian and normal. Then Det.K1 .S ŒG// D Det.K1 .RŒG//:

5.1 Character Action on K1 We let G0 .Zp ŒG/ denote the Grothendieck group of finitely generated Zp ŒGpr modules and let G0 .Zp ŒG/ denote the Grothendieck group of finitely generated Zp ŒG-modules which are projective over Zp . From 38.42 and 39.9 in [CR2] we have: Proposition 5.1. We have pr

G0 .Zp ŒG/ Š G0 .Zp ŒG/ Š G0 .Qp ŒG/ with the first isomorphism induced by the forgetful map and the second isomorphism induced by the extension of scalars map ˝Zp Qp . Proposition 5.2. Let R be an integral domain containing Zp . Then the ring pr G0 .Zp ŒG/ and hence, by the previous proposition, G0 .Qp ŒG/, acts naturally on K1 .RŒG/ via the rule that for an Zp ŒG-lattice L and for an element of K1 .RŒG/ represented by a pair .P; ˛/ (where P is a projective RŒG-module and ˛ is an RŒG-automorphism of P ), then L .P; ˛/ D ..L ˝Zp P /; .1 ˝Zp ˛//: The functor G 7! K1 .RŒG/ is a Frobenius module for the Frobenius functor pr G 7! G0 .Zp ŒG/ (see page 4 in [CR2] and also [L]). Then G 7! SK1 .RŒG/ is a Frobenius submodule of K1 .RŒG/ and therefore the action of G0 .Qp ŒG/ on K1 .RŒG/ induces an action on Det.GL.RŒG//. (See Ullom’s Theorem in 2.1 of [T1], and see also below for his explicit description of this action).


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Proof. From Ex 39.5 in [CR2] we know that G 7! K1 .Zp ŒG/ is a Frobenius modpr pr ule for G 7! G0 .Zp ŒG/. Moreover, the extension of scalars map G0 .Zp ŒG/ ! pr pr G0 .RŒG/ gives a morphism of Frobenius modules over G 7! G0 .Zp ŒG/ by 38.11 loc. cit.. (Here the RŒG-modules are R-projective.) This then establishes the first part of the proposition. Let N denote the field of fractions of R: Because the base pr change G0 .RŒG/ ! G0 .N c ŒG/ gives similarly a morphism of Frobenius modules pr over G 7! G0 .RŒG/, it follows that the kernel SK1 .RŒG/ D ker.K1 .RŒG/ ! K1 .N c ŒG// pr

is also a Frobenius module over G 7! G0 .RŒG/.

t u

Next we recall Ullom’s explicit formula for the action of the character ring G0 .Qp ŒG/ on Det.GL.RŒG//. We view G0 .Qp ŒG/ as the ring of virtual characters of finitely generated Qp ŒGmodules and we let 2 K0 .Qcp ŒG/ (which we identify with the ring of virtual Qcp -valued characters of G), and let r 2 GL.RŒG/ correspond to the pair .P; ˛/ under the two descriptions of K1 .RŒG/. Then Ullom has shown that the induced Frobenius action of G0 .Qp ŒG/ is given explicitly by .P; ˛/ D . 7! Det.r/.// D . 7! Det.r/.// where denotes the contragredient of the character . In particular we have G H .IndG H / Det.r/ D IndH . .ResG .Det.r////

(16)

which is one of the standard identities for Frobenius modules. The proofs of these standard facts are exactly the same as the proofs in Chap. 2, pages 21–25 in [T1].

5.2 Brauer Induction For a given positive integer m, m denotes the group of roots of of unity of order m in Qcp . We then identify Gal.Qp .m /=Qp / as a subgroup of .Z=mZ/ in the usual way. Let l be a prime number. Recall that a semi-direct product of a cyclic group C (of order m, say, which is coprime to l) by an l-group L, C Ì L, is called Qp -l-elementary (see page 112 in [S]) if for given 2 L there exists t D t./ 2 Gal.Qp .m /=Qp / .Z=mZ/ such that for all c 2 C c1 D c t :

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Theorem 5.3. For a given finite group G, there exists an integer m coprime to l such that X mG0 .Qp ŒG/

IndG J .G0 .Qp ŒJ // J

where J ranges over the Qp -l-elementary subgroups of G. t u

Proof. See Theorem 28 in [S].

5.3 Qp -p-Elementary Groups Suppose now that G is a Qp -p-elementary group, so that G may be written as a semi-direct product C Ì P , where C is a cyclic normal subgroup of order s, which is prime to p, and where P is p-group. We decompose the commutative group ring Zp ŒC according as the divisors m of s Zp ŒC D

Y

Zp Œm;

(17)

m

where Zp Œm is the semi-local ring Zp Œm D ZŒm ˝Z Zp and where m is a primitive mth root of unity. We set Rm D R ˝Zp Zp Œm, although Rm is not an integral domain, by Sect. 6 in [CPT1] we know that Rm decomposes as a product of integral domains each of which satisfies the Standing Hypotheses. For each m the conjugation action of P on C induces a homomorphism ˛m W P ! Authm i and we let Hm D ker.˛m / and Am D Im.˛m /. When m has been fixed we shall feel free to drop the index m. Tensoring the decomposition (17) with ˝Zp ŒC RŒG gives a decomposition of R-algebras Y RŒG D RŒm ı P (18) m

where RŒG ı P denotes the natural twisted group ring. We shall study the group Det.RŒG / by studying the various subgroups Det.RŒm ı P /. Note that the twisted group ring RŒm ı P contains the standard group ring RŒmŒHm . We therefore have the inclusion map i W RŒmŒHm ! RŒm ı P . We also have a restriction map defined by choosing a transversal fai g of P =Hm , which induces a restriction homomorphism res W RŒm ı P ! GLjAj .RŒmŒHm / and hence using Theorem 5.1 above: rm W Det.RŒm ı P / ! Det.GLjAj .RŒmŒHm // Š Det.RŒmŒHm /:


65

Since for 2 P , x 2 RŒm ı P , we know that Det. x 1 / D Det.x/, we see that rm W Det.RŒm ı P / ! .Det.RŒmŒHm //Am : Here Am acts via ˛m on RŒm and by conjugation on Hm . One can then relatively easily show (see Theorem 6.2 in [CPT1]): Theorem 5.4. The map rm gives an isomorphism

rm W Det.RŒm ı P / ! Det.RŒmŒHm /Am : Finally we show: Theorem 5.5. Let S and R be as stated in Theorem 3.4 and let G be a finite Qp -p-elementary group, then Det.S ŒG / D Det.RŒG /: Proof. By (18) and Theorem 5.4 above Det.S ŒG / D ˚m Det.S Œm ı P / D ˚m .Det.S ŒmŒHm /Am / : As the actions of and Am commute on S ŒmŒHm D S ˝R RŒmŒHm ( acts via the first term and Am acts via the second term), we see that Det.S ŒG / D ˚m .Det.S ŒmŒHm / /Am and so by Theorem 3.4 we have equalities Det.S ŒG / D ˚m Det.RŒmŒHm /Am D ˚m Det.RŒm ı P / D Det.RŒG /:

t u

Corollary 5.1. For an arbitrary finite group G the quotient group Det.S ŒG / =Det.RŒG / has order prime to p. Proof. By Theorem 5.3 we can find an integer m which is coprime to p, Qp -pelementary subgroups Hi of G, integers ni ; and i 2 K0 .Qp ŒHi /, such that m 1G D

X i

ni IndG Hi .i /:

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Thus, given Det.x/ 2 Det.S ŒG / , by the Frobenius module structure of Det.S ŒG / over K0 .Qp ŒG/ (see (16)) Det.x/m D

Y

Hi ni IndG Hi .i ResG .Det.x/// :

i

However, by Theorem 5.5, we know that i i ResH G .Det.x// 2 Det.S ŒHi / D Det.RŒHi /:

Hence Det.x/m 2 Det.RŒG /.

t u

In order to prove Theorem 5.2 in full generality one now has to consider the Qp l-elementary subgroups of G for the primes l ¤ p and then use Brauer induction to show that for each such prime l the quotient Det.S ŒG / =Det.RŒG / has order prime to l. In fact the argument for the primes different from p is much more straightforward; the reader is referred to 6.b in [CPT1] for the details.

6 Applications of the Group Logarithm The group logarithm has numerous applications in both algebra and arithmetic. Although here we concentrate principally on the arithmetic, we begin our account by describing some algebraic applications. Historically these came first, and also this will help us develop the notation that we will need for the subsequent arithmetic applications. The intention is to give only a very brief overview of each chosen topic, but to highlight the role played by the group logarithm.

6.1 Some Algebraic Applications We denote by K0 .ZŒG/ the Grothendieck group of finitely generated projective ZŒG-modules. There is a natural notion of ZŒG-rank which induces a surjective homomorphism rk W K0 .ZŒG/ ! Z; the kernel of the rank map is called the class group of the group ring ZŒG and is denoted C l.ZŒG/. In [F2] Fröhlich gave a description of C l.ZŒG/ in terms of adelic K1 -groups and described a natural isomorphism Q0

Cl.ZŒG/ Š

K1 .Qp ŒG/ Q I K1 .QŒG/[ p K1 .Zp ŒG/[ p

(19)


67

Q [ here K1 .QŒG/ Q denotes [the image of K1 .QŒG/ Qin K1 .Qp ŒG/ Q (product over all primes p); K .Z ŒG/ denotes the image of K .Z ŒG/ in K1 .Qp ŒG/; and 1 p 1 p Q0 K .Q ŒG/ denotes the restricted product, which is the subgroup of elements in Q 1 p K1 .Qp ŒG/ with almost all but a finite number of the entries lying in K1 .Zp ŒG/[ . Let M denote a maximal order in QŒG which contains ZŒG; then Cl.M/ may be defined similarly and extension of scalars yields a surjective homomorphism Cl.ZŒG/ ! Cl.M/; the kernel is called the kernel group and is denoted D.ZŒG/; it is independent of the particular choice of the maximal order M. Let Mp D M ˝Z Zp : The kernel group can also be written in terms of K1 -groups: Q

D.ZŒG/ Š

[ p K1 .Mp / Q : K1 .QŒG/[ p K1 .Zp ŒG/[

(20)

However, it is often more practicalQ to use the identification, deriving from the isomorphism Det W K1 .Qp ŒG/ ! Qp . / (where the product extends over the irreducible Qcp -characters of G). We let Qc denote a chosen algebraic closure of Q and set ˝ D Gal.Qc =Q/; we let Qcp denote a chosen algebraic closure of Qp and set ˝p D Gal.Qcp =Qp /; and once and for all we fix an embedding Qc ! Qcp : We let RG , resp. RG;p , denote the ring of virtual Qc -characters, resp. of virtual Qcp characters of G. The embedding Qc ,! Qcp yields an isomorphism RG Š RG;p and we have natural identifications (using the Hasse-Schilling norm theorem) K1 .Zp ŒG/[ D Det.Zp ŒG /;

K1 .M/[ D Det.M / D HomC ˝ .RG ; OQc /;

K1 .Mp /[ D Det.M p / D Hom˝p .RG ; OQcp /I here HomC ˝ .RG ; OQc / denotes the subgroup of Hom˝ .RG ; OQc / of homomorphisms whose values on the symplectic characters of G are totally positive, in the sense that they are real and positive at each Archimedean place of Qc . In particular, if G is a p-group, then for l ¤ p, we know that Zl ŒG is a maximal Zl -order and so Det.M l / D Det.Zl ŒG /; and hence in this case we have the very useful and practical isomorphism:

D.ZŒG/ Š

Hom˝p .RG;p ; OQ c / p HomC ˝ .RG ; OQc /Det.Zp ŒG /

:

(21)

In particular note that this isomorphism shows how, for calculations, we can represent a class by a character function. The exact sequence resulting from extension of scalars 1 ! D.ZŒG/ ! Cl.ZŒG/ ! Cl.M/ ! 1

(22)

is crucial in understanding the class group Cl.ZŒG/. On the one hand the image Cl.M/ can readily be described as a product of various kinds of class groups of rings

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of integers (see Theorem 3.7 in [F1] for details); and, on the other hand, the group logarithm is an excellent tool in the study of the kernel group D.ZŒG/. To illustrate this latter point we now suppose that G is a p -group. Early on in the study of such kernel groups Ullom showed (see for instance 50.19 in [CR2] or Theorem 3.1 in [U]): Theorem 6.1. If G is a non-trivial p-group, then D.ZŒG/ is a finite p-group whose exponent divides jGj p 1 if p > 2 and whose exponent divides jGj =4 if p D 2 and if jGj > 2. This leads naturally to the question of when this bound for the exponent of D.ZŒG/ is attained. To begin to answer this question we introduce the notion of Swan modules: Definition 6.1. Let m be an integer which is coprime to p: The Swan module .m; ˙/ is the left P (and in fact two-sided) ZŒG-ideal generated by m and the trace element ˙ D g2G g. Then .m; ˙/ is a projective ZŒG-module whose class in C l.ZŒG/ lies in D.ZŒG/. These classes form a subgroup of D.ZŒG/ which is denoted T .ZŒG/ and is called the Swan subgroup of D.ZŒG/. (See [U].) The class of .m; ˙/ is denoted by Œm; ˙. Recall G is a p-group; then this class is represented under the isomorphism (22) by the character function fm : fm . / D m. ;"/

(23)

where " denotes the trivial character of G and . ; "/ denotes the standard character inner product of with " (see 3.1 in [T1]). Definition 6.2. Recall that for n 4 the semi-dihedral group of order 2n has presentation n1 n2 h; j 2 D 1 D 2 ; D 1C2 i: A 2-group is called exceptional if it is dihedral, quaternion or semi-dihedral. Theorem 6.2 (Kulakoff, Alperin, Feit, Thompson). If G is a p-group, then the cardinality of the set fg 2 G j g p D 1g is divisible by p 2 unless G is cyclic or p D 2 and G is exceptional. S. Ullom conjectured the following result (see [U]) which was proved by the third named author in [T2]. Recall that G is a p-group. Theorem 6.3. If p > 2 and G is not cyclic, then T .ZŒG/ is a cyclic group of order jGj p 1 . If p D 2 and G is neither cyclic nor exceptional, then T .ZŒG/ is a cyclic group of order jGj =4. Remark 6.1. If G is cyclic, then by Corollary 1.3 in Chap. 3 of [T1] we know that T .ZŒG/ D f1g. If G is an exceptional 2-group, then from Theorems 2.5, 2.6 loc.cit. we know that T .ZŒG/ D D.ZŒG/ and has order two if G is semi-dihedral or quaternion, and, if G is dihedral, then T .ZŒG/ D D.ZŒG/ D f1g.


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Proof (Sketch). Let denote the regular character of G. We define a map h W Hom˝p .RG;p ; OQ c / ! pZp mod p jGj Zp p by the rule that for 2 Hom˝p .RG;p ; OQ c /, p

h./ D log .p

p

/ mod p jGj Zp :

We note that, because p p is a rational valued character, for in the group p Hom˝ .RG ; OQ / D ˙1 and so log .p p / D 0. c /, it follows that .p For D Det.z/ 2 Det.Zp ŒG / by Lemma 3.1, h./ D log .p

p

/ mod p jGj Zp

D .pL.Det.z/// mod p jGj Zp and by Theorem 3.1, .pL.Det.z/// 2 .pZp ŒConjŒG/ p jGj Zp . It therefore follows that h induces a map (denoted by the same symbol) h W D.ZŒG/ ! pZp mod p jGj Zp : We now evaluate h on the class Œ1 C p; ˙ by evaluating h.f1Cp /: h.f1Cp / D log.f1Cp .p D log..1 C p/.p and the character inner product .p by

p

p

// mod jGj Zp

p ;"/

/ mod jGj Zp

; "/ D .p; "/.

p

; "/ can be evaluated

1 X p.g/".g 1 / D p g jGj X 1 X p 1 X .g/".g 1 / D .g p / D 1: . p ; "/ D g g gjg p D1 jGj jGj .p; "/ D

The result then follows at once from the Kulakoff-Alperin-Feit-Thompson theorem. t u The involution g 7! g 1 of the group G induces involutions denoted c 7! c on both C l.ZŒG/ and D.ZŒG/. We define D.ZŒG/C D fd 2 D.ZŒG/ j d D d g D.ZŒG/ D fd 2 D.ZŒG/ j d D d g:

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For the sake of simplicity we shall suppose for the remainder of this subsection that p > 2. We then have the direct decomposition D.ZŒG/ D D.ZŒG/C ˚ D.ZŒG/ : In [O6] Oliver constructs a map W D.ZŒG/C ! K0 .QŒG/T .ZŒG/ where K0 .QŒG/ acts on classes as in 5.1. Let A.QŒG/ denote the Artin ideal of K0 .QŒG/ obtained by inducing up the rational characters K0 .QŒC / of cyclic subgroups C of G. Oliver then shows further that K0 .QŒG/T .ZŒG/ Š

K0 .QŒG/ A.QŒG/

and that if furthermore p is a regular prime number, then in fact is an isomorphism (see Theorem 3.4 in [O6]). This then underlines the central role played by Swan modules in the study of class groups of group rings. To conclude this subsection we give a very brief indication of some of Oliver’s further work in this area. In Theorem 3.8 of [O6] he gives a closed formula for the cardinality of the group D.ZŒG/ (see also [OT] in this regard). In the important paper [O7], when G is cyclic, he describes the group structure of D.ZŒG/ for p > 2 and of the full group D.ZŒG/ when p D 2.

6.2 The Fröhlich Conjecture For a number field K we let OK denote the ring of algebraic integers of K. We consider a finite Galois extension N=K of number fields and write G D Gal.N=K/. The group ring ZŒG acts on ON in the natural way, and by Noether’s theorem ON is a projective ZŒG-module if, and only if, N=K is at most tamely ramified. We suppose henceforth that the extension N=K is tame, and so we may consider the virtual class ŒON ŒOK ŒG 2 C l.ZŒG/: From the previous subsection we recall the K-theoretic description Q0

C l.ZŒG/ D

l K1 .Ql ŒG/ Q K1 .QŒG/[ l Det.Zl ŒG /

(24)

and for each prime l we have K1 .Ql ŒG/ D Det.Ql ŒG / D

Y

Ql . / :

(25)


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Let .s; / denote the Artin L-function associated to the character of G (see [M] for details); then .s; / satisfies a functional equation .s; / D W . /.1 s; / where the constant W . /, which is called the Artin root number, lies on the unit circle in the complex plane. Note that if is a real valued character of G, then W . /2 D 1 and so W . / D ˙1: We then define the class ŒWN=K 2 C l.ZŒG/ to be the class, represented under the isomorphism (24) and the description of K1 .Qp ŒG/ in (25), by the element Y0 Y Y w ;l 2 K1 .Ql ŒG/ l

l

where for irreducible Q -characters of G W . /; if is symplectic and p j jGj w ;l D 1; otherwise. c

Based on a considerable number of known examples, Fröhlich conjectured the following result, which was proved by the third author (see [T4]): Theorem 6.4. There is an equality in C l.ZŒG/ ŒON ŒOK ŒG D ŒWN=K : We now very briefly indicate the role of the group logarithm in the proof of this result. For illustrative purposes we henceforth suppose G to be a p-group. In Theorem 6 of [F1] Fröhlich gave an explicit representative Y l

l 2

Y0 l

K1 .Ql ŒG/

for the class ŒON ŒOK ŒG ŒWN=K with l D 1 if l ¤ p: Step 1. A key-ingredient in the definition of the l is the Galois Gauss sum N=K which is related to the Artin root number W . / by the formula (see page 126 in [F1]) W1 . /W . /N f. /1=2 D N=K . /: As in the corollary to Theorem 18 in [F1], we know that the N=K . / can be written naturally as a product Y N=K . / D l . /: l

and almost all l . / D 1. Recall that we have fixed an embedding Qc ! Qcp . For x 2 Qc we write .x/p for the image of x in Qcp . By using Frohlich’s Galois action formula for Gauss sums (see Theorem 20B in [F1]), it is relatively straightforward to identify where for l ¤ p the element .l /p lies, and we get

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Y

.l . //p 2

Y

Qp .l /. / D K1 .Qp .l /ŒG/ D Det.Qp .l /ŒG /:

For those l ¤ 1, it is convenient to make an adjustment and work with the modified local Gauss sums l D l l for a root of unity valued character function l 2 HomC ˝ .RG ; / where the precise formula for l is given on page 151 of [F1]. (This change is very similar to the change from Deligne’s "-constants to "0 -constants in [De].) Moreover, if Mp;G denotes a maximal Zp Œl -order in Qp .l /ŒG, then, since l ¤ p, the l . / are all p-units, and it follows that .l /p 2 K1 .Mp;G /[ D Det.M p;G / : The key-point in the proof of Theorem 6.4 is that we can show that .l /p lies in the subgroup Det.Zp Œl ŒG / and for this we make crucial use the group logarithm L. (Note that this result would not be true for the unmodified Galois Gauss sum.) Since Det.Zp Œl ŒG / has finite index in Det.M p;G / and since the group logarithm takes values in the uniquely divisible group Qp .l /ŒConj.G/, extends naturally to a map e

W Det.M p;G / ! Qp .l /ŒConj.G/: We wish to show that e

..l /p / lies in the integral lattice Zp Œl ŒConj.G/. To see this we write X e

..p /l / D tc.g/ c.g/ 2 Qp .l /ŒConj.G/: c.g/2Conj.G/

Then, by Fourier inversion, we may calculate tc.g/ : tc.g/ D

1 X

.e

.p /l / .g 1 /

jc.g/j

summing over the irreducible characters of G. By the explicit formula (8) in Lemma 3.1 this may be written as tc.g/ D

1 X 1 Œlog.p /l . / log.pF /l .

p jc.g/j

p

/ .h1 /

where F denotes the Frobenius of Qp .l /=Qp . Using again the Galois action formula for Gauss sums loc. cit. we see that pF . / differs from p . / by a root of unity. So by the above this expression can be written as tc.g/ D

X 1 Œlog.p /l .p

p jc.g/j

p

/ .g 1 /:

This sum can then be readily evaluated using the Hasse-Davenport Gauss sum formulas, and we find that each tc.g/ 2 Zp Œl , as required. See Chap. IV, Sect. 5 in [F1] for details.


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Step 2. The above work essentially shows that we can find zl 2 Zp Œl ŒG so that Det.z/l 2 kere

. As G is a p-group, Zp Œl ŒG D 0 .1 C JG / where 0 is the group of roots of unity in Zp Œl of order to prime to p, and we extend

from Det.1 C JG / to 0 on Det.Zp Œl ŒG / by setting 0 to be 0 on Det.0 /. Then, with this convention, by Theorem 3.2 we now get that ker 0 D 0 Det.G/; however, by the reasoning after in Lemma 3.1, we see that kere

is the much larger group Det.M / consisting of all the elements in Det.M t or p;G p;G / of finite order. The next step in the proof consists of finding virtual characters i of G and ideals ai of Zp Œl ; i with the property that the following sequence is exact: 1 ! ker ! kere

!

Y i

Zp Œl ; i mod ai

where the right-hand arrow is induced by the product of evaluation of determinants on i followed by reduction mod ai . We then conclude the proof by showing that for each i Det.zl /. i /l . i / 1 mod ai : This then allows us to deduce that l 2 Det.Zp Œl ŒG /, as required. Step 3. Fröhlich’s description of the representative p allows us to write it as p D p

Y p¤l

.l . //p :

Here the term p is made up of the corestriction of non-abelian resolvents and the Gauss sum p . /. Using some elementary algebra and the Galois action formula for Gauss sums and resolvents it is relatively straightforward to show that p 2 Det.OT ŒG / for some tame extension T of Qp ; and the work in Steps 1 and 2 shows that Y .l . //p 2 Det.ON ŒG / p¤l

for some non-ramified extension N of Ql . Increasing T if necessary to a larger tame and Galois extension of Ql , we have shown that p 2 Det.OT ŒG / \ K1 .Qp ŒG/: If we set D Gal.T =Qp /, then this implies that p 2 Det.OT ŒG / and, by an easy extension of the fixed point result Theorem 5.5 (see for instance Chap. 8 in [T1]) we then have p 2 Det.OT ŒG / D Det.OT ŒG / D Det.Zp ŒG / and this then shows that the representative p lies in the denominator of the description of C l.ZŒG/ given in (24).

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In summary, note that the group logarithm has played two essentially different roles in the proof of the Fröhlich Conjecture: firstly, in showing that the adjusted Gauss sum is a group ring determinant; and secondly through the use the fixed point theorem for determinants.

6.3 Equivariant Second Chern Classes and Adelic Riemann-Roch Next we give a flavor of some new applications of the above results on Det and SK1 for the calculation of Euler characteristics of equivariant sheaves on arithmetic surfaces. These are then also, implicitly, applications of the group logarithm. Let G be a finite group and consider an irreducible, two dimensional regular scheme Y with structure map f W Y ! Spec.Z/ which is projective and flat. Let E denote an OY ŒG-vector bundle of rank n, i.e. a coherent sheaf of OY ŒG-modules on Y which is locally free of rank n. We can then form the (reduced) projective Euler characteristic of E, denoted .Y; E/, which lies in C l.ZŒG/. This class is constructed by first finding a perfect ZŒG-complex, which is quasi-isomorphic to a ˇ Cech complex which computes the cohomology of E. One then forms the alternating sum of the classes of the terms in the perfect complex; see [CE] for details. We now indicate briefly the role of group logarithmic techniques in forming a second Chern class and showing an adelic Riemann-Roch theorem, which can be used to calculate such projective Euler characteristics. We start by recalling some constructions due to Beilinson and Parshin. For i D 0, 1, 2, we let i denote a point of Y of codimension i . A Parshin pair is a pair of points .i ; j / with i < j and with j on the closure of i . A Parshin triple is a triple of points .0 ; 1 ; 2 / with 2 on the closure of 1 . (Since Y is irreducible there is only one generic point 0 .) We can then form the completed ring OO Y;i ; following Parshin and Beilinson for a Parshin pair .i ; j /, resp. a Parshin triple .0 ; 1 ; 2 /, we can form the multicompletion OO Y;i j , resp. OO Y;0 1 2 . We then form the products of (Quillen) K2 -groups K2 .AY;012 ŒG/ D K2 .AY;ij ŒG/ D

Y Y

K2 .OO Y;0 1 2 ŒG/; K2 .OO Y;i j ŒG/:

Here the first product extends over all Parshin triples .0 ; 1 ; 2 / on Y , and the second product extends over all Parshin pairs .i ; j / on Y for some fixed choice 0 i < j 2. For any triple .0 ; 1 ; 2 / and each choice of i , j with 0 i < j 2, there is a natural map OO Y;i j ! OO Y;0 1 2 and so we have maps K2 .AY;ij ŒG/ ! K2 .AY;012 ŒG/.


75

Following Parshin and Beilinson, we define various restricted adelic subgroups K20 .AY;012 ŒG/, K20 .AY;ij ŒG/ (see [CPT3] Sect. 2). Let us write K20 .AY;ij ŒG/[ for the image of K20 .AY;ij ŒG/ in K2 .AY;012 ŒG/. We now define the second equivariant (adelic) Chow group CH2A .Y ŒG/

K20 .AY;012 ŒG/ Q D

Q

0 [ 0 i;j 2 K2 .AY;ij ŒG/ : 0 [ 0 i;j 2 K2 .AY;ij ŒG/

(26)

This name is justified since a similar construction for surfaces over a field in the nonequivariant case G D f1g produces the classical Chow group of dimension 0 cycles; this follows from the Gersten resolution. There is a corresponding definition of a first equivariant adelic Chow group CH1A .Y ŒG/. In fact, we can interpret the class group C l.ZŒG/ as the first equivariant Chow group CH1A .Spec.Z/ŒG/ of Spec.Z/. Now let us assume in addition that the group ring is split, i.e. that QŒG Š ˚i Mni .Li / where Li are (commutative) fields. Suppose also that all the fibers of f W Y ! Spec.Z/ are reduced with smooth irreducible components and that the fibers over all the divisors of jGj are irreducible and smooth. Then in [CPT3] we describe a pushdown (Gysin) map f W CH2A .Y ŒG/ ! CH1A .Spec.Z/ŒG/ D C l.ZŒG/ obtained by using tame symbols. For a suitable equivariant vector OY ŒG-bundle E of rank n we now sketch the construction of a second Chern class c2 .E/ 2 CH2A .Y ŒG/. For each point i on Y we choose a basis fei g of the completed stalk E ˝OY OO Y;i over OO Y;i ŒG; we then have the adelic transition matrices i j 2 GLn .OO Y;i j ŒG/ defined by ei D i j ej . Definition 6.3. We say that E has elementary structure if the bases can be chosen so that all the i j considered in GL.OO Y;i j ŒG/ actually belong to the subgroup of elementary matrices E.OO Y;i j ŒG/. We suppose henceforth that E has elementary structure. The importance of elementary structure is that it allows us to form a second Chern class as follows: For each triple .0 ; 1 ; 2 / and 0 i < j 2 we have the corresponding Steinberg sequence (see [R]) associated to the ring OO Y;i j ŒG: 1 ! K2 .OO Y;i j ŒG/ ! S t.OO Y;i j ŒG/ ! E.OO Y;i j ŒG/ ! 1: We can now choose lifts e i j of i j in S t.OO Y;i j ŒG/ and we define the “2cocycle” 0 2 .e 1 2 /1 .e 0 1 /1 2 K2 .OO Y;0 1 2 ŒG/: z.0 ; 1 ; 2 / D e

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Q In fact one can show that these liftsQcan be chosen so that z.0 ; 1 ; 2 / lies in the restricted product K20 .AY;012 ŒG/ 0 i;j 2 K20 .AY;ij ŒG/[ . The second equivariant Chern class c2 .E/ 2 CH2A .Y ŒG/ is then defined to be the class represented in (26). Here of course, among other things, one needs to check that the class is independent of the bases fei g and of the lifts of the corresponding transition matrices used. The main result of [CPT3] then is: Theorem 6.5 (Non-commutative adelic Riemann-Roch). Under the above assumptions, there is an equality in C l.ZŒG/

.Y; E/ D f .c2 .E//: Note here that the assumption that E has an elementary structure implies that the appropriate generalization of the first Chern class of E is trivial; this then explains the shape of the identity above. Indeed, this agrees with the shape of the classical Riemann-Roch formula for vector bundles of virtual rank zero and trivial determinant. Let us admit however a limitation of our method: If E does not have an elementary structure, we cannot, in general, define a second adelic Chern class. Apparently, a general adelic Riemann-Roch theorem for arithmetic surfaces is currently not known even in the non-equivariant case; Deligne’s functorial RiemannRoch can be thought of as providing a result of similar flavor but this is not expressed via adeles. The group logarithm is vital in this work for dealing with elementary structures. Notice that given transition matrices fi j g as above, the results on Det obtained via the group logarithm can help determine when we have Det.i j / D 1; this then gives i j 2 SLn .OO Y;i j ŒG/, where SLn .OO Y;i j ŒG/ denotes the kernel of Det on GLn .OO Y;i j ŒG/. In order to determine whether i j 2 E.OO Y;i j ŒG/ we need to understand the quotient group SK1 .OO Y;i j ŒG/ D

SL.OO Y;i j ŒG/ : E.OO Y;i j ŒG/

Here we use our results on the description of SK1 .RŒG/ for more general p-adic rings R, which are also obtained via the group logarithm, to describe some of the groups SK1 .OO Y;i j ŒG/ and show that the bundles we use have elementary structure.

References [CE] T. Chinburg, B. Erez, Equivariant Euler-Poincaré characteristics and tameness. Journées Arithmétiques, 1991 (Geneva). Astérisque 209 13, 179–194 (1992) [CPT1] T. Chinburg, G. Pappas, M.J. Taylor, K1 of a p-adic group ring I. The determinantal image. J. Algebra 326, 74–112 (2011)


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[CPT2] T. Chinburg, G. Pappas, M.J. Taylor, K1 of a p-adic group ring II. The determinantal kernel SK1 , to appear [CPT3] T. Chinburg, G. Pappas, M.J. Taylor, Higher adeles and non-abelian Riemann-Roch, to appear, preprint, arXiv:1204.4520 [CR1] C. Curtis, I. Reiner, Methods of Representation Theory. Vol. I. With Applications to Finite Groups and Orders. Reprint of the 1981 original. Wiley Classics Library. A WileyInterscience Publication (Wiley, New York, 1990) [CR2] C. Curtis, I. Reiner, Methods of Representation Theory. Vol. II. With Applications to Finite Groups and Orders. Pure and Applied Mathematics (New York). A WileyInterscience Publication (Wiley, New York, 1987) [De] P. Deligne, Les constantes des e´ quations fonctionnelles des fonctions L. Modular functions of one variable, II (Proceedings of International Summer School, University of Antwerp, Antwerp, 1972). Lecture Notes in Mathematics, vol. 349 (Springer, Berlin, 1973), pp. 501–597 [D] A. Dress, Induction and structure theorems for orthogonal representations of finite groups. Ann. Math. 102(2), 291–325 (1975) [F1] A. Fröhlich, Galois Module Structure of Algebraic Integers. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3, vol. 1 (Springer, Berlin, 1983) [F2] A. Fröhlich, Arithmetic and Galois module structure for tame extensions. J. Reine Angew. Math. 286/287, 380–440 (1976) [IV] D. Izychev, O. Venjakob, Galois invariants of K1 -groups of Iwasawa algebras, Preprint, arXiv:1006.5357 [Ka1] M. Kakde, K1 of some Iwasawa algebras, Preprint, arXiv:1005.4152 [Ka2] M. Kakde, K1 of some non-commutative group rings, Preprint, arXiv:1003.3772 [Ka3] M. Kakde, The main conjecture of Iwasawa theory for totally real fields, Preprint, arXiv:1008.0142 [K] K. Kato, K1 of some non-commutative completed group rings. K-Theory 34(2) 99–140 (2005) [L] T.Y. Lam, Induction theorems for Grothendieck groups and Whitehead groups of finite ´ groups. Ann. Sci. Ecole Norm. Super. 1(4), 91–148 (1968) [M] J. Martinet, Character theory and Artin L-functions, in Algebraic Number Fields: LFunctions and Galois Properties (Proceedings of symposium at University of Durham, Durham, 1975) (Academic, London, 1977), pp. 1–87 [O1] R. Oliver, SK1 for finite groups rings. I. Invent. Math. 57(2), 83–204 (1980); correction ibid 64(1), 167–169 (1981) [O2] R. Oliver, SK1 for finite group rings. II. Math. Scand. 47(2), 195–231 (1980) [O3] R. Oliver, SK1 for finite group rings. III, in Algebraic K-Theory, Evanston 1980 (Proceedings of the Conference at Northwestern University, Evanston, 1980), Lecture Notes in Mathematics, vol. 854 (Springer, Berlin, 1981), pp. 299–337 [O4] R. Oliver, SK1 for finite group rings. IV. Proc. Lond. Math. Soc. (3) 46(1), 1–37 (1983) [O5] R. Oliver, Whitehead Groups of Finite Groups. London Mathematical Society Lecture Note Series, vol. 132 (Cambridge University Press, Cambridge, 1988) [O6] R. Oliver, Projective Class Groups of Integral Group Rings: A Survey. Orders and Their Applications (Oberwolfach, 1984). Lecture Notes in Mathematics, vol. 1142 (Springer, Berlin, 1985), pp. 211–232 [O7] R. Oliver, Class groups of cyclic p-groups. Mathematika 30(1), 26–57 (1983) [OT] R. Oliver, L.R. Taylor, Logarithmic descriptions of Whitehead groups and class groups for p-groups. Mem. Am. Math. Soc. 76(392), vi+97 (1988) [RW1] J. Ritter, A. Weiss, The integral logarithm in Iwasawa theory: an exercise. J. Théor. Nombres Bordeaux 22(1), 197–207 (2010) [RW2] J. Ritter, A. Weiss, On the “main conjecture” of equivariant Iwasawa theory. J. Am. Math. Soc. 24(4), 1015–1050 (2011) [R] J. Rosenberg, Algebraic K-Theory and Its Applications. Graduate Texts in Mathematics, vol. 147 (Springer, New York, 1994)

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K1 of Certain Iwasawa Algebras, After Kakde Peter Schneider and Otmar Venjakob

Abstract This paper contains a detailed exposition of the content of Sect. 5 in Kakde’s preprint “The main conjecture of Iwasawa theory for totally real fields”. We proceed in a slightly more axiomatic way to pin down the exact requirements on the p-adic Lie group under consideration. We also make use of our conceptual theory of the completed localization of an Iwasawa algebra as developed in Schneider and Venjakob (Am. J. Math. 132, 1–36, 2010). This simplifies some of the arguments. Otherwise, with the exception of the notation at certain places, we follow Kakde’s paper.

MSCs: 19B28, 11S23, 11R23 Keywords Completed group ring • Iwasawa algebra • algebraic K-group • skew Laurent series • integral group logarithm Let, for simplicity in this introduction, G be a one dimensional compact p-adic Lie group and .G / D lim Zp ŒG =N its Iwasawa algebra, where N runs through all open normal subgroups of G : The main purpose is to establish a description of the K-group K1 ..G // in terms of the groups of units .U / for a suitable system of abelian subquotients U of G . The strategy for achieving this relies on the following commutative diagram P. Schneider () Universität Münster, Mathematisches Institut, Einsteinstr. 62, 48291 Münster, Germany e-mail: [email protected]; http://www.uni-muenster.de/math/u/schneider/ O. Venjakob Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany e-mail: [email protected]; http://www.mathi.uni-heidelberg.de/venjakob/ J. Coates et al. (eds.), Noncommutative Iwasawa Main Conjectures over Totally Real Fields, Springer Proceedings in Mathematics & Statistics 29, DOI 10.1007/978-3-642-32199-3 4, © Springer-Verlag Berlin Heidelberg 2013

79

80

1

P. Schneider and O. Venjakob

/ .O/ G ab D

1

/ .O/ G ab

/ K10 ..G //

L

/ ˚

L

!

/

/ G ab

/ 1

D

Š

ˇ

ı

/ OŒŒConj.G /

!ıˇ 1

/ G ab

/ 1

which to explain in too much detail is not possible at this point. We only mention that OŒŒConj.G / is an additive version and K10 ..G // WD K1 ..G //=SK1..G // with SK1 ..G // being defined in (5) is a quotient of K1 ..G //, L is the integral logarithm of Oliver and Taylor [Oli, T, CPT], and Y Y .U ab / ˚

.U ab / and

are the description we want to achieve and its additive version, respectively, where U runs through a certain system of open subgroups of G : Their precise definition is technical and involves quite a bit of notation, which will be developed only in the course of this article. But for the convenience of the reader we give a collection in Sect. 7 at the very end of the article. The upper row is exact by work of Oliver (and Fukaya/Kato). In a first step (Sect. 3) the somewhat easier additive isomorphism ˇ will be established. Then it will be a major point to define the map L which makes the middle square commutative. Finally it will be shown that the lower row is exact as well. For both see Sect. 4. Once all of this is done it is a formal consequence that the map is an isomorphism. Some of this will be generalized to the completed localization B.G / of .G /, additively in Sect. 3 and multiplicatively in Sect. 6. The latter requires an extension of the integral logarithm to the B.G /-setting (Sect. 5).

1 Skew Laurent Series All Iwasawa algebras under consideration will have coefficients in the ring of integers O of a finite extension of Qp . We fix a prime element 2 O. Let G be any compact p-adic Lie group. We assume that G contains a closed normal subgroup H such that G =H Š Zp . Further assumptions are only made later when necessary. In order to give a description of .G / relative to .H / we choose a closed Š

subgroup G such that ! G =H and we pick a topological generator of (in particular, G D H Ì ). We then have the ring automorphism W .H / ! .H / a 7! a 1

K1 of Certain Iwasawa Algebras, After Kakde

81

as well as the left -derivation ı WD id on .H /. We recall that the latter means that ı.ab/ D ı.a/b C .a/ı.b/

for any a; b 2 .H /:

Proposition 1.1 (Venjakob). We have the isomorphism .H /ŒŒtI ; ı D f

X

Š

ai t i W ai 2 .H /g ! .G /

i 0

sending t to 1 between the .; ı/-skew power series ring over .H / and the Iwasawa algebra .G /; the multiplication on the left hand side is determined by the rule ta D .a/t C ı.a/ for any a 2 .H / and by continuity. According to [CFKSV] the set S WD S.G / WD ff 2 .G / W .G /=.G /f is finitely generated over .H /g is an Ore set in .G / consisting of regular elements. We then may form the localization A.G / WD .G /S as well as its Jac..H //-adic completion

2

B.G / WD .G /S : Theorem 1.1 ([SV1] Theorem 4.7 and Proposition 2.26 (i)). i. The isomorphism in Proposition 1.1 extends to an isomorphism between .H / tI ; ı WD f

X

ai t i W ai 2 .H /; lim ai D 0 in .H /g i !1

i 2Z

and B.G /. ii. B.G / is noetherian and pseudocompact with Jac.B.G // D f

X

ai t i 2 B.G / W ai 2 Jac..H //g :

i 2Z

iii. B.G / is flat over A.G / and hence over .G /. We point out that the commutation rules in the ring .H / tI ; ı are considerably more complicated. For example, one has at 1 D

X

t i ı i 1 .a/

i 1;

where the sum is over all ideals a of OFU which are prime to ˙U , the Artin symbol of a in U ab is denoted by ga and the absolute norm of the ideal a is denoted by N.a/. A well known theorem of [Kli62] and [Sie70] says that the function .ı .x/; s/ has analytic continuation to the whole complex plane except for a simple pole at s D 1, and that .ı .x/ ; 1 k/ is a nonzero rational number for every even positive integer k.

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M. Kakde

If $ is a locally constant function on U ab with values in a Q-vector space V , say for a large enough j $

X

$.x/ı .x/ :

x2U ab =Z

pj

Then the special value L˙U .$; 1 k/ can be canonically defined as L˙U .$; 1 k/ D

X x2U ab =Z

$.x/.ı .x/ ; 1 k/ 2 V:

(6)

pj

If $ is an Artin character of degree 1, then L˙U .$; 1 k/ is of course the value at 1 k of the complex L-function associated to $ with Euler factors at primes in ˙U removed. The p-adic cyclotomic character of F is denoted by i.e. it is a homomorphism W Gal.F =F / ! Z p defined by g D .g/ ; for all g 2 Gal.F =F / and all 2 p1 : If k is an integer divisible by ŒF .p / W F , then k factors through Gal.F1 =F /. It is for this reason that throughout the article we have this assumption on k. If $ is a locally constant Qp -valued function on U ab , then for any positive integer k divisible by ŒF .p / W F and any u 2 U ab , we define uU .$; 1 k/ D L˙U .$; 1 k/ .u/k L˙U .$u ; 1 k/;

(7)

where $u is a locally constant function defined by $u .g/ D $.ug/, for all g 2 U ab .

5 Approximation to p-Adic Zeta Functions We get a sequence of elements in certain group rings which essentially approximate the abelian p-adic zeta functions U . These group rings are obtained as follows. Recall that is the p-adic cyclotomic character of F . Let f be a positive integer such that p1 .Z/ D 1 C p f Zp . Definition 5.1. Let U V be in S.G; Z/ such that U is normal in V . Define the map V W Zp Œ U ab =Z p =.p f Cj / ! Zp Œ U ab =Z p =.p f Cj /; U;j j

j

Congruences Between Abelian p-Adic Zeta Functions

given by

X

x 7!

135

gxg 1 :

g2V =U N U

V V D im.U;j / and denote TU;jG simply by TU;j . Put TU;j V The following commutative diagram relates U;j and the map UV defined by Sect. 1 UV

.U ab /

/ .U ab /

pj

Zp Œ U =Z =.p ab

f Cj

/ Zp Œ U =Z pj =.p f Cj / ab

/ V U;j

We omit proof of the following simple lemma. Lemma 5.1. For any U 2 S.G; Z/, we have an isomorphism

.U ab / ! lim Zp Œ U ab =Z p =.p f Cj /: j

j 0

If U V in S.G; Z/ are such that U is normal in V , then the above isomorphism V TU;j . maps TUV isomorphically to lim j

Proposition 5.1. (Ritter-Weiss) For any j 0, any positive integer k divisible by ŒF .p / W F and any U 2 S.G; Z/ and any u 2 U , the natural surjection of .U ab / j onto Zp Œ U ab =Z p =.p f Cj /, maps .1 u/U 2 .U ab / to X x2U ab =Z p

uU .ı .x/ ; 1 k/ .x/k x

.mod p f Cj /:

j

In particular, we claim that the inverse limit is independent of the choice of k. Also note that since x is a coset of Z f Cj in U ab , the value .x/k is well defined only modulo p f Cj . Proof. Since U is a pseudomeasure, .1 u/U lies in .U ab / (see Theorem 4.2, [Coa11]). We prove the proposition in three steps: first we show that the sums form an inverse system. Then we show that the inverse limit is independent of the choice of k. Finally we show that it interpolates the same values as .1 u/U . Step1:

Let j 0 be an integer. Let W Zp Œ U ab =Z p

j C1

=.p f Cj C1 / ! Zp Œ U ab =Z p =.p f Cj /; j

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M. Kakde

denote the natural projection. Then

X

x2U ab =Z p

uU .ı .x/ ; 1 k/ .x/k x j C1

X

D

y2U ab =Z p

j

y2U ab =Z p

y2U ab =Z p

x2yZ p =Z p

j C1

uU .ı .x/ ; 1 k/ .mod p f Cj /

X

.y/k y

j

X

D

uU .ı .x/ ; 1 k/ .x/k .x/ .mod p f Cj /

X j

X

D

j

x2Z p =Z p

j C1

uU .ı .y/ ; 1 k/ .y/k y.mod p f Cj /: j

j C1

Here the second equality is because for any x 2 U ab =Z p we have .x/k .y/k .mod p f Cj / if .x/ D y. This shows that the sums form an inverse system. Step2: The inverse limit would be independent of the choice of k if we show that for any two positive integers k and k 0 divisible by ŒF .p / W F , we have X

uU .ı .x/ ; 1 k/ .x/k x ab

X

0

uU .ı .x/ ; 1 k 0 / .x/k x.mod p f Cj /: ab

x2 Uj

x2 Uj

Zp

Zp

Or equivalently that, 0

uU .ı .x/ ; 1 k/ .x/k uU .ı .x/ ; 1 k 0 / .x/k .mod p f Cj /;

(8)

for all x 2 U ab =Z p . Choose a locally constant function W U ab ! Z p such that ŒF .p /WF ŒF .p /WF .mod p f Cj /. Define the functions $k and $k 0 from U ab to Qp by j

$k D and $k 0 D

1 p f Cj 1 p f Cj

.x/k ı .x/ ;

0

.x/k ı .x/ :

Then Theorem 0.4 [Del80] implies that uU .$k ; 1 k/ uU .$k 0 ; 1 k 0 / 2 Zp : We now note that


137

p f Cj uU .$k ; 1 k/ D .x/k uU .ı .x/ ; 1 k/ .x/k uU .ı .x/ ; 1 k/.mod p f Cj /

and 0

0

p f Cj uU .$k0 ; 1 k 0 / D .x/k uU .ı .x/ ; 1 k 0 / .x/k uU .ı .x/ ; 1 k 0 /.mod p f Cj /:

Hence the congruence (8) holds. Step3: Let u D lim

j 0

uU .ı .x/ ; 1 k/ .x/k x.mod p f Cj / 2 .U ab /:

X x2U ab =Z p

j

Let $ be a locally constant function on U ab factoring through U ab =Z p for some non-negative integer j . Then for every i j X

j

uU .ı .x/ ; 1 k/$.x/

i x2U ab =Z p

D

X x2U ab =Z

D

pi

X

x2U ab =Z

i

X

x2U ab =Z p

i

1 x/

.u/k L˙U .ı .u

x2U ab =Z p

; 1 k/$.x/

i

X

L˙U .ı .x/ ; 1 k/$.x/

x2U ab =Z p

.x/

.u/k L˙U .ıu ; 1 k/$.x/ pi

X

L˙U .ı .x/ ; 1 k/$.x/

x2U ab =Z p

D

X

L˙U .ı .x/ ; 1 k/$.x/

.u/k L˙U .ı .x/ ; 1 k/$.ux/ i

D L˙U .$; 1 k/ .u/k L˙U .$u ; 1 k/ D uU .$; 1 k/:

Hence by the definition of u , for any i j , we have u . k $/

X x2U ab =Z

uU .ı .x/ ; 1 k/$.x/.mod p f Ci / pj

uU .$; 1 k/.mod p f Ci /: On the other hand, by definition of the p-adic zeta function or the construction of Deligne-Ribet (see discussion after Theorem 0.5 in [Del80]) we have .1 u/U . k $/ D uU .$; 1 k/: Hence .1 u/U D u because they interpolate the same values on all cyclotomic twists of locally constant functions. This finishes the proof. t u

138

M. Kakde

6 A Sufficient Condition to Prove the Basic Congruences j

Lemma 6.1. Let y be a coset of Z p in U ab . Then for any u 2 Z and for any g 2 G, we have uU .ı .y/ ; 1 k/ D ugUg1 .ı .gyg

1 /

; 1 k/:

Proof. By definition uU .ı .y/ ; 1 k/ D .ı .y/ ; 1 k/ k .u/.ıu.y/ ; 1 k/ and ugUg1 .ı .gyg

1 /

; 1 k/ D .ı .gyg

1 /

; 1 k/ k .u/.ıu.gyg

1/

; 1 k/:

Since 1 y/

ıu.y/ D ı .u

ıu.gyg

and

1 /

1 gyg 1 /

D ı .u

1 yg 1 /

D ı .gu

;

it is enough to prove that .ı .y/ ; 1 k/ D .ı .gyg

1 /

; 1 k/:

Note that for Re.s/ > 1 .ı .gyg

1 /

; s/ D

X ı .gyg1 / .ga / a

D

N.a/s

X ı .y/ .gag / a

N.ag /s

D .ı .y/ ; s/: 1

The meromorphic functions .ı .gyg / ; s/ and .ı .y/ ; s/ agree on the right half plane 1 and hence .ı .gyg / ; 1 k/ D .ı .y/ ; 1 k/, as required. t u Remark. In Proposition 5.1 we expressed p-adic zeta functions (multiplied by a simple factor) as an inverse limit of elements of group rings. A congruence between U ’s, for various U ’s, may be expressed as an appropriate congruences between elements of group rings appearing in Proposition 5.1. Congruences in Propositions 3.1–3.3 are rewritten in Propositions 6.1–6.3 as congruences between uU ’s. Proposition 6.2 is proven by Ritter and Weiss [Rit08] and subsequent propositions are mild generalisations of it. Notation: For the rest of the article we fix an element u in Z f1g.


139

Proposition 6.1. To prove the congruence (1) in Proposition 3.1 it is sufficient to j prove the following: for any integer j 1 and any coset y of Z p in U ab fixed by V p uU .ı .y/ ; 1 k/ uV .ı .y/ ı verVU ; 1 pk/.mod pZp / (9) for a positive integers k divisible by ŒF .p / W F . To prove congruence (2) in Proposition 3.1 it is sufficient to prove the following: for j any integer j 1 and any coset y of Z p in V p piV

iV

uV p .ı .y/ ; 1 k/ uV .ı .y/ ı 'V ; 1 pk/.mod p iV Zp /

(10)

for a positive integer k divisible by ŒF .p / W F . Proof. By Proposition 5.1 the image of .1up /U in Zp ŒU ab =Z p =.p f Cj 1 / is X p uU .ı .y/ ; 1 k/ .y/k y.mod p f Cj 1 /: (11) j

y2U ab =Z p

j

j 1

The image of .1 u/V in Zp Œ V ab =Z p =.p f Cj 1 / is X uV .ı .x/ ; 1 pk/ .x/pk x.mod p f Cj 1 /: x2V ab =Z p

j 1

Let V 0 be the kernel of the homomorphism verVU W V ab ! U ab . Then V 0 \ Z D f1g which implies that the map V ab =V 0 Z p

j 1

! U ab =Z p

j

induced by verVU is injective. Moreover k .V 0 / D f1g. Hence the image of verVU ..1 u/V / D .1 up /verVU .V / in Zp Œ U ab =Z p =.p f Cj 1 / is j

X x2V ab =V 0 Z p

uV .ı .x/ ; 1 pk/ .x/pk verVU .x/.mod p f Cj 1 /: j 1

This sum can be written as X uV .ı .y/ ı verVU ; 1 pk/ .y/k y.mod p f Cj 1 / y2U ab =Z p

(12)

j

because if y … Im.verVU /, then ı .y/ ı verVU 0 and if y D verVU .x/, then .y/k D .x/pk . Subtracting (12) from (11) gives

140

M. Kakde

p

uU .ı .y/ ; 1 k/ uV .ı .y/ ı verVU ; 1 pk/ .y/k y.mod p f Cj 1 /:

X y2U ab =Z p

j

If y is fixed by V then

p uU .ı .y/ ; 1

k/

uV .ı .y/

ı

verVU ; 1

pk/ .y/

k

(13) y

py 0.mod under Eq. (9). On the other hand if y is not fixed by V , then the full orbit of y under the action of V in the above sum is X p 1 1 .uU .ı .gyg / ; 1 k/ uV .ı .gyg / ı verVU ; 1 pk// .gyg1 /k gyg 1 V TU;j /

g2V =U

p

X D uU .ı .y/ ; 1 k/ uV .ı .y/ ı verVU ; 1 pk/ .y/k gyg 1 g2V =U V 2TU;j : V .mod p f Cj 1 /. By The equality is by Lemma 6.1. Hence the sum in (13) lies in TU;j Lemma 5.1 .1 up /.U verVU .V // 2 TUV . Note that .1 up / 2 .A.U ab / /V =U . V Hence .U verVU .V // 2 TU;S . This proves the first part. Proof of the second part 0 is similar. Note that if V 2 C.G; Z/ is such that V V 0 and ŒV 0 W V D iV , then V 0 =V p acts trivially on y. t u

Proposition 6.2. To prove congruence (3) in Proposition 3.2 it is sufficient to show j the following: for any j 0 and any coset y of Z p in U whose image in U=Z is a generator of U=Z dU

uU .ı .y/ ; 1 k/ 0.mod

j.WG U /y j Zp /; iU

(14)

for a positive integers k divisible by ŒF .p / W F . Recall that dU D ŒFU W FNG U and .WG U /y is the stabiliser of y in WG U . Proof. Let v D udU . Then by Lemma 5.1 the image of .1 v/U !Um ..1 v/U / j in Zp Œ U=Z p =.p f Cj / is X y2U=Z

vU .ı .y/ ; 1 k/ .y/k .y !Um .y/y/.mod p f Cj /:

(15)

pj

If the image of y in U=Z is not a generator of U=Z, then y !Um .y/y D 0. For y whose image in U=Z is a generator of U=Z, we look at the P WD WG U orbit of y in expression (15). It is X

vU .ı .gyg

1 /

; 1 k/ .gyg 1/k .gyg 1 !Um .gyg 1 /gyg 1 /.mod p f Cj /

g2P =Py

DvU .ı .y/ ; 1 k/ .y/k

X g2P =Py

.1 !Um .gyg 1 //gyg 1 :


141

This sum lies in pTU;j = iU under Eq. (14) and then the sum in expression (15) lies in pTU;j = iU . Then by Lemma 5.1 .1 v/.U !Uk .U // 2 pTU = iU . As .1 v/ 2 .A.U ab / /NG U=U congruence (3) holds. t u Proposition 6.3. To prove congruence (5) in Proposition (3.3) it is sufficient to j prove the following: for any integer j 0 and any coset y of Z p in U whose image in U=Z is a generator of U=Z X dU dU =p uU .ı .y/ ; 1 k/ uV .ı .y/ ı 'V ; 1 pk/.mod j.WG U /y jZp /; (16) V 2Pc .U /

for a positive integers k divisible by ŒF .p / W F . Proof. Let v D udU =p . By Lemma 5.1 the image of .1 vp /U in X

vU .ı .y/ ; 1 k/ .y/k y.mod p f Cj 1 /: p

y2U=Z

j

Zp ŒU=Z p .pf Cj 1 /

is (17)

pj j 1

Let V 2 Pc .U /. The image of .1 v/V in Zp Œ V =Z p =.p f Cj 1 / is X vV .ı .x/ ; 1 pk/ .x/pk x.mod p f Cj 1 /: x2V =Z p

j 1

Let V 0 be the kernel of the homomorphism 'V W V ! U . Then V 0 \ Z D f1g whence the map j 1 j ! U=Z p V =V 0 Z p induced by 'V is injective. Moreover, k .V 0 / D f1g. Hence the image of 'V ..1 v/V / D .1 vp /'V .V / in Zp Œ U=Z p =.p f Cj 1 / is X vV .ı .x/ ; 1 pk/ .x/pk 'V .x/.mod p f Cj 1 /: j

x2V =V 0 Z p

j 1

This sum can be written as X vV .ı .y/ ı 'V ; 1 pk/ .y/k y.mod p f Cj 1 / y2U=Z p

(18)

j

because if y … Im.'V /, then ı .y/ ı 'V 0 and if y D 'V .x/, then .y/k D .x/pk . Put X p xU .ı .y/ / WD vU .ı .y/ ; 1 k/ vV .ı .y/ ı 'V ; 1 pk/: V 2Pc .U /

Adding (18) over all V 2 Pc .U / and subtracting it from (17) gives

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M. Kakde

X y2U=Z

xU .ı .y/ / .y/k y.mod p f Cj 1 /

(19)

pj

For any 0 m p 1, consider the expression (19)—!Um(19). X xU .ı .y/ / .y/k .1 !Um .y//y.mod p f Cj 1 /

(20)

j y2U=Z p

If the image of y in U=Z is not a generator of U=Z, then y !Um .y/y D 0. If image of y in U=Z is a generator of U=Z, then we take the orbit of y in the sum in (20) under the action of P WD WG U . It is X 1 xU .ı .gyg / / .gyg 1 /k .1 !Um .gyg 1 //gyg 1 g2P =Py

D xU .ı .y/ / .y/k

X

.1 !Um .gyg 1 //gyg 1 :

g2P =Py

This sum lies in pTU;j .mod p f Cj 1 / under Eq. (16) and hence the sum in expression (20) lies in pTU;j .mod p f Cj 1 /. Therefore by Lemma (5.1) .1 vp /.xU !Um .xU // lies in pTU . As .1 vp / 2 .A.U ab / /NG U=U congruence (5) holds. t u Remark. The congruences in Propositions 6.1–6.3 are required for some positive integer k divisible by ŒF .p / W F . However, we prove them for all positive integers k divisible by ŒF .p / W F .

7 Hilbert Modular Forms In this section we briefly recall the basic notions in the theory of Hilbert modular forms. Let L be an arbitrary totally real number field of degree r over Q. We denote the Hilbert upper half plane of L by HL . The p-adic cyclotomic character of L is denoted by . Define OLC to be the totally positive elements in the ring of integers OL of L.

7.1 Definition Let f be an integral ideal of L. We put GLC 2 .L ˝ R/ for the group of all 2 2 matrices with totally positive determinant. For any even positive integer k, the group GLC 2 .L ˝ R/ acts on functions f W HL ! C by a C d ab /; f jk ./ D N .ad bc/k=2 N .c C d /k f . c d c C d


143

where N W L ˝ C ! C is the norm map. Set ab 00 .f/ D f 2 SL2 .L/ W a; d 2 1 C f; b 2 D1 ; c 2 fDg; c d where D is the different of L=Q. A Hilbert modular form f of weight k on 00 .f/ (or of weight k and level f) is a holomorphic function f W HL ! C (which we assume to be holomorphic at 1 if L D Q) satisfying f jk M D f

for all M 2 00 .f/:

The space of all Hilbert modular forms of weight k on 00 .f/ is denoted by Mk .00 .f/; C/. Since f is invariant under the translation 7! C b (for b 2 D1 ), we may expand f as a Fourier series to get the standard q-expansion X c.; f /qL ; f ./ D c.0; f / C

where runs through all totally positive elements in OL and qL D e 2 it rL=Q . / .

7.2 Restrictions Along Diagonal Let L0 be a totally real extension of L of degree r 0 . The inclusion of L in L0 induces HL0 and SL2 .L ˝ R/ ! SL2 .L0 ˝ R/. For a holomorphic function maps HL ! f W HL0 ! C, we define the “restriction along diagonal” RL0 =L f W HL ! C by RL0 =L f ./ D f . /: We then have

.RL0 =L f /jr 0 k M D RL0 =L .f jk M /;

for any M 2 SL2 .L ˝ R/. Let f be an integral ideal of L, then RL0 =L induces a map RL0 =L W Mk .00 .fOL0 /; C/ ! Mr 0 k .00 .f/; C/: If the standard q-expansion of f is c.0; f / C

X

c. ; f /qL 0 ;

2OLC0

then the standard q-expansion of RL0 =L f is X X c.0; f / C 2OLC

c. ; f / qL :

Wt rL0 =L . /D

Here OLC and OLC0 denotes totally positive elements of OL and OL0 respectively.

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M. Kakde

7.3 Cusps

1

Let AL be the ring of finite adeles of L and 00.f/ be the adelisation of 00 .f/ i.e. closure of 00 .f/ in SL2 .AL /. Then by strong approximation

1

SL2 .AL / D 00 .f/ SL2 .L/:

1

Any M 2 SL2 .AL / can be written as M1 M2 with M1 2 00 .f/ and M2 2 SL2 .L/. We define f jk M to be f jk M2 . Any ˛ in A L determines a cusp. We let ˛0 : f j˛ D f jk 0 ˛ 1 The q-expansion of f at the cusp determined by ˛ is defined to the standard q-expansion of f j˛ . We write it as X c.; ˛; f /qL ; c.0; ˛; f / C

where the sum is restricted to all totally positive elements of L which lie in the square of the ideal “generated” by ˛. Lemma 7.1. Let f be an integral ideal in L. Let f 2 Mk .00 .fOL0 /; C/. Then the constant term of the q-expansion of RL0 =L f at the cusp determined by ˛ 2 A L is equal to the constant term of the q-expansion of f at the cusp determined by ˛ 2 A L0 i.e. c.0; ˛; RL0 =L f / D c.0; ˛ ; f /: Proof. The q-expansion of f at the cusp determined by ˛ is the standard qexpansion of f j˛ . Similarly, the q-expansion of RL0 =L f at the cusp determined by ˛ is the standard q-expansion of .RL0 =L f /j˛ . But .RL0 =L f /j˛ D RL0 =L .f j˛ /. t u

7.4 A Hecke Operator The following lemma is well-known. Lemma 7.2. Let ˇ 2 OL be a totally positive element. Assume that f ˇOL . Then there is a Hecke operator Uˇ on Mk .00 .f/; C/ so that for f 2 Mk .00 .f/; C/ the standard q-expansion of f jk Uˇ is c.0; f / C

X

c.ˇ; f /qL :


145

Proof. The claimed operator Uˇ is the one defined by

ˇ0 . Note that 0 1

ˇ0 1b 00 .f/ 00 .f/ D [b 00 .f/ : 0 1 0ˇ where b ranges over all coset representatives of ˇD in D and the union is a disjoint union. Define X 1b ./; f jk Uˇ ./ D N .ˇ/k=21 f jk 0ˇ b

where b runs through the set of coset representatives of ˇD in D. Then f jk Uˇ ./ D N .ˇ/k=21

X

f jk

b

1b 0ˇ

./

D N .ˇ/k=21 N .ˇ/k=2 N .ˇ/k

X b

f.

Cb / ˇ

X X 1 1 D N .ˇ/1 c.0; f / C c.; f /e 2 it rL=Q ..ˇ Cˇ b// b

D c.0; f / C N .ˇ/1

X

c.; f /e 2 it rL=Q .=ˇ/ .

X

e 2 it rL=Q .b=ˇ/ /

b

P 2 it r .b=ˇ/ L=Q The sum D 0 unless 2 ˇOL . On the other hand, if 2 ˇOL , P 2 itb re .b=ˇ/ L=Q then b e D N .ˇ/. Hence we get f jk Uˇ ./ D c.0; f / C

X

c.ˇ; f /qL :

t u

7.5 Eisenstein Series The following proposition is proven by Deligne and Ribet ([Del80], Proposition 6.1). Proposition 7.1. Let ˙ be a finite set of finite primes of L and L˙ be the maximal abelian totally real extension of L unramified outside ˙. If $ is a locally constant C-valued function on Gal.L˙ =L/, then for every even positive integer k (i) There is an integral ideal f of L with all its prime factors in ˙, and a Hilbert modular form Gk;$ in Mk .00 .f/; C/ with standard q-expansion

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2r L.$; 1 k/ C

X X

$.ga /N.a/k1 qL ;

a

where the first sum ranges over all totally positive 2 OL , and the second sum ranges over all integral ideals a of L containing and prime to ˙. Here ga is the image of a under the Artin symbol map. N.a/ denotes norm of the ideal a. (ii) The q-expansion of Gk;$ at the cusp determined by any ˛ 2 A L has constant term N k ..˛//2r L.$g ; 1 k/; where .˛/ is the ideal of L generated by ˛ and N..˛// is its norm. The element g is the image of .˛/ under the Artin symbol map (see for instance 2.22 in Deligne-Ribet [Del80]). The locally constant function $g is given by $g .h/ D $.gh/

for all h 2 Gal.L˙ =L/:

Remark. Deligne-Ribet prove this proposition for all k 2 N. For our purpose k even suffices.

7.6 The q-Expansion Principle Let f 2 Mk .00 .f/; Q/ i.e. c.; ˛; f / 2 Q for all 2 OLC [ f0g and all ˛ 2 A L. Suppose the standard q-expansion of f has all non-constant coefficients in Z.p/ (localisation of Z at the prime ideal .p/) and let ˛ 2 A L be a finite adele. Then c.0; f / N.˛p /k c.0; ˛; f / 2 Zp : Here ˛p 2 L ˝Q Qp is the pth component of ˛ and N W L ˝Q Qp ! Qp is the norm map. This is the q-expansion principle of Deligne-Ribet (see [Del80] 0.3 and 5.13–5.15). Remark 7.1. If u is the image in Gal.L˙ =L/ of an idèle ˛ under the Artin symbol map, then using the equation N..˛//k N.˛p /k D .u/k , we get c.0; Gk;$ / N.˛p /k c.0; ˛; Gk;$ / D2r u .$; 1 k/ WD2r .L.$; 1 k/ .u/k L.$u ; 1 k//; for any positive even integer k. In particular, the q-expansion principle can be used to prove congruences between u .$; 1 k/ by proving congruences between nonconstant terms of Gk;$ ’s (for various $’s and k’s). This is used in the next section.


147

8 Proof of the Sufficient Conditions in Section 6 Recall that we have fixed an element u in Z f1g in Sect. 6. Proposition 8.1. The sufficient conditions in Proposition 6.1 for proving Proposition 3.1 holds. In particular, M3 holds. Proof. We first prove condition (9). We must show that for any U V in S.G; Z/ j such that ŒV W U D p and any integer j 0, any coset y of Z p in U ab fixed by V , we have the congruence p

uU .ı .y/ ; 1 k/ uV .ı .y/ ı verVU ; 1 pk/.mod pZp /; for a positive integers k divisible by ŒF .p / W F . Choose an integral ideal f of FV such that the Hilbert Eisenstein series Gk;ı.y/ and Gpk;ı.y/ ıverV , given by U Proposition 7.1, on HFU and HFV respectively have levels fOFU and f respectively. Moreover, we may assume that all prime ideals dividing f lie in ˙FV and f pOFV . Define E by E D RFU =FV .Gk;ı.y/ /jpk Up Gpk;ı.y/ ıverV : U

A FV

Then E 2 Mpk .00 .f/; Q/. Let ˛ 2 whose image in V ab under the Artin symbol map coincides with u. Then by Lemma 7.1 and Remark 7.1 c.0; E/ N.˛p /pk c.0; ˛; E/ D 2rU uU .ı .y/ ; 1 k/ 2rV uV .ı .y/ ı verVU ; 1 pk/: p

Note that the image of ˛ in U ab under the Artin symbol map is up . Since 2rU 2rV .mod p/ it is enough to prove, using the q-expansion principle, that the non-constant terms of the standard q-expansion of E all lie in pZ.p/ i.e. for all 2 OFCV c.; E/ D c.p; RFU =FV .Gk:ı.y/ // c.; Gpk;ı.y/ ıverV / U X X .y/ k1 .y/ pk1 D ı .gb /N.b/ ı .gaOFU /N.a/ 2 pZ.p/ a

.b;/

Here the pair .b; / runs through all integral ideals b of FU which are prime to ˙FU containing the totally positive element 2 OFU such that t rFU =FV ./ D p. The ideal a runs through all integral ideals of FV prime to ˙FV containing . The group V =U acts trivially on the pair .b; / if and only if there is an ideal a such that aOFU D b and 2 OFV . In this case ı .y/ .gb /N.b/k1 ı .y/ .gaOFP /N.a/pk1 Dı .y/ .gb /.N.a/p.k1/ N.a/pk1 / 2pZ.p/ :

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On the other hand, if V =U does not act trivially on the .b; /, then the orbit of .b; / under the action of V =U in the above sum is

X ı .y/ .ggb g 1 /N.bg /k1 g2V =U

DjV =U jı .y/ .gb /N.b/k1 2pZ.p/ : Here we use ı .y/ .ggb g 1 / D ı .y/ .gb / because y is fixed under the action of V . This proves condition (9). Proof of condition (10) is similar. We just note that if V 0 2 C.G; Z/ is such that ŒV 0 W V D iV , then in the proof we must restrict the Eisenstein series over FV and FV p to the ones over FV 0 . t u Lemma 8.1. Let U 2 C.G; Z/. If N is a subgroup of NG U containing U but not contained in C.G; Z/, then the image of the transfer homomorphism ver W N ab ! U is a proper subgroup of U . Proof. We recall definition of the transfer homomorphism. For g 2 N let fx1 ; : : : ; xn g be a set of double coset representatives of hginN=U . Define m to be the smallest positive integer such that g m 2 U . Then a set of left coset representatives of U in N is f1; g; : : : ; gm1 ; x1 ; gx1 ; : : : ; gm1 x1 ; : : : ; xn ; gxn ; : : : ; gm1 xn g: For all 0 i m 1 and 1 j n, we define hij .g/ 2 U by 0

g.g i xj / D g i xj 0 hij .g/: 0 0 for Q a unique 0 i m 1 and 1 j n. Then ver.g/ is defined to be i;j hij .g/. It is clear that

( hij .g/ D

1 if i m 2 1 m xj g xj if i D m 1:

Q Hence ver.g/ D nj D1 xj1 g m xj . As N=Z is not a cyclic subgroup n > 1. Since U=Z and N=Z are p-groups, the action of N=Z on the subgroup of order p of U=Z r1 is trivial. If p r is the order of g m in U=Z, then N acts trivially on g mp .mod Z/. Therefore Y Y r1 r1 r1 ver.g/p D xj1 g mp xj D g p 2 Z: xj

j

Hence ver.g/ is not a generator of U=Z. This proves the lemma.

t u


149

Proposition 8.2. The sufficient condition in Proposition 6.2 for proving Proposition 3.2 holds. j

Proof. We must show that for any U 2 C.G; Z/, any j 0, any coset y of Z p in U whose image in U=Z is a generator of U=Z and any u in Z we have dU

uU .ı .y/ ; 1 k/ 0.mod

j.WG U /y j Zp /; iU

for a positive integer k divisible by ŒF .p / W F . Choose an integral ideal f of OFNG U such that the Hilbert Eisenstein series Gk;ı.y/ over HFU , given by Proposition 7.1 has level fOFU . Define E D RFU =FNG U .Gk;ı.y/ /: Then E 2 MdU k .00 .f/; Q/. Let ˛ be a finite idèle of FNG U whose image under the Artin symbol map coincides with u. By Lemma 7.1 and Remark 7.1, we have c.0; E/ N.˛p /dU k c.0; ˛; E/ D 2rU uU .ı .y/ ; 1 k/: dU

Hence, using the q-expansion principle, it is enough to prove that the non-constant j.W U / j terms of the standard q-expansion of E all lie in GiU y Z.p/ i.e. for any 2 OFCN U , G

c.; E/ D

X

ı .y/ .gb /N.b/k1 2

.b; /

j.WG U /y j Z.p/ ; iU

where .b; / runs through all integral ideals b of FU which are prime to ˙FU and

2 b is totally positive with t rFU =FNG U . / D dU . The group .WG U /y acts on the pairs .b; /. Let V be the stabiliser of .b; /. Then there is an integral ideal c of FV WD FUV and a totally positive element of OFV such that cOFU D b and D . If V D N=U for a subgroup N of NG U such that N … C.G; Z/, then ı .y/ .gb / D 0 by Lemma 8.1. On the other hand, if V D N=U for a subgroup N of NG U such that N 2 C.G; Z/, then jV j D ŒN W U iU and the .WG U /y -orbit of .b; / in the above sum is X

ı .y/ .ggb g 1 /N.bg /k1

g2.WG U /y =V

Dj.WG U /y =V jı .y/ .gb /N.b/k1 2

j.WG U /y j Z.p/ : iU

Here we use ı .y/ .ggb g 1 / D ı .y/ .gb / for any g 2 .WG U /y . This proves the proposition. u t

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Proposition 8.3. The sufficient conditions in Proposition 6.3 for proving Proposition 3.3 hold. Proof. We use the notation from Proposition 6.3. We must show that for any U 2 j C.G; Z/ fZg and V 2 Pc .U /, for any integer j 0, any coset y of Z p in U whose image in U=Z is a generator of U=Z dU

xU .ı .y/ / D uU .ı .y/ ; 1 k/

X

dU =p

uV

.ı .y/ ı 'V ; 1 pk/ 2 j.WG U /y jZp ;

V 2Pc .U /

for a positive integer k divisible by ŒF .p / W F . Put L D FNG U . Choose an integral ideal f of L such that the Hilbert Eisenstein series Gk;ı.y/ and Gpk;ı.y/ ı'V , given by Proposition 7.1, on HFU and HFV respectively have levels fOFU and fOFV respectively (for each V 2 Pc .U /). Moreover, we may assume that all prime factors of f are in ˙L and f dU OL . Define X

E D RFU =L .Gk;ı.y/ /jdU k UdU

RFV =L .Gpk;ı.y/ ı'V /jdU k UdU =p :

V 2Pc .U /

Then E 2 MdU k .00 .f/; Q/. Let ˛ be a finite idèle of L whose image under the Artin symbol map coincides with u. By Lemma 7.1 and Remark 7.1 c.0; E/ N.˛p /iV p c.0; ˛; E/ D X dU dU =p 2rU uU .ı .y/ ; 1 k/ 2rV uV .ı .y/ ı 'V ; 1 pk/: V 2Pc .U /

As 2rU 2rV .mod rU /, for each V 2 Pc .U /, and rU j.WG U /y j, 2rU uU .ı .y/ ; 1 k/ dU

X

2rV uV .ı .y/ ı 'V ; 1 pk/ iV

V 2Pc .U /

2rU xU .ı .y/ /.mod j.WG U /y jZp /: Hence, using the q-expansion principle, it is enough to prove that the non-constant terms of the standard q-expansion of E all lie in j.WG U /y jZ.p/ i.e. for all totally positive in OL we have X

c.; E/ D c.dU ; RFU =L .Gk;ı.y/ // D

X .b;/

c.dU =p; RFV =L .Gpk;ı.y/ ı'V //

V 2Pc .U /

ı .y/ .gb /N.b/k1

X X

ı .y/ .gaOFU /N.a/pk1 2 j.WG U /y jZ.p/ :

V 2Pc .U / .a; /

Here the pairs .b; / run through all integral ideals b of FU which are prime to ˙FU and 2 b is a totally positive element with tr FU =L ./ D dU . For each


151

V 2 Pc .U / the pairs .a; / run through all integral ideals a of FV which are prime to ˙FV and 2 a is a totally positive element with t rFV =L . / D dU =p. The group P WD .WG U /y acts on the pairs .b; / and .a; /. Let W P be the stabiliser of .b; /. Then there is an integral ideal c of FW WD FUW and a totally positive element of OFW such that cOFU D b and D . In W D N=U for a subgroup N of NG U such that N … C.G; Z/, then ı .y/ .gb / D 0 by Lemma 8.1. On the other hand if W D N=U for a subgroup N of NG U such that N 2 C.G; Z/ and N ¤ U , then there is a unique V 2 Pc .U / such that V N . Moreover, for the corresponding pair .a; / in the above expression we have cOFV D a and D . This implies that aOFU D b and D . The orbit of the pair .b; / in the above sum is

X ı .y/ .ggb g 1 /N.bg /k1 ı .y/ .ggb g 1 /N.bg /pk1 g2P =W

DjP =W jı .y/ .gb / N.b/k1 N.b/pk1

DjP =W jı .y/ .gb / N.c/jW j.k1/ N.c/jW j.pk1/=p 2jP jZ.p/ D j.WG U /y jZ.p/ : Finally assume that W is a trivial group then the P orbit of the pair .b; / in the above sum is X ı .y/ .ggb g 1 /N.bg /k1 D jP jı .y/ .gb /N.b/k1 2 jP jZ.p/ D j.WG U /y jZ.p/ : g2P

In both cases the first equality uses ı .y/ .ggb g 1 / D ı .y/ .gb / for g 2 P . In the first case we also use the fact that N.c/jW j N.c/jW j=p .mod jW j/. This proves the proposition. t u

9 Proof of M4: From the Basic Congruences We have proved the basic congruences in previous subsections. We want to deduce M4 from these congruences. However, we cannot do it directly for the extension F1 =F . We use the following trick: we extend our field slightly to FQ1 F1 such that FQ1 =F is an admissible p-adic Lie extension satisfying the Iwasawa conjecture and Gal.FQ1 =F / D GQ with GQ Š HQ G, where HQ is a cyclic group of order jG=Zj. We know the basic congruences for FQ1 =F which we use to deduce the M4 for FQ1 =F . This proves the main conjecture for FQ1 =F and hence implies the main conjecture for F1 =F (main conjecture for an extension trivially implies main conjecture for a subextension).

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9.1 The Field FQ1 Choose a prime l large enough such that l 1.mod jG=Zj/ and Q.l / \ F1 D Q. Let K be the extension of Q contained in Q.l / such that ŒK W Q D jG=Zj. Define FQ D KF and FQ1 D FQ F1 . Then Q Gal.FQ1 =F / D Gal.FQ =F / Gal.F1 =F / DW HQ G DW G:

FQ1 D FQ F1 tt HQ tt t t tt tt G F1 GQ G FQ D FK HQ sss s ss sss ss F

ss ss ss s s ss ss

K

HQ

Q

9.2 A Key Lemma We replace the field F1 by FQ1 as we need the following key lemma. Recall that Q Z/, we have defined the integer iU by for any U 2 C.G; iU D maxV 2C.G;Z/ fŒV W U such that U V g Q Q Z/. If U ¤ Z, then Lemma 9.1. Let U 2 C.G; TU p iU2 .U /:


If U D Z, then

153

Q TZ D jG=Zj.Z/:

c Similar statements hold for TU;S and T U. Proof. Case 1: U=Z HQ . Then iU D ŒHQ W .U=Z/ and NGQ U D GQ acts trivially on .U /. Hence TU D ŒGQ W U .U / D jG=ZjŒHQ W .U=Z/.U /: If U ¤ Z, then jG=Zj p iU . Hence the claim. Q h/, with hQ 2 HQ and h 2 G=Z. Case 2: U=Z ª HQ . Let U=Z be generated by .h; Q By assumption h ¤ 1. Let V 2 C.G; Z/ such that ŒV W U D iU . Let .hQ 0 ; h0 / be a generator of V =Z such that hQ i0U D hQ and hi0U D h. Now note that .U=Z/ HQ hh0 i NG=Z Q acts trivially on .U /. As U=Z HQ hh0 i this implies that TU D

jHQ hh0 ij .U / jU=Zj jHQ jjhh0 ij .U / jU=Zj

D jHQ jiU .U / p iU2 .U /: The last containment holds because jHQ j p iU . The assertion about TZ is clear. t u

9.3 Completion of the Proof Q Q Z/. If .xV /V 2 Lemma 9.2. Let U 2 C.G; V 2Pc .U / TV;S is fixed under the action of NGQ U , then X 'V .xV / 2 TU;S =p V 2Pc .U /

and

Y

'V .xV / 2 TU;S =p:

V 2Pc .U /

Proof. Let ŒV be the orbit of V under the action of NGQ U . Then X W 2ŒV

'W .xW / D

X g2NGQ U=NGQ V

g'V .xV /g 1 2 TU;S =p:

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M. Kakde

t u

This proves the first claim. The second assertion is clear by Lemma 9.1.

For an element x 2 B.U / such that x 1.mod p/, the series log.x/ D

X .x 1/n n1

n

makes sense and we use this expression in the following. In the following U 2 Q Z/ and U ¤ Z. Using congruence (3) we get C.G; log.

!Uk .U / ! k .U / 2 c /1 U .mod .pT U = iU / /: U U

This implies Pp1 Qp1 ! k . /

pU kD0 !Uk .U / kD0 U U 2 c log .mod p.T U = iU / /: p U U Putting this together with a similar expression for all V 2 Pc .U /, we obtain log

Q

V 2Pc .U / 'V .˛V .V // p Qp1 U = kD0 !Uk .U /

X

p'V .V / Pp1 ! k .'V .V //

kD0

X

p'V .V / Pp1 ! k .'V .V //

kD0

X V 2Pc .U /

U

'V .V /

V 2Pc .U /

U

'V .V /

V 2Pc .U /

.p'.V /

Pp1 k0

pU Pp1 ! k .U /

kD0 U U

X

p'V .V / Pp1 ! k .V /

V 2Pc .U /

kD0

U

U

!Uk .'V .V ///.U 'V .V // c .mod p T U /: U 'V .V /

2 c c Here we use p.T U = iU / p TU as implied by Lemma 9.1. The second congruence above uses congruence (5). Now using congruences (2) and (3) for V we get

U 'V .V / 2 .p iV / and p'V .V /

p1 X kD0

!Uk .'V .V // 2 p'V .TV;S = iV /:


155

Hence

p'V .V /

p1 X

!Uk .'V .V // U 'V .V /

kD0

2p.'V .TV;S /= iV / .p iV / p 2 'V .TV;S /: Putting this together for all V 2 Pc .U /, we get p1

X X .p'V .V / !Uk .'.V ///.U 'V .V // V 2Pc .U /

k0

2pTU;S

.by Lemma 9.2/: Q

Hence


log

c 2 pT U:

c c As log induces an isomorphism between 1 C p T U and p TU , we have Q


c 2 1 C pT U:

By congruence (3) Y

p1 p U =

!Uk .U / 2 1 C pTU;S = iU :

kD0

Similarly, for any V 2 Pc .U / ˛V .V / 2 1 C pTV;S = iV : Hence Y

'V .˛V .V // 2 1 C pTU;S = iU

(by Lemma 9.2):

V 2Pc .U /

Q

Therefore


2 1 C pTU;S = iU :

We now observe that bU \ 1 C pTU;S = iU D 1 C pTU;S : 1 C pT

156

M. Kakde

Q

Hence

V 'V .˛V .V // p Qp1 k U = kD0 !U .U /

2 1 C pTU;S :

This proves congruences in M4 for all U ¤ Z. For Z we first note that ˛Z .Z / 1.mod p/: U 2Pc .Z/ 'U .˛U .U //

Q

and the image .aU /U WD L..U /S / Hence we can apply L to the tuple .U /U 2S.G;Z/ Q lies in B ˝ Qp (for the definition of L we refer to [Sch11] where it appears after Lemma 4.1. For B see Sect. 3 of loc. cit.). Let a denote the inverse image of .aU /U Q ab ˝Zp Qp (see Remark 3.7 loc.cit.). under ˇB in B.G/ Q Z/ fZg, we get As we have proven M4 for all V 2 C.G; bV : V .LV ..U /U // 2 T (Lemma 4.5 in loc. cit.). Therefore a lies in Q ab ˝Zp Qp : B.Z/ ˝Zp Qp ˚ .˚g¤12R B.Z/g/ B.G/ For the definition of R and the above direct sum decomposition see (H3) in loc. cit.. The claim about a follows directly from the definition of ı (it is defined after Q ab if and only if its image, denoted by a, Lemma 3.2 in loc. cit.). Hence a 2 B.G/ ab ab Q Q Q ab then ˇB .a/ 2 B and hence in in B.G / ˝Zp Qp lies in B.G /. If a 2 B.G/ bZ and M4 hold for Z. Therefore it is enough to prove particular LZ ..U /U / 2 T that LGQ ..U /U / D a lies in B.GQ ab /. Lemma 9.3. Put D GQ 2 .GQ ab / S . Then LGQ ..U /U / D

p 1 log. /: p './

Proof. This is proven in Lemma 4.7 in [Sch11] (though the lemma is stated for tuples satisfying M1–M4, only properties M1 and M2 are used in the proof). We can prove it more directly using the interpolation properties of p-adic L-functions. It suffices to prove that Q

jG=Zj D Z

Y

˛U .U /

jU=Z j p

:

(21)

Q U 2C.G;Z/fZg

Q Z/ we define U Note that the expression is viewed in A.G ab /. For each U 2 C.G; to be the following function on U

U .h/ D

jU=Zj if h generates U 0 otherwise


157

P Q Q Then jG=Zj D U 2C.G;Z/ IndGU U (see, for example, Proposition 27, Sect. 9.4 in Q Q Denote its restriction [Ser77]). Let be a one dimensional Artin character of G. to U by U . Let U and denote the p-adic cyclotomic character of FU and F respectively. Then for any positive integer k divisible by ŒF .p / W F Frobenius reciprocity together with the above equation gives X

GQ k k Q Ind U .U U U / D jG=Zj :

Q U 2C.G;Z/

Now note that ˛U .U /.U Uk / D L˙U .U U ; 1 k/: Hence Eq. (21) holds (as the two sides interpolate the same values on a dense set of Q This proves the lemma. representations of G). t u Therefore a D LGQ ..U /U / 2 B.GQ ab / and M4 holds for Z. This finishes proof of the main conjecture. Acknowledgements I would like to thank the organisers of the workshop and all the participants. I am very grateful to the referee whose comments were most useful.

References [Coa05] J. Coates, T. Fukaya, K. Kato, R. Sujatha, O. Venjakob, The GL2 main conjecture for elliptic curves without complex multiplication. Publ. Math. IHES 101, 163–208 (2005) [Coa11] J. Coates, D. Kim, Introduction to the work of M. Kakde on the non-commutative main conjectures for totally real fields, This volume (2011) [Del80] P. Deligne, K.A. Ribet, Values of abelian L-functions at negative integers over totally real fields. Invent. Math. 59, 227–286 (1980) [Kak10] M. Kakde, K1 of some noncommutative p-adic group rings, Preprint, March (2010) ¨ [Kli62] H. Klingen, Uber die werte der Dedekindschen Zetafunktionen. Math. Ann. 145, 265–272 (1962) [Oli88] R. Oliver, Whitehead Groups of Finite Groups. London Mathematical Society Lecture Note Series, vol. 132 (Cambridge University Press, Cambridge, 1988) [Rit08] J. Ritter, A. Weiss, Congruences between abelian pseudomeasures. Math. Res. Lett. 15(4) 715–725 (2008) [Sch10] P. Schneider, O. Venjakob, A splitting for K1 of completed group rings. http://arxiv.org/ abs/arXiv:1006.1493 (2010) [Sch11] P. Schneider, O. Venjakob, K1 of certain Iwasawa algebras, after Kakde, This volume (2011) [Ser77] J.-P. Serre, Linear Representation of Finite Groups (Springer-Verlag, New York, 1977) ¨ [Sie70] C. Siegel, Uber die Fourierschen Koeffizienten von Modulformen. Gött. Nachr. 3, 15–56 (1970) [Suj11] R. Sujatha, Reductions of the main conjecture, This volume (2011)

On the Work of Ritter and Weiss in Comparison with Kakde’s Approach Otmar Venjakob

Abstract In this chapter a survey of the proof by Ritter and Weiss of the Iwasawa Main Conjecture over totally real fields for the Tate motive in comparison with Kakde’s approach is provided. In particular, we shall compare the two different descriptions of K1 ..G//: one explicitly given in Kakde’s work as presented in the contribution Schneider and Venjakob [SV] of this volume, the other one derived from Ritter and Weiss’ work; the latter description given in Sect. 6 is new.

MSCs: 11R23, 19B28, 11S23, 20C10, 11S40 Keywords Completed group ring • Iwasawa algebra • algebraic K-group • explicit Brauer induction • main conjecture • L-functions

1 Introduction Almost simultaneously Ritter and Weiss [RW10] on the one hand and Kakde [Kak3] on the other hand gave a proof of the non-commutative Iwasawa main conjecture over totally real fields for the Tate motive under the assumption that a certain -invariant vanishes as has been conjectured also by Iwasawa. Actually the article of Ritter and Weiss appeared slightly earlier on the arXive and deals with the case of one-dimensional p-adic Lie-extensions, which can be combined with Burns’ wellknown insight in [Bur, Theorem 2.1] (this crucial idea has been presented by Burns already during a seminar at the University of Kyoto in early 2006) based on Fukaya and Kato’s result [FK, Proposition 1.5.1] to obtain the conjecture for general p-adic (admissible) Lie-extensions. Kakde’s paper, which got its final version practically O. Venjakob () Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, 69120 Heidelberg, Germany e-mail: [email protected]; http://www.mathi.uni-heidelberg.de/venjakob/ J. Coates et al. (eds.), Noncommutative Iwasawa Main Conjectures over Totally Real Fields, Springer Proceedings in Mathematics & Statistics 29, DOI 10.1007/978-3-642-32199-3 6, © Springer-Verlag Berlin Heidelberg 2013

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during this workshop, leads up directly to a full proof in the general situation, using the same result of Fukaya and Kato. Various special cases have been known by the work of (in alphabetical order) Hara [Har], Kakde [Kak1], and Kato [Kat] as well as Ritter and Weiss [RW7]. Since in both approaches the reduction steps from the general case of onedimensional extensions to the l-elementary extensions (in the language of Ritter and Weiss) or to essential pro-p-extensions (in the language of Kakde) are based on the same principles (even though at different places: for Ritter and Weiss with respect to the Hom-description while in Kakde’s case for the K-theory) and since again the generalisation from the pure pro-p case to this slightly more general case follows certain standard techniques, see [Suj], we restrict in this survey from the very beginning to the one-dimensional pro-p-case. In these notes we shall use the same notation as in [SV], but for the convenience of the reader we have collected the crucial notation from [RW10] in a glossary below comparing it with Kakde’s and our notation, respectively. Since during the workshop and hence in this volume Kakde’s approach has been discussed in great detail and we may hence assume greater familiarity with his techniques, many comments in this comparison will be made from the perspective of Kakde’s point of view. So we want to stress that this certainly does not reflect the historical development as Kakde has probably been influenced by a couple of ideas from Ritter and Weiss. For example, the analytic techniques for the actual proof that the abelian p-adic L-functions satisfy the required conditions in order to induce the “non-commutative” p-adic L-function has been applied first by Ritter and Weiss as well as by Kato (in an unpublished preprint about Heisenberg type p-adic Lie group extensions). For that reason we won’t discuss this analytic part at all in this comparison because again the methods are essentially the same, see [RW10, Sect. 3] and [Kak3, Sect. 6], respectively. One evident difference between the two works consists of the way of presentation: While Kakde, who also had partial results in previous publications, delivers an almost self-contained account of a full proof of the main conjecture (even dealing with arbitrary admissible p-adic Lie extensions) in one ingenious paper, the impressive work of Ritter and Weiss is spread over at least ten articles [RW1, RW2, RW3, RW5, RW4, RW8, RW9, RW6, RW7, RW10], viz naturally in the way as their theory has been developed over the recent decade. The last article, which contains the general result, is rather an instruction how to modify and extend the proofs of earlier results (in less general cases) in previous publications combined with an extensive discussion of the new Möbius-Wall-congruence in order to complete the proof in the general case than a self-contained proof. A general outline of the overall strategy of the proof is missing and the reader is forced even to collect the notation from all the other articles. This is somewhat unfortunate as otherwise the strategy of their nice proof could have been much more easily accessible for the reader.1

1

In order to remedy this Ritter and Weiss are apparently thinking about writing a lecture note volume about their approach.

On the Work of Ritter and Weiss in Comparison with Kakde’s Approach

161

Another obvious difference is the fact that while Kakde describes K1 ..G // modulo SK1 ..G // in terms of certain relations (some call them congruences) among elements in the units of Iwasawa algebras for certain abelian subquotients of G Ritter and Weiss are only interested in whether the particular system of abelian pseudomeassures is in the image of K1 : But in the last section of this survey we try to clarify what can be proved by the methods of Ritter and Weiss towards describing the image of K1 in the product of the K1 ’s of appropriate subquotients of G in the manner of Kakde. I am very grateful to Cornelius Greither and Mahesh Kakde for reading an earlier version and suggesting a couple of improvements.

2 Glossary In the following the notation in the different columns usually denotes the same object, but sometimes it only indicates that they are just closely related. Kakde p¤2 F1 =F G D G.F1 =F / F cyc H D G.F1 =F cyc / A WD A.G / WD .G /S D .G /.p/ B WD B.G / D .G /S .G /S D A.G /Œ p1 O=Zp unramified; . D O / finite set of ˙ ˙ram .F1 =F / primes of F K0 ..G /; .G /S / D K0 .HS / D K0 .CS / ŒR .G.F˙ =F1 /; Qp =Zp /_ F1 =F

2

OŒŒC ong .G / D O .G /ab D 0 , There exists an open prop subgroup H 0 H such that XF D G.F˙ =F/ab .p/ is finitely H0 generated over Zp for F D F1 and F˙ the maximal extension of F unramified outside ˙.

Ritter=Weiss l ¤2 K=k G D G.K=k/ k1 k H .G/ ^ .G/ QG O

finite set of S Sram .K=k/ [ S1 primes of k K0 T .G / Ã D ÃS (“mhO” or “agemO”) ; F1 =F .pseudomeasure/ if F1 =F is abelian TO .G /

0 .XF1 / D 0 for some open subgroup 0

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Kakde - Schneider/Venjakob or this article VU ; NVU Q UV Tr

Ritter/Weiss U V V U

prVU

deflVU resVU ResVU L Q L; U LU

L; LB L

3 2-Extensions Versus Perfect Complexes The statement of the Main Conjecture (MC) affirms in both approaches the existence of an element ; respectively 2 K1 ..G /S / satisfying firstly a certain interpolation property we shall discuss later and which, secondly, is mapped under the connecting homomorphism of the localisation sequence of K-theory K1 ..G /S / ! K0 .HS / Š K0 .CS / to the class ÃS ; respectively ŒR .G.F˙ =F1 /; Qp =Zp /_ : To be more precise, one uses in general the localisation with respect to the Ore set S for the statement of the MC, but assuming the vanishing of ; it is easily reduced in both approaches to the above statement. Thus the first question which naturally arises is: Why is the class of R .G.F˙ =F1 /; Qp =Zp /_ the inverse of the class of Ã D Ã˙ in the relative K0 ? Let ˝ denote the maximal p-extension of F1 unramified outside ˙; here we fixed a splitting G of the projection G : We denote the Galois group G.˝=F / by G˙0 and note that this group is well-known to be (topologically) finitely generated. Hence, choosing d generators we obtain the following commutative diagram

1

1

N

N

1

/ R

/ Fd

/ G

/ 1

1

/ H0 ˙

/ G0 ˙

/ G

/ 1;

1

1


163

in which Fd denotes the free pro-finite group on d elements and all the other groups are defined by exactness of the diagram. By Neukirch et al. [NSW, Proposition 5.6.6] there is a canonical complex associated to the above diagram / N ab .p/

0

/ Zp ŒŒG 0 ˙

/ Zp ŒŒG 0 d ˙

/ 0

which forms a resolution of Zp by projective Zp ŒŒG˙0 -modules, whence flat Zp ŒŒH˙0 -modules, i.e., the complex has only one nonzero cohomology group, Zp ; in degree zero, where we put Zp ŒŒG˙0 in degree zero. Therefore, the complex / Zp ŒŒG d

/ N ab .p/H 0 ˙

0

/ Zp ŒŒG

/ 0;

which arises by taking H˙0 -coinvariants, represents R .G.F˙ =F1 /; Qp =Zp /_ Š R .H˙0 ; Qp =Zp /_ by Pontryagin duality and using a theorem of Neumann. By Neukirch et al. [NSW, Proposition 5.6.7] and using the weak Leopoldt statement H2 .H˙0 ; Zp / D 0 there is a quasi-isomorphism 0

/ N ab .p/H 0 ˙

/ Zp ŒŒG d

0

/ 0

/ Y

dY

/ Zp ŒŒG

/ 0

/ Zp ŒŒG

/ 0;

where Y WD IG˙0 =IH˙0 IG˙0 and the map dY W Y ! Zp ŒŒG factorises over the augmentation ideal as Y IG ,! Zp ŒŒG : Now consider the exact sequence of (vertical) complexes 0

/ Zp ŒŒG

/ Y dY

0

/ Zp ŒŒG

0

/ Zp ŒŒG

/ M

/ 0

/ N

/ 0;

where is a suitably chosen homomorphism such that definition image in IG ; is injective. Then, by definition

0

WD dY ı ; which has by

Ã D ŒM ŒN D ŒM ! N 2 K0 .HS / Š K0 .Chb .HS //;

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where HS denotes the exact category of S -torsion .G /-modules with finite projective dimension while Chb .HS / denotes the category of bounded complexes in HS and the above identification is shown in [Wei, II. Theorem 9.2.2]. We also shall write CS WD Chperf;S ..G // for the category of perfect complexes of .G /modules which become acyclic after tensoring with .G /S : Under the identification K0 .Chb .HS // D K0 .Chperf;S ..G /// (cp. [Wei, II Exercise 9.2, V Exercise 3.14]) we then have ŒM ! N D ŒY ! Zp ŒŒG ŒZp ŒŒG ! Zp ŒŒG D ŒY ! Zp ŒŒG D ŒR .G.F˙ =F1 /; Qp =Zp /_ because the complex Zp ŒŒG ! Zp ŒŒG is acyclic and quasi-isomorphisms induce identities in K0 : Thus we have shown the following Proposition 3.1. Ã D ŒR .G.F˙ =F1 /; Qp =Zp /_

in

K0 .HS / Š K0 .Chperf;S ..G ///:

When this survey was prepared for the workshop this result—although certainly well-known to the experts—was not available in the literature. Before the proofreading I learned that also Andreas Nickel had the same intention to fill this gap in [Nic, Theorem 2.4]. Actually I am grateful to him for pointing out a missing sign.

4 Hom-Description We shall write Irr.G / for the set of Qp -valued irreducible representations of G with finite image. Considering elements of K1 ..G // as maps on (irreducible) representations we obtain a natural homomorphism

Det W K1 ..G // ! Maps.Irr.G /; Zp / into maps from Irr.G / to the units of the ring of integers Zp of Qp ; which allows for example requiring interpolation properties for elements of the K-group. Of course, the target is “much bigger” than the image and one crucial question for the proof of the main conjecture consists of finding the most appropriate intermediate target in which to work. Ritter and Weiss follow the philosophy of Fröhlich’s so called Hom-description:

HomGQp .R.G /; Zp / Hom.R.G /; Zp / D Maps.Irr.G /; Zp /; R.G / being the group of virtual characters, i.e., the free abelian group generated by Irr.G /; and they refine it slightly by an Iwasawa-theoretic variant


K1 ..G /

Det

165

/ HomG ; R. / .R.G /; . / / HomG .R.G /; Zp / Qp Qp Zp

whose definition will be recalled in the case of the localised Iwasawa algebra .G /S at the end of this section. In contrast, Kakde uses a version which is closer to the side of K-theory

K1 ..G //

/

Q

ab U 2S.G ;Z / K1 ..U //

G

D

Q

ab U 2S.G ;Z / .U /

G

where G acts by inner conjugation on S.G ; Z /; the set of all subgroups Z U

G for a (fixed) central open subgroup Z G , and among the Iwasawa algebras: .xU /U 7! .g 1 xgUg1 g/U : Recall from [SV, Sect. 4] that the component U of is the composite NUG

U W K1 ..G // ! K1 ..U // ! K1 ..U ab // D .U ab / of the norm map and the homomorphism induced by the canonical surjection U U ab : By explicit Brauer induction [Bol, Sna] we obtain a splitting aG of the canonical map .˚U 2S.G ;Z / R.U ab =Z //G

Ind

/ R.G =Z / ;

which is induced by sending a tuple .nU U / of one-dimensional characters U of P G =Z U=Z and integers nU to nU IndU=Z . U /I by abuse of notation we shall also write U for the associated character inflUU=Z . / of U by inflation. Again, G is acting by inner conjugation: g.nU U /U 2S.G ;Z / WD .ngUg1 gUg1 .g g 1 //U 2S.G ;Z / ; where denotes the missing argument. Hence, using also the fact [Kak3, Lemma 92] that we have a surjection Irr. / Irr.G =Z / Irr.G /;

G . ; / 7! inflG . / inflG =Z ./

and hence a surjection R. / ˝Z R.G =Z / R.G /; we may define a homomorphism 0 BrInd W @

Y

U 2S.G ;Z /

1G .U ab / A ! HomGQp ; R. / .R.G /; Zp . / /

(1)

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such that BrInd..xU /U /./ WD

Y

DetU ab .xU /. U /nU ;

U

if aG ./ is represented by .nU U /U 2S.G ;Z / (and is induced from G =Z via inflation). Here DetU ab denotes the natural map DetU ab W K1 ..U ab // ! HomGQp ; R.U / .R.U ab /; Zp .U / / with U the image of U under the fixed projection G and we use the embedding Zp .U / Zp . / in order to consider the values of DetU ab in Zp . / : Lemma 4.1. The map BrInd is well-defined and BrInd..xU /U / is GQp - and R. /invariant, where these action are recalled at the end of this section. Proof. Assume first that Q arises by inflation from Q 2 Irr.G =Z /: First we check that the defining term U DetU ab .xU /. U /nU of BrInd..xU /U /./ is independent of the choice of representatives .nU U /U 2S.G ;Z / for aG ./ D aG ./ Q (using our lax notational convention): For any conjugate g.nU U /U 2S.G ;Z / D .ngUg1 gUg1 .g g 1 //U 2S.G ;Z / we have by the G -invariance of .xU /U Y

DetU ab .xU /. gUg1 .g g 1 //ngUg1

U

D

Y

DetU ab .g 1 xgUg1 g/. gUg1 .g g 1 //ngUg1

U

D

Y

gUg1 .g.g 1 xgUg1 g/g 1 /ngUg1

U

D

Y

gUg1 .xgUg1 /ngUg1

gUg 1

D

Y

DetU ab .xU /. U /nU :

U

Next we show that our (partial) definition sofar is invariant under characters of the kind W G =Z ! Zp : Indeed,

Y

DetU ab .xU /. U /nU D

U

Y

nU ResG U . / DetU ab .xU /. U /

U

D

Y

nU DetU ab .xU /.ResG U . / U /

U

by the R.U /-invariance of DetU ab .xU /: But by Lemma 4.2 below .nU ResG U . / U / represents aG . /; whence we have shown that


167

BrInd..xU /U /./ D BrInd..xU /U /. / as claimed. If 2 Irr.G / is arbitrary, using (1) we choose some 0 W G ! Zp such that 0 ˝ is inflated from G =Z : We then define BrInd..xU /U /./ WD . 0 /1 BrInd..xU /U /. 0 ˝ /: If 00 is a second such character, we conclude that WD . 0 /1 00 comes from a character of G =Z : Using the invariance of BrInd..xU /U / with respect to such characters as shown above we conclude that BrInd..xU /U / is well-defined on arbitrary irreducible representations of G and hence extends to a homomorphism on R.G / being R. /-invariant by construction. Finally the GQp -invariance follows from the GQp -invariance of all the DetU ab .xU / and clearly BrInd is a homomorphism. For a finite group G let RC .G/ denote the free abelian group on the G-conjugacy classes of characters W U ! Zp ; where U is any subgroup of G: We shall write .U; /G for the G-conjugacy class of .U; /: We clearly have a natural isomorphism RC .G/ Š .˚U R.U ab //G ; where U runs through all subgroups of G and usually explicit Brauer induction is defined in terms of a section aG of the canonical map induced by summing up inductions RC .G/ i.e.,

X

aG ./ D

Ind

/ / R.G/ ;

˛.U;/G ./.U; /G

.U;/G

for some integers ˛.U;/G ./. Note that there is a natural action of any character G

W G ! Zp on RC .G/; sending .U; /G to .U; /G WD .U; ResG U . // : Lemma 4.2. For every 2 R.G/ we have ˛.U;ResG . //G . / D ˛.U;/G ./; i.e. U aG . / D aG ./: Proof. According to [Sna, Theorem 2.3.15] the explicit formula for aG is given by ˛.U;/G ./ D

jU j jGj

X

.U 0 ; 0 /;.U 00 ; 00 / < 00 ; ResG U 00 ./ >;

.U 0 ; 0 /2.U;/G ; .U 0 ; 0 / .U 00 ; 00 /

where denotes the Möbius function on the partially ordered set MG consisting of the characters on subgroups .U; / of G with the .U; / .U 0 ; 0 / if and 0 only if U U 0 and ResUU . 0 / D while < ; > denotes the usual Schur inner

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product (loc. cit., def. 1.2.7). Since sending .U; / to .U; ResG U . // induces an G-equivariant isomorphism W MG ! MG ; of partially ordered sets, we obtain that .U 0 ; 0 /; .U 00 ; 00 / D .U 0 ; 0 /;.U 00 ; 00 / : G G 00 00 Taking into account that < ResG U . / ; ResU 00 . / >D< ; ResU 00 ./ > it follows that

˛.U;ResG . //G ./D U

jU j jGj

X

00 G .U 0 ; 0 /; .U 00 ; 00 / < ResG U . / ; ResU 00 . / >;

.U 0 ; 0 /2.U;/G ; .U 0 ; 0 / .U 00 ; 00 /

equals ˛.U;/G ./: By construction, the definition of and the functorial behaviour of Det with respect to norm maps and induction ([RW3, Lemma 9]) imply that: G DetG ./.IndG U / D DetU .NU .//. / D DetU ab .U .//. /;

whence we have a commutative diagram DetG

K1 ..G /

Q

U 2S.G ;Z /

.U ab /

G

/ HomGQ ; R. / .R.G /; Z . / / p p 3 O h h h BrInd hhhh hh brInd hhhh h h h h Q

/ DetU ab

Q

U 2S.G ;Z /

G HomGQp ; R.U / .R.U ab /; Zp .U / / ;

where brInd is defined in an analogous way as BrInd. Q While all of Kakde’s congruences are among the tuples .xU /U 2 U 2S.G ;Z / .U ab / , in the work of Ritter and Weiss some are in a similar product (as speculated below, but Ritter and Weiss do not formalise this) and others are expressed within their Hom-description (with the ultimate aim of constructing an integral logarithm). Now we are going to describe the setting of Ritter and Weiss’ approach in more detail. First we set QQp . / WD Quot.Zp ŒŒ / and note that the homomorphism Det. /. / W K1 ..G /S / ! QL . / QQp . / nr

in [RW3, Sect. 3] defined using the reduced norm K1 ..G /S / ! Z..G /S / coincides with ˚ of [CFKSV] by Burns [Bur, Lemma 3.1] and defines a homomorphism Det

K1 ..G /S / ! HomGQp ;R. / .R.G /; QQp . / /


169

into the group of GQp - and R. /-invariant homomorphism, where: (a) 2 GQp acts coefficientwise on QQp . / and R.G /,

(b) ' 2 R. / acts on R.G / via the tensor product 7! inflG .'/ and on QQp . / via the homomorphism induced by twisting T w' W Zp . / ! Zp . / 7! '. /: In particular, from RW’s construction (see Remark E in [RW3]): kerDet D ker.nr.G /S / D SK1 ..G /S /: By Ritter and Weiss [RW3, Lemma 9] Det behaves functorially with respect to open subgroups U G and the map induced by Induction .IndUG / , as well as to factor groups G G and the map induced by inflation .infl G G / on the Homdescription.

5 The Interpolation Property Versus LF1 =F By LF1 =F . / WD L ;˙ D

G ;˙ . 1/ 2 QQp . / H . 1/

we denote the .˙-truncated) p-adic Artin L-function for F cyc =F attached to

2 R.G /, for a fixed topological generator of [RW3, Sect. 4]. Then LF1 =F belongs to HomGQp ;R. / .R.G /; QQp . / / and is independent of the choice of (loc. cit. Proposition 11). Applying the extended localised augmentation map ' 0 W QQp . / ! Qp [f1g, which is induced by sending any 2 to 1; (see before [Suj, Theorem 3.1]) one sees that giving LF1 =F is morally the same as requiring the usual interpolation property: Det./ D LF1 =F for some 2 K1 ..G /S / corresponds to . / D ' 0 .Det./. // D ' 0 .LF1 =F . // D L˙ . ; 1/: Extending this interpolation property also to the cyclotomic character , it even determines LF1 =F uniquely. Recall that for any one-dimensional p-adic Lie group G we write B.G / for the p-adic completion of .G /S I we also set BZp .G / WD Zp ˝Zp B.G /: In fact, Ritter and Weiss show that

170

O. Venjakob .1/

LF1 =F 2 HomGQ

p;

R. / .R.G /; BZp .

/ /;

where .1/ indicates that f satisfies the congruence f . /p 1 mod p f . p /

(2)

with the ring endomorphism W BZp . / ! BZp . / being induced by sending to p , while p W R.G / ! R.G / denotes the p t h Adams operator, i.e., .

p /.g/

D .g p /

for any character . This result uses explicit Brauer induction twice: Firstly to generalise a theorem of Snaith [Sna, Theorem 4.1.6] saying that the image .1/ of Det lies in HomGQ ; R. / .R.G /; BZp . / /: Due to the existence of Serre’s p pseudomeasures which can be interpreted as elements in K1 ..G /S / for G abelian, it follows that LF1 =F satisfies the generalised Snaith congruences. But secondly by explicit Brauer induction the values of LF1 =F in the general case can be expressed by the values of suitable abelian L’s, hence implying the congruences. In both cases it is crucial that the Brauer induction can be arranged in a compatible way with respect to the p t h Adams operator, which in general does not behave well under induction. Theorem 5.1 ([RW5] Theorem B^ in Sect. 6). For G pro-p we have Det K1 .B.G // \ HomGQp ; R. / .R.G /; Zp . / / DetK1 ..G //: in HomGQp ; R. / .R.G /; BZp . / /: This should be compared to ˚B \

Y

.U ab / D ˚

U 2S.G ;Z /

in Kakde’s work [SV, Theorem 6.1, Theorem 6.2] and to Burns’ result [Bur, Theorem 6.1, Remark 6.2]—the latter approach avoids the analysis of integral logarithms. Thus, by the usual argument (compare [Kak4] Proposition 2.1, Theorem 2.1) the Main Conjecture is equivalent to showing that LF1 =F 2 Det K1 .B.G //:

(3)

In order to verify the latter condition, Ritter and Weiss introduce a new integral group logarithm, which makes it possible to translate this multiplicative statement in a additive statement plus a statement about the kernel of the integral group logarithm


171

which is referred to as the “torsion” part (since for a finite p-group G this kernel is actually a torsion group) even though in this setting it may contain a torsionfree part! Set TB WD B=ŒB; B for any ring B, where ŒB; B denotes the set of additive commutators ab ba; a; b 2 B, and consider the diagram

K1 .B.G // Det .1/

HomGQ

p ;R. /

L

/ TB.G / TB.G /Œ 1 p Š

.R.G /; BZp . / / f 7! L.f /. / D

L

Tr

/ HomG ;R. / .R.G /; B . /Œ 1 / Qp Zp p

f . /p 1 log. / p .f . p //

in which Lf is well-defined due to the congruence (2) (note that due the Galois invariance, for any f in HomGQp ;R. / .R.G /; BZp . / / the value f ./ belongs to some BOL . / for some finite extension L of Qp / and which defines the integral logarithm L D LG in the upper row following the approach of Snaith in [Sna]. The trace map Tr is the analogue of Det in the additive setting, see [RW5]. Question: Does L coincide with Kakde’s integral logarithm LB in [SV, Sect. 5]? As mentioned above now Ritter and Weiss divide the condition (3) into the question whether the “additive” element L.LF1=F / lies in Tr.im.L// and under which conditions a (torsion) element in ker.L/; viz the defect of LF1=F not being determined by L.LF1=F /; is in the image of Det? To this aim they introduce tF1=F WD T r 1 L.LF1=F / 2 TB.G /

1 ; p

and call it the logarithmic pseudomeasure. Theorem 5.2. LF1 =F 2 DetK1 .B.G // if and only if (i ) tF1 =F 2 TB.G / (integrality) and V (ii) verVU F ŒV;V =F V F1 =F1 U mod im. / (torsion congruence) U 1

1

for all U V G with U abelian, ŒV W U D p and where verVU is induced from verVU W V ab ! U ab by linear extension. Remark 5.1. Ritter and Weiss’ strategy of decomposing the problem into an additive/logarithmic and torsion part by various diagram chases should be compared with the use of the 5-lemma in Kakde’s approach with respect to the following diagram in [SV, between (6) and Lemma 4.6]

172

1

O. Venjakob

D

1

/ .O / G ab

L

/ K10 ..G //

/ O ŒŒConj.G /

ı

/ .O / G ab

L

/ ˚

/

/ G ab

/ 1

D

Š

ˇ

!

!ıˇ 1

/ G ab

/ 1: (4)

combined with diagram [SV, (9)] for the B-situation. Idea of Proof: (for the converse direction). If tF1 =F 2 TB.G /, then there exists y 2 K1 .B.G // such that tF1 =F D L.y/ (5) and y 7! F ŒG ;G =F

(6)

1

under the canonical map prG W K1 .B.G // ! K1 .B.G ab //. This follows immeG ab diately from the abelian case LG ab .F ŒG ;G =F / D tF ŒG ;G =F and the following 1 1 commutative diagram [RW10, Lemma 7.1 (ii)] with exact rows 1

/ 1Ca

0

/ .a/

/ / B.G ab /

/ B.G /

/ 0

LG ab

LG

/ TB.G ab /

/ TB.G /

tF1 =F

/ 0

/ t ŒG ; G F =F 1

where a WD ker.B.G / ! B.G ab // and .a/ is the image of a with respect to the canonical map W B.G / TB.G /. Setting ! WD Det.y/1 LF1 =F we have by (6) that !jR.G ab / 1 (7) and by the definition of tF 1 =F whence

L.!/ D 0; !. /p D 1; !. p / n

as logj1Cp BO . / is injective. Therefore !. /p D n !. pn / D n .!.1// .1/ D 1 L for n sufficiently big such that pn R.G =Z/ D f1g. That means that ! is a torsion element. In [RW4, Proposition 2.4] even uniqueness of ! is shown. We want to show that ! D 1. Assume first that G contains


173

an abelian subgroup G 0 of index p. Since then any irreducible representation of G is either inflated from an abelian character ˛ of G ab

D inflG G ab ˛ or induced from an abelian character ˇ of G 0 0

D indG G 0 .ˇ /;

it suffices by (7) to verify that 0 G !jIndG G 0 R.G / 1 or .IndG 0 / ! 1: But by the functoriality properties .IndG G 0 / LF1 =F

NGG0 .y/

D

.IndG G 0 / Det.y/

IndG G0

(8)

D LF1 =F G 0 and Det 1

we obtain

! G F1 =F1 G 0 IndG 0 LF1 =F D Det !D Det NGG0 .y/ NGG0 .y/

where NGG0 W K1 .B.G // ! K1 .B.G 0 // denotes the norm map. Since Det is injective on K1 .B.G 0 //, we see that F1 =F1 G 0 NGG0 .y/

is a torsion element. By the Wall-congruence ([RW5, proof of Lemma 12]) ab G G NGG0 .y/ verG G 0 prG ab .y/ mod im .G 0 / (which corresponds in this special case to (M3) in Kakde’s work, see [SV]: ab G NGG0 .G 0 .x// verG G 0 .G .x// mod im .G 0 /) it follows that by our choice of y G with prG ab .y/ D F1 ŒG ;G =F1 ab G F1 =F1 G 0 F1 =F1 G 0 verG G 0 prG ab .y/

D e WD G G ab NG 0 .y/ NG 0 .y/ F1 ŒG ;G =F1 verG G0 F1 =F G 0 .i i / 1 1 mod im.GG0 / ab G verG 0 .F ŒG ;G =F / 1

1

by our assumption. The Theorem follows from the Claim: If e 2 B.G 0 / is torsion and satisfies e 1 mod im.GG0 /, then e D 1; cp. with [SV, proof of Lemma 4.8]. By Higman’s theorem we have: e D h 1 mod im.GG0 / with h 2 H 0 (for the group ring B.G 0 / D B. 0 /ŒH 0 , if G 0 decomposes as 0 H 0 ). The augmentation

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map $ W B.G 0 / ! B. 0 / induces 1 mod pB. 0 /, whence D 1 and h0 1 2 0 im.GG /. Thus h0 D 1; e D 1 and (8) holds. By an inductive argument this argument can be extended to arbitrary G , see proof of Theorem in [RW4, Sect. 3]. t u The hard part is now to inductively show that tF1 =F 2 TB.G /

(9)

holds, which requires the new Möbius-Wall congruence X

G G =Z .U=Z /verUA .F ŒU;U =F U / 2 im.A / ab

A U G

1

1

for any abelian normal open subgroup A E G (actually for each such onedimensional subextension) introduced in [RW10], more precisely in (loc. cit.) only a similar relation for units in B.G / is called Möbius-Wall congruence. Recall that for a finite p-group G, the Möbius-function is defined inductively as follows G .1/ D 1 P G .U / D V Ü G .V / for 1 ¤ U G: How this condition enters the proof will be explained in the next section, in which we try first to abstract and formalise what the methods of Ritter and Weiss actually prove.

6 The Abstract Setting: A Reinterpretation of Ritter and Weiss’ Approach Fix a one dimensional pro-p-group G with projection onto Š Zp ; let H denote its kernel and define the following index sets S WD SG WD fU one-dimensional subquotient of G g D fU jU G =C open for some C E G with C H g and S ab WD SGab D fU 2 SG jU abeliang:


175

Note that for any U 2 S the quotientQ U ab also belongs to S and in particular to S ab : 2 e B to be the subgroup of Define ˚ consisting of those .U /U 2SGab ab B.U / U 2SG satisfying the following conditions: (RW1)

For every surjection U V in S ab we have prUV U D V ; where prUV W K1 .B.U // ! K1 .B.V // is the natural map induced by the projection, and for every inclusion V U in S ab we have NVU U D V ;

(RW2) (RW3)

where NVU denotes the norm map. For all U 2 SG the (sub-)tuple .V ab /V U is U -invariant. (Möbius-Wall congruence) For all U 2 SG and all abelian normal open subgroups A E U we have X

U U=A .V =A /verVA .V ab / 2 A .B.A //:

A V U

In particular the torsion congruence (RW3a) (Torsion congruence) For all U 2 SG and all abelian normal open subgroups A E U of index p we have U verUA .U ab / A 2 A .B.A //:

Actually Ritter and Weiss show that (RW3) holds for every tuple which arises from an element # 2 K1 .B.G //; see [RW10, Theorem 2]. Strictly speaking, they use the term Möbius-Wall congruence only for such #, but we extend this notation e B : The proof generalises Wall’s proof of (RW3a) by analysing the to tuples in ˚ Leibniz formula for determinants; for combinatorial reasons the Möbius function shows up. Is it by chance that this or a similar Möbius function also shows up in the explicit formula of Brauer induction? While the proof of (RW3) is rather tedious it is straightforward to check (RW1) and (RW2). Using explicit Brauer induction (as at the beginning of Sect. 4), for any U 2 SG e B ; or rather to its sub-tuple .V /V U ; elements one can assign to a tuple .V / 2 ˚ .1/

&U 2 HomGQ

2

p ;R.U /

.R.U /; BOL .U / /

As M. Kakde kindly pointed out to me it seems quite difficult to check directly that the set satisfying the conditions below forms a subgroup, so this becomes only clear a posteriori once it is shown that this set is the full image of K1 :

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and tU WD T r

1

1 ; .LU .&U // 2 TB.U / p

such that tU D LU .U / 2 TB.U / for all U 2 SGab : Indeed, for every onedimensional character of U (which is trivial on some central subgroup ZU of U ) aU ./ is represented by itself under explicit Brauer induction by Boltje [Bol]. Finally we require (RW4)

For any U 2 SG ; the definition of &U does not depend on the above chosen way by explicit Brauer induction, i.e., &U ./ D

Y

DetV ab .V ab /. V /nV ;

V

whenever D

X

nV IndVU V

in R.U / for certain subgroups V U; one-dimensional representations

V of V and (finitely many nonzero) integers nV : This conditions looks a little weird, but whenever one is interested in p-adic L-functions, it is completely harmless, as it is always satisfied by the usual behaviour of L-functions under induction. Questions: Do conditions (RW1–3) imply already (RW4)? Is there a way of proving the next Lemma 6.1 without requiring (RW4)? All what one needs to extend the proof of property (9) given by Ritter and Weiss to Theorem 6.2 below are the following functoriality properties. Lemma 6.1. prUV .&U / D &V and NVU .&U / D &V as well as similarly prUV .tU / D Q UV .tU / D tV for all possible U; V 2 SG ; where the modified trace Tr Q UV is tV and Tr introduced in (10) below. Proof. While it is well-known that explicit Brauer induction (given by aG as above) behaves well under inflation, it does not behave well under induction. Therefore we need at present condition (RW4) here to prove the correct behaviour under the norm: For V U; let X D nW IndW V W W V

in R.V /: Then, by the transitivity of induction we obtain Ind./ D

X

nW IndW U W

W V

in R.U /: By the definition of the norm on the Hom-description we thus have


177

.NVU &U /./ D &U .Ind.// Y DetW ab .W ab /. W /nW D W V

D &V ./: For pr W U V we obtain inflUV ./ D

X

0

0

W nW IndW U .inflW W /;

W V

where W 0 WD pr 1 .W / is the full preimage of W under pr: Hence prUV .&U /./ D &U .inflUV .// Y 0 nW DetW 0ab .W 0ab /.inflW D W W / W V

D

Y

0 ab

nW DetW ab .prW W ab .W 0 ab //. W /

W V

D

Y

DetW ab .W ab /. W /nW

W V

D &V ./: (the last property does not require (RW4)!). The corresponding statements for tU follow from the functorial properties of T r 1 ı LU , see [RW10, Lemma 7.2]. t u e B there exist G 2 K1 .B.G // such that Theorem 6.1. For every .U /U 2 ˚ DetG .G / D &G : Question: How big is the kernel of DetG ‹ In order to obtain (3) from this Theorem we just have to observe that for the tuple .U /U 2S ab consisting of the abelian pseudomeasures U D F1 C =F V if U D V =C 2 1 S ab for some subgroup V G D G.F1 =F / and some normal subgroup C E G contained in H the associated element &G equals LF1 =F , supposed of course e B : While (RW1), (RW2) and (RW4) are well-known that .U /U 2S ab belongs to ˚ properties, (RW3) forms a completely new property which is proved by Ritter and Weiss using the q-expansion principle of Deligne-Ribet. Now we are going to prove the Theorem: using (RW3a) and the analogue of Theorem 5.2, which also can be proved in this general setting, it suffices to prove Theorem 6.2. In the situation of the above Theorem we have tG 2 TB.G /:

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Idea of proof: Actually we shall show that tU 2 TB.U / for all U 2 SG by contraposition (alternatively one could formulate the argument using induction): If this statement is false we firstly may consider among the counterexamples those U for which the order of ŒU; U is minimal (this order must be different from one as the claim of the proposition holds for all abelian U ). Among those U we may assume that the order of ŒU W Z.U / is minimal, where Z.U / denotes the centre of U: Without loss of generality we may and do assume that these minimality conditions already hold for G itself. Then we choose a central element c 2 ŒG ; G of order p and set C WD< c > : Furthermore we choose a maximal (with respect to inclusion) abelian normal subgroup A of G ; which then automatically contains C: Also we fix a central subgroup Z Š Zp of G contained in A : In order to arrive at a contradiction we will use the following lemma of Ritter Q UV for V U denotes the modified trace map defined by and Weiss, in which Tr Ritter and Weiss (extending the definition in [Oli] to their Iwasawa theoretic Homdescription) to make the following diagram, in which NVU denotes the norm map, commutative LU

/ TB.U /

K1 .B.U //

(10)

QU Tr V

NVU

K1 .B.V //

/ TB.V /:

LV

Note that here we encounter another significant difference in comparison to Kakde’s approach, who uses in a similar situation the original trace map TrVU ; but then has to take a modified integral logarithm map e LV in order to obtain a similar commutative diagram (which then induces the diagram in [SV, Proposition 4.1 and (9)] involving ˚ and ). G

Q U for U in S.G ; A / WD fU 2 Lemma 6.2. Defining ˇQ D .ˇQU / by ˇQU D prUU ab Tr SG jA U g we obtain a commutative diagram of injective homomorphisms TB.G / _

ˇQ

/

Q

U 2S.G ;A / B.U

_

ab

/

Q 1 ˇŒ p / Q ab 1 TB.G /Œ p1 U 2S.G ;A / B.U /Œ p : For the desired contradiction it thus suffices to show that there exist a t 2 TB.G / Q D .tU ab /U 2S.G ;A / ; because then t D tG as tG is mapped to the same such that ˇ.t/ tuple under ˇQ by Lemma 6.1. We first will search for a t such that t 7! tA ; tG ab :

(11)


179

Setting GN WD G =C and AN WD A =C we define the ideals a and b by exactness of rows in the following commutative diagram 0

/ a O

/ B.G / O

/ B.GN/ O

/ 0

0

/ b

/ B.A /

/ B.AN/

/ 0:

Ritter and Weiss show that one has a commutative diagram / a

0

/ TB.G /

/ TB.GN/

QG Tr A

QG Tr AN

/ 0

G A .b/ _

/ B.A /

/ b

0

N

/ B.AN/

/ 0;

By the minimality of ŒG ; G we know that tGN belongs to TB.GN/: We choose any lift t D LG .#/ 2 TB.G / of tGN in the image of LG (which is possible by the same reasoning as in the proof of Theorem 5.2 with the additional property that prA N G .#/ D AN). Then the condition (RW3) for the tuple .U /U reads AN A X

G G =A .V =A /verVA .V ab / 2 A .B.A //

A V G

while one easily shows—using the fact that NAG .#/1 A lies in the kernel of B.A / ! B.AN/ —that for the tuple .prUU ab NUG .#//U induced by # it reads NAG .#/ C

X

G G =A .V =A /verVA .V ab / 2 A .B.A //

A ¤V G

since

verVA .prVV ab NVG .#// D verVA .V ab / for V ¤ A :

This implies that

G NAG .#/1 A 2 1 C A .B.A //;

from which Ritter and Weiss derive that G

G G G Q A .t/ tA D LG .NAG .#/1 x WD Tr A / 2 b \ A .B.A // D A .b/

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QG lies in the image of .a/ under Tr A : Upon modifying t accordingly, we may and do thus assume that QG Tr A .t/ D tA : as claimed above. In fact, it follows easily that then t is also mapped to tU ab under ˇQU for all U such that c is contained in ŒU; U : Otherwise we have C \ ŒU; U D 1; i.e., the order of ŒU; U is strictly smaller than that of ŒG ; G ; whence tU 2 TB.U / by assumption. We set G

Q U .t/ tU 2 TB.U /: xU WD xU .t/ WD Tr By our contraposition the support, i.e., the set supp.t/ WD fU 2 S.G ; A /jC \ ŒU; U D 1; xU .t/ ¤ 0g; is non-empty for all choices of t satisfying (11). Let t be such a choice with the order of supp.t/ being minimal and let U 2 supp.t/ have minimal ŒU W A : Explicit Q using [RW10, Lemma 5.1], the minimality of ŒG W Z.G / and calculations with Tr the uniqueness of expressing elements in TB.U / as linear combinations over B.Z / similarly as for the proof of ! cyc being injective in [SV, Lemma 3.4] show firstly that U=A must be cyclic and secondly that in this cyclic case one can modify t to an element t 0 (using [RW10, Sect. 6, claim 6.A]) with strictly smaller support than t; a contradiction. t u At the end we want to investigate whether for .G / itself instead of B.G / the e in terms of ./ satisfying again (RW1–4) describes, i.e., analogous group ˚ equals the image of K1 ..G //. From Kakde’s result we know that his condition (M4) (together with (M1–M3) at least) imply the Möbius-Wall congruence (RW3), even though there does not seem to be a direct link between them. For example (M4) involves only cyclic subgroups while (RW3) ranges over all subgroups among A and G : In fact we will now prove the converse, viz that (RW1–4) also implies (M1–4). Theorem 6.3. Q K10 ..G // WD K1 ..G //=SK1..G // Š ˚: Proof. Consider the commutative diagram with exact rows 0

/ G ab

0

/ ˚Q \ Q. U ab /

/ K 0 ..G // 1 Q

/ ˚Q

L

Q

LU

/ im.L/

/ 0

Q ˇjim.L/

/ im.˚/ Q

/ 0


181

analogous to (4). The map is an isomorphism by (RW3a) (cp. [SV, Lemma 4.8]) while ˇQ is an isomorphism by Theorem 6.2 combined with (5) and Lemma 6.2 above. Note that in contrast to Kakde’s approach an explicit description of the image of ˇQ is avoided in the above proof. As Kakde pointed out to me such description would be very messy, because the description of the image of Q is—among others—in terms of the additive congruences (RW3) which would translate into terms of the Q logarithm applied to them for the image of ˇ. Recently, also the kernel of LB.G / has been determined by Kakde [Kak2] using the result for abelian G by Ritter and Weiss in [RW9]: ker.LB.G / / D SK1 .B.G // G ab ; 1 SK1 .B.G // WD ker K1 .B.G // ! K1 .B.G /Œ / : p

where

Hence the same proof as above also shows the following Theorem 6.4. K10 .B.G // WD K1 .B.G //=SK1 .B.G // Š ˚Q B : Question: Does an analogous statement hold for A.G /‹ Finally we want to remark that—fixing a central subgroup Z Š Zp of G contained in —we may replace the infinite index sets SG ; SGab by the finite subsets SG ;Z WD fU jZ U G =C for some C E G with C H g (note that Z can be considered canonically as subgroup of G =C; because Z \C D 1) and SGab;Z WD fU 2 SG ;Z jU abeliang: e we obtain the same statements as in Theorems 6.3 For the corresponding ˚ and 6.4.

References [Bol] R. Boltje, A canonical Brauer induction formula. Astérisque 181–182(5), 31–59 (1990) [Bur] D. Burns, On main conjectures in non-commutative Iwasawa theory and related conjectures, Preprint (2011) [CFKSV] J. Coates, T. Fukaya, K. Kato, R. Sujatha, O. Venjakob, The GL2 main conjecture for elliptic curves without complex multiplication. Publ. Math. IHES. 101(1), 163–208 (2005)

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[FK] T. Fukaya, K. Kato, A formulation of conjectures on p-adic zeta functions in non-commutative Iwasawa theory, in Proceedings of the St. Petersburg Mathematical Society, vol. XII (AMS, Providence, 2006), pp. 1–86 (Amer. Math. Soc. Transl. Ser. 2, vol. 219) [Har] T. Hara, Iwasawa theory of totally real fields for certain non-commutative p-extensions. J. Number Theory 130(4), 1068–1097 (2010) [Kak1] M. Kakde, Proof of the main conjecture of noncommutative Iwasawa theory for totally real number fields in certain cases. J Algebr. Geom. 20, 631–683 (2011) [Kak2] M. Kakde, Integral logarithm for completions of a localisation of Iwasawa algebras, preprint (2011) [Kak3] M. Kakde, The main conjecture of Iwasawa theory for totally real fields, preprint (2011) [Kak4] M. Kakde, Congruences between abelian p-adic zeta functions, this volume (2012) [Kat] K. Kato, Iwasawa theory for totally real fields for Galois extensions of Heisenberg type, preprint (‘Very preliminary version’) (2007) [NSW] J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields. Grundlehren der mathematischen Wissenschaften, vol. 323 (Springer, Berlin/New York, 2000) [Nic] A. Nickel, Equivariant Iwasawa theory and non-abelian Stark-type conjectures, Preprint, arXiv:1109.5525 (2011) [Oli] R. Oliver, Whitehead Groups of Finite Groups. London Mathematical Society Lecture Note Series, vol. 132 (Cambridge University Press, Cambridge, 1988) [RW1] J. Ritter, A. Weiss, The lifted root number conjecture and Iwasawa theory. Mem. Am. Math. Soc. 157(748), viii+90 (2002) [RW2] J. Ritter, A. Weiss, Toward equivariant Iwasawa theory. Manuscr. Math. 109(2), 131–146 (2002) [RW3] J. Ritter, A. Weiss, Toward equivariant Iwasawa theory II. Indag. Math. (N.S.) 15(4), 549–572 (2004) [RW4] J. Ritter, A. Weiss, Toward equivariant Iwasawa theory IV. Homol. Homotopy Appl. 7(3), 155–171 (2005) [RW5] J. Ritter, A. Weiss, Toward equivariant Iwasawa theory III. Math. Ann. 336(1), 27–49 (2006) [RW6] J. Ritter, A. Weiss, Congruences between abelian pseudomeasures. Math. Res. Lett. 15(4), 715–725 (2008) [RW7] J. Ritter, A. Weiss, Equivariant Iwasawa theory: an example. Doc. Math. 13, 117–129 (2008) [RW8] J. Ritter, A. Weiss, Non-abelian pseudomeasures and congruences between abelian Iwasawa L-functions. Pure Appl. Math. Q. 4(4, Special Issue: In honor of Jean-Pierre Serre. Part 1), 1085–1106 (2008) [RW9] J. Ritter, A. Weiss, The integral logarithm in Iwasawa theory: an exercise. J. Théor. Nombres Bordeaux 22(1), 197–207 (2010) [RW10] J. Ritter, A. Weiss, On the ‘main conjecture’ of equivariant Iwasawa theory. J. Am. Math. Soc. 24, 1015–1050 (2011) [SV] P. Schneider, O. Venjakob, K1 of certain Iwasawa algebras – after Kakde, this volume (2012) [Sna] V.P. Snaith, Explicit Brauer induction. Cambridge Studies in Advanced Mathematics, vol. 40 (Cambridge University Press, Cambridge, 1994). With applications to algebra and number theory [Suj] R. Sujatha, Reductions of the main conjecture, this volume (2012) [Wei] C.A. Weibel, An introduction to algebraic K-theory (2011), http://math.rutgers.edu:80/ weibel/Kbook.html, in progress

Noncommutative Main Conjectures of Geometric Iwasawa Theory Malte Witte

Abstract In this chapter we give a survey on noncommutative main conjectures of Iwasawa theory in a geometric setting, i.e. for separated schemes of finite type over a finite field, as stated and proved by Burns and the author. We will also comment briefly on versions of the main conjecture for function fields.

MSCs: 14G10, 14G15, 11R23 Keywords Iwasawa theory • varieties over finite fields • L-functions

In this chapter we give a survey on noncommutative main conjectures of Iwasawa theory in a geometric setting, i.e. for a separated scheme of finite type over a finite field, as stated and proved in [Wit3] and [Bur2]. We begin by formulating the conjecture in Sect. 1 and then give a sketch of the proof in Sect. 2. In Sect. 3 we will comment on the special case that the scheme is smooth, geometrically connected, and of dimension 1, which corresponds to an analogue of the main conjecture for function fields.

1 Formulation of the Conjecture As a motivation, let us begin by recalling the formulation of the main conjecture in the number field case from [CK, Theorem 5.1]. Since the letter p is conventionally reserved to denote the characteristic of the base fields appearing later in the text, we

M. Witte () Universität Heidelberg, Mathematisches Institut, Im Neuenheimer Feld 288, Heidelberg, Germany e-mail: [email protected] J. Coates et al. (eds.), Noncommutative Iwasawa Main Conjectures over Totally Real Fields, Springer Proceedings in Mathematics & Statistics 29, DOI 10.1007/978-3-642-32199-3 7, © Springer-Verlag Berlin Heidelberg 2013

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will denote by ` the prime which is conventionally denoted by p in Iwasawa theory. Fix a totally real number field F and an admissible `-adic Lie extension F1 =F in the sense of [CK, Sect. 1]. (In particular, F1 is also totally real.) Let F denote the cyclotomic character. Theorem 1.1. Let F1 =F be unramified outside the finite set of primes ˙ (with ` 2 ˙) and assume its Galois group G contains no element of order `. If the generalized Iwasawa conjecture holds for F1 =F , then there exists a F1 =F 2 K1 ..G/S / satisfying the interpolation property F1 =F . Fn / D L˙ .; 1 n/ for all Artin representations of G and the equation @F1 =F D ŒX .F1 / ŒZ` in K1 ..G/; .G/S /. Our aim is to formulate an analogue of this theorem for a separated scheme X of finite type over the field Fq with q elements. For this, we need to introduce for any prime ` an appropriate notion of admissible `-adic Lie extensions, a replacement for the .G/-module X .F1 /, and the corresponding L-functions. This will be the content of the following paragraphs. To avoid technicalities, we will additionally assume that X is geometrically connected. However, we will not exclude the case ` D p, where p denotes the characteristic of Fq .

1.1 Admissible Lie Extensions In the situation of Theorem 1.1, let OF denote the ring of integers of F and W the open subscheme of Spec OF given by the open complement of ˙. The `-adic Lie group G is a factor group of the Galois group Gal.F˙ =F / of the maximal extension F˙ of F unramified outside ˙ within a fixed algebraic closure FN of F . The Galois group Gal.F˙ =F / is in geometric terms the e´ tale fundamental group 1eK t .W; / of the scheme W with respect to the geometric point W Spec FN ! W corresponding to the algebraic closure FN . The e´ tale fundamental group 1eK t .X; / with respect to a fixed geometric base point of X is in fact defined for any connected scheme. The open normal subgroups U Co 1eK t .X; / are in 1-1-correspondence with the isomorphism classes of finite connected pointed Galois coverings f W .Y; / ! .X; /

Geometric Iwasawa Theory

185

with Galois group Gal.Y =X / D AutX .Y / D 1eK t .X; /=U [Mil1, Chap. 1, Sect. 5]. In the following, we will allow ourselves to neglect the base points in our notation. We extend the above correspondence to closed normal subgroups V of 1eK t .X; / by writing f WY ! X for the projective system of X -schemes .fU W YU ! X / where U runs over the open normal subgroups of 1eK t .X; /=V and YU denotes the Galois cover associated to the preimage of U in the fundamental group of X . The pro-scheme Y will then be called a Galois extension of X with Galois group Gal.Y =X / D 1eK t .X; /=V There exists a precise analogue of the cyclotomic Z` -extension of a number field in the geometric setting, namely the unique Z` -extension Fq `1 =Fq of the base field. If ` ¤ p and if we suppose that X is connected and normal, this is in fact the only Z` -extension by the Katz-Lang finiteness theorem [KS, Theorem 2.8]. There are examples of non-normal X with additional Z` -extensions. For ` D p one can find in general infinitely many other Zp -extensions of X . The analogue of an admissible `-adic Lie extension of F is then given by the following Definition 1.1. An Galois extension Y ! X is defined to be an admissible `-adic Lie extension if the Galois group Gal.Y =X / is an `-adic Lie group and if Y ! X factors through X Fq Fq `1 . Comparing with the definition in the number field case [CK, Sect. 1], we note that the extension Y ! X is by definition (pro-)étale and therefore unramified. Of course, there is no analogue of a totally real extension for our scheme X and we may simply drop this extra condition.

1.2 Algebraic K-Theory We chose a slightly different approach to algebraic K-theory than the one given in [Suj, Sect. 2]. Let R be any ring and S R a left denominator set, i.e. a multiplicatively closed subset satisfying 1. (Ore condition) for each s 2 S , b 2 R there exist s 0 2 S , b 0 2 R such that b 0 s D s 0 b, and

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2. (Annihilator condition) for each s 2 S , b 2 R with bs D 0 there exists s 0 2 S with s 0 b D 0. Then the left quotient ring RS with respect to S exists and one can define a long exact localisation sequence @

K1 .R/ ! K1 .RS / ! K0 .R; RS / [WY]. If R is left noetherian, then the annihilator condition is implied by the Ore condition. A left denominator set is then also referred to as left Ore set. We can obtain the following explicit description of the sequence in terms of perfect complexes. Recall that a complex of modules over a ring R is said to be strictly perfect if it is bounded and all modules are finitely generated and projective. The complex is called perfect if it is quasi-isomorphic to a strictly perfect complex. Theorem 1.2. Let R be any ring and S R be a left denominator set. 1. The group K1 .R/ is as abelian group generated by symbols Œf where f is a quasi-automorphism of a perfect complex of R-modules P . Moreover, the following (possibly incomplete) list of relations is satisfied: a. ŒgCŒh D Œgıh if g and h are quasi-automorphisms of the same complex P . b. Œf D Œf 0 if there exists a quasi-isomorphism a such that the diagram f

P

/ P

a

Q

a

/ Q

f0

commutes in the derived category of complexes of R-modules. c. Œb D Œa C Œc if there exists an exact sequence 0 ! A ! B ! C ! 0 of perfect complexes such that the diagram 0

/ A

0

/ A

/ B a

/ B

/ C b

/ C

/ 0 c

/ 0

commutes in the non-derived category of complexes. 2. The group K1 .RS / is generated by symbols Œf where f is a morphism of a perfect complex of R-modules P such that the localisation fS is a


187

quasi-automorphism. The symbols Œf satisfy the same relations as the symbols ŒfS in description (1). 3. The group K0 .R; RS / is presented by generators ŒP for each perfect complex P of R-modules such that the localisation PS has vanishing cohomology and the relations a. ŒP D ŒQ if P and Q are quasi-isomorphic. b. ŒB D ŒA C ŒC if there exists a distinguished triangle A ! B ! C ! AŒ1 in the derived category of R-modules. 4. @W K1 .RS / ! K0 .R; RS / is given by @Œf D ŒCone.f / where Cone.f / denotes the cone of f . In the above description, one may replace every occurrence of perfect complex by strictly perfect complex. Proof. This follows from the construction of the localisation sequence from a cofibre sequence of certain Waldhausen categories [WY], the algebraic description of the first Postnikov section of the associated topological spaces [MT], the identicalness of K-theory and ‘derived’ K-theory in degrees 0 and 1 [Mur, Theorem 5.1], and the explicit description of the boundary homomorphism [Wit3, Theorem A.5]. (The normalisation of the boundary homomorphism differs from loc. cit. by 1 in order to agree with the sign convention in the other chapters of this book.) t u Remark 1.1. We stress that Theorem 1.2 might not give a presentation of K1 .R/ or K1 .RS /. We have only listed the relations which are obvious from the description as the kernel of a certain group homomorphism given in [MT]. The precise set of relations will not be needed in the subsequent arguments. The derived tensor product with an R0 -R-bimodule M which is finitely generated and projective as R0 -module defines a group homomorphism K1 .R/ ! K1 .R0 /. In order to define this map in terms of the above presentation with perfect complexes, we can use the following Lemma 1.1. Let f W P ! P be a endomorphism of a perfect complex P of R-modules. There exists a endomorphism f 0 of a strictly perfect complex Q and a quasi-isomorphism q such that the diagram Q

q

f0

Q

q

/ P / P

f

commutes in the derived category of complexes of R-modules.

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Proof. This follows from the well-known fact that the morphisms from a strictly perfect complex Q to a complex P in the derived category are the same as the set of homotopy classes of complex homomorphisms Q ! P . t u We may now define K1 .R/ ! K1 .R0 / by mapping a generator Œf to ŒM ˝R f 0 for any choice of f 0 as in the lemma. The construction extends to our presentation of K1 .RS / and K0 .R; RS / in the obvious way. Given an admissible `-adic Lie extension Y ! X we shall put as in the number field case G D Gal.Y =X /;

D Gal.Fq `1 =Fq /;

H D ker.G ! /:

In particular, G is a semidirect product of H and . We will write O .G/ for its Iwasawa algebra with coefficients in the valuation ring O of a finite field extension of Q` and O .G/S for the quotient ring with respect to Venjakob’s canonical Ore set S D ff 2 .G/W O .G/=O .G/f is finitely generated as .H /-moduleg: In most situations, we omit the subscript O. Considering the localisation sequence for S .G/, we can even prove that it gives rise to a split exact sequence @

0 ! K1 ..G// ! K1 ..G/S / ! K0 ..G/; .G/S / ! 0: [Wit5, Corollary 3.4]. We also recall that K1 ..G// D lim K1 .O=`n OŒG=U / n;U Co G

carries a natural structure of a profinite group [FK, Proposition 1.5.3]. As in [Kak1], we define SK1 ..G// D lim SK1 .OŒG=U /; U Co G

K10 ..G//

D K1 ..G//=SK1 ..G//;

K10 ..G/S /

D K1 ..G/S /=SK1..G//:

If W G ! GLn .O/ is a continuous character for the valuation ring O of any finite field extension of Q` , we obtain an evaluation map tw

pr

det

˚ W K1 ..G/S / ! K1 ..G/S / ! K1 .. /S / ! . / S: Š


189

Here, . /S is the localisation of . / at the prime ideal generated by the maximal ideal of O, the map pr is the canonical projection, and the map tw is given by f

id˝f

! P D Œ ˝O P ! ˝O P tw ŒP on morphisms f of strictly perfect complexes P . If W G ! GLn .O/ has finite image, it is immediate that ˚ .SK1 ..G/// D 1 and hence, ˚ factors through K10 ..G/S /. We stress that none of the results of this section are specific to the geometric nature of our conjecture. They apply equally well to the number field case.

1.3 A Crash Course in Etale Cohomology Next, we need to find an analogue of the .G/-module X .F1 / that features in the main conjecture for number fields. For this, it is necessary to take a step back and have a look at the larger picture. In the following, we will make heavy use of the formalism of e´ tale cohomology. ´ We have to refer the reader to [Mil1] for a thorough introduction. Etale cohomology is a cohomology theory in the spirit of sheaf cohomology on topological spaces. It is related to the cohomology of the e´ tale fundamental group just as this sheaf cohomology is related to the cohomology of the classical fundamental group. For any noetherian ring R and any scheme X (of finite type over a sufficiently nice base scheme, e. g. Spec Fq or Spec Z) one defines a certain abelian category of constructible e´ tale sheaves of R-modules Sh.X; R/ together with • A global section functor eK t .X; / assigning an R-module to any sheaf in Sh.X; R/, • Constructions of inverse image functors f W Sh.X 0 ; R/ ! Sh.X; R/ and direct image functors f W Sh.X; R/ ! Sh.X 0 ; R/ for morphisms f W X ! X 0 , • An extension-by-zero functor jŠ W Sh.X; R/ ! Sh.X 0 ; R/ for open immersions j W X ! X 0, • A tensor product F ˝R G of sheaves F and G in Sh.X; R/, • For any sheaf F on X a Godement resolution GX .F /, i.e. a complex of flasque sheaves that may be used to define higher derived functors. We may apply the section functor to each degree of the Godement resolution to define a total derived section functor ReK t .X; F / D eK t .X; GX .F // and likewise, we may also define a total higher derived image functor Rf . If f W X ! Y is a separated morphism of finite type over a noetherian scheme Y , we can fix a commutative diagram

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j

X

/ Xc BB BB f c BB BB f Y

with an open immersion j and a proper morphism f c . We may then define a total derived image functor with proper support RfŠ F D fc GX c .jŠ F / as an analogue of the total derived image functor with compact support from topology. (One can prove that this construction is independent of the choice of X c up to quasi-isomorphism, see [Del, Arcata, IV, Sect. 5].) If X is separated and of finite type over Fq , we may apply this construction to the structure morphism sW X ! Spec Fq to define a total derived section functor with proper support Rc .X; F / D eK t .Spec Fq ; RsŠ F /: This construction works fine for finite rings R, but it does not give the right cohomology groups if we apply it directly to rings such as Z` or .G/. Instead, we have go a step further and define a continuous version of it. The following definition is a straight-forward generalisation of [Del, Rapport, Definition 2.1] (see also [Gro, Exp. VI]). Definition 1.2. Let R be a profinite ring. A flat R-sheaf on X is a projective system .FI / indexed by the open (two-sided) ideals of R such that FI is a flat sheaf in Sh.X; R=I / and such that for I J the transition morphism FI ! FJ factorises through an isomorphism R=J ˝R=I FI Š FJ : O We denote the category of flat R-sheafs by Sh.X; R/. The above constructions of Rf and RfŠ extend to R-sheaves by applying them to each element of the projective system individually. We redefine the total derived section functor by additionally taking the total derived inverse limit of the resulting projective system of complexes. In order to give an R-sheaf on X , it suffices to know it on a cofinal system of open ideals. In particular, if F D .F.`n / / is a flat Z` -sheaf and f W Y ! X is an admissible `-adic Lie extension with Galois group G, we obtain a flat .G/-sheaf FG D .fU fU F.`n / /U Co G;n>0 :


191

To check the conditions in Definition 1.2, it suffices to note that the stalk .fU fU F.`n / / in a geometric point of X is isomorphic to Z=`n ZŒG=U ˝Z=`n Z .F /. Moreover, we may assign to each continuous Z` -representation W 1eK t .X; / ! GLk .Z` / a flat Z` -sheaf M ./ D .1 ˝. Ket .X;// fUn fUn Z=`n Z/n0 1

by choosing Un D ker. 1eK t .X; / ! GLk .Z=`n Z//. In this way, the category O Z` /. More of continuous Z` -representations becomes a full subcategory of Sh.X; generally, we may replace Z` by any compact noetherian commutative ring. If X is the e´ tale analogue of a K. ; 1/-space, e.g. a smooth affine curve over a finite field of characteristic prime to ` or an open dense subscheme of Spec OF Œ 1` , then the e´ tale cohomology ReK t .X; M .// agrees with the continuous group cohomology of , but in general, it is the e´ tale cohomology and not the group cohomology that leads to the right constructions. The following complex is our replacement for the module X .F1 /. We will explain this in more detail in Sect. 3. Definition 1.3. For any flat Z` -sheaf F we set C.Y =X; F / D Rc .X; FG / By construction C.Y =X; F / is a complex of .G/-modules. We can say even more: Proposition 1.1. The complex C.Y =X; F / is a perfect complex of .G/-modules, i.e. there exists a bounded complex Q of finitely generated, projective .G/modules and a quasi-isomorphism qW Q ! C.Y =X; F /. Proof. By Deligne [Del, p. 95, Theorem 4.9], the complex Rc .X; fU fU F / is a perfect complex of Z=`n ZŒG=U -modules for any flat e´ tale sheaf of Z=`n Zmodules F . To pass from this statement to the statement of the proposition is not completely straightforward. We give some details, following [Gro, XV, p. 472– 492] and clarifying [Bur2, Proposition 3.1.(ii)]. A slightly different proof is given in [Wit1, Proposition 5.2.3 C Definition 5.4.13]. Both proofs use that .G/ is compact for the topology defined by the powers of the Jacobson radical Jac..G//: .G/ D lim .G/=Jac..G//n : n In particular, G admits a fundamental system of neighbourhoods .Un / of open normal subgroups indexed by the positive integers such that the kernel of

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.G/ ! Z=`ZŒG=U1 is contained in Jac..G//. The Künneth formula for the trivial product of X with the spectrum of its base field implies that there exists a quasi-isomorphism Z=`n ZŒG=Un ˝LZ=`nC1 ZŒG=U

nC1

Rc .X; fUnC1 fUnC1 F.`nC1 / /

# Rc .X; Z=`n ZŒG=Un ˝Z=`nC1 ZŒG=UnC1 fUnC1 fUnC1 F.`nC1 / / k Rc .X; fUn fUn F.`n / /: By successively applying [Gro, XV, 3.3, Lemma 1] (the proof of which uses lifting of idempotents for .G/ as a central ingredient) one finds a quasi-isomorphism of projective system of complexes .qn /W .Qn / ! .Rc .X; fUn fUn F.`n / // with Qn being a strictly perfect complex of Z=`n ZŒG=Un -modules such that ! Qn Z=`n ZŒG=Un ˝Z=`nC1 ZŒG=UnC1 QnC1

is an isomorphism (not merely a quasi-isomorphism) of complexes. We set Q D lim Qn : n

Using that projective systems of finite modules are lim-acyclic [Jen, Corollary 7.2], we obtain a quasi-isomorphism qW Q ! C.Y =X; F /: Each Qk is projective as a compact .G/-module [Bru, Corollary 3.3] and the complex Q is bounded. Moreover, since the transition morphisms in the system .Qn / are surjective, we have O .G/ Q D lim Z=`ZŒG=U1 ˝Z=`n ZŒG=Un Qn D Q1 Z=`ZŒG=U1 ˝ n [Bru, Lemma A.4]. Hence, each Qk is finitely generated by the topological Nakayama lemma. In particular, Qk is also projective as abstract .G/-module. t u The Künneth-formula for the trivial product as in the proof above shows that C.Y =X; F / behaves well with respect to derived tensor products. In particular, we have quasi-isomorphisms


193

.Gal.Y 0 =X // ˝L.Gal.Y =X // C.Y =X; F / ' C.Y 0 =X; F / for any subextension Y 0 =X of Y =X . Moreover, ˝LZ` C.Y =X; F / ' C.Y =X; M ./ ˝Z` F / for any continuous `-adic representation of Gal.Y =X /.

1.4 L-Functions In this subsection we will recall Grothendieck’s and Deligne’s fundamental results on L-functions for Z` -sheaves on the scheme X over Fq . In the case ` D p this is complemented by a result of Emmerton and Kisin [EK]. Let X be a geometrically connected scheme of finite type over Fq . We let X 0 denote the set of closed points of X . For any x 2 X 0 we let deg.x/ denote the degree of the residue field k.x/ of x over Fq and N.x/ D q deg.x/ the order of k.x/. N Furthermore, we fix an algebraic closure k.x/ of k.x/ and let denote Fx 2 1eK t .X; / 1 N its geometric Frobenius element acting by a 7! aN.x/ for a 2 k.x/. For any representation W 1eK t .X; / ! GLn .OE / on the ring of integers OE of a number field E, we may define a complex function LA .; s/ D

Y

Ex .N.x/s /1 ;

Ex .t/ D det.1 .Fx /t/

x2X 0

as direct analogue of an Artin L-function for a number field. Different from the number field case, we obtain LA .; s/ by evaluating the formal power series with OE -coefficients Y Ex .t deg.x/ /1 L.; t/ D x2X 0

in q s . In particular, we may view L.; t/ as an element of OŒŒt for any completion O of OE at a prime of E. (We warn the reader that this is not yet the precise analogue of the Iwasawa power series of an `-adic L-function.) More generally, one may define L.F ; t/ 2 OŒŒt for any compact commutative noetherian Z` -algebra O and any flat O-sheaf F by taking the product over the inverses of the characteristic polynomials of the Frobenius operation on the stalks of F . There is a second power series that we may attach to the O-sheaf F . Recall that the determinant induces an isomorphism detW K1 .OŒŒt/ ! OŒŒt :

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Let sW X ! Spec Fq denote the structure morphism. By the same argument as in the proof of Proposition 1.1, the complex eK t .FN q ; RsŠ F / is a perfect complex of O-modules with an action of the Frobenius F 2 Gal.FN q =Fq /. We may set LŠ .F ; t/ D detŒ1 tF W OŒŒt ˝O eK t .FN q ; RsŠ F /1 2 OŒŒt : The interesting property of LŠ .F ; t/ is that it is in fact a rational function. To see this, we use Lemma 1.1 to replace F by an endomorphism f on a strictly perfect complex Q . By the relations given in Theorem 1.2 we may then write LŠ .F ; t/ D

Y

detŒ1 tf W OŒŒt ˝O Qn .1/

nC1

:

n2Z

We introduce the power series v.F ; t/ D L.F ; t/=LŠ .F ; t/ to measure the difference. Furthermore, we set Ohti D lim O=Jac.O/n Œt n

and write Oftg for the localisation of Ohti at the multiplicatively closed subset of those elements which become a unit in OŒŒt. In particular, we have LŠ .F ; t/ 2 Oftg : Theorem 1.3. Let ` be any prime, O a compact, commutative, noetherian Z` -algebra and F an O-sheaf on X . 1. (Grothendieck-Deligne) If ` ¤ p then v.F ; t/ D 1. 2. (Emmerton-Kisin) If ` D p then v.F ; t/ 2 Ohti . In particular, we have L.F ; t/ 2 Oftg in both cases. Proof. Assertion .1/ follows from [Del, Fonctions L mod `, Theorem 2.2] by passing to the projective limit. Assertion .2/ for q D p is [EK, Corollary 1.8]. It remains true for q D p n because the v for a scheme X=Fq considered as a scheme over Fp is obtained from the original v by replacing t by t n . t u In fact, Theorem 1.3.(1) remains valid for non-commutative coefficients [Wit2]. The formulation of a reasonable non-commutative version of Theorem 1.3.(2) poses additional technical difficulties related to the fact that for non-commutative O, the map K1 .Ohti/ ! K1 .OŒŒt/ might not be injective. We use the above theorem to construct the true analogue of the Iwasawa power series of the classical `-adic L-function.


195

Lemma 1.2. Let 2 Š Z` be a topological generator, O a compact, commutative, noetherian Z` -algebra. The assignment t 7! 1 defines a ring homomorphism Oftg ! . /S : Proof. Clearly, t 7! 1 defines a homomorphism Ohti ! . /. Let f .t/ 2 Ohti be a unit in OŒŒt. We need to show that f . 1 / 2 S . The set S consists of those elements in . / whose reduction modulo every maximal ideal m of O is not zero. Now the reduction of f . 1 / modulo m is a polynomial in 1 whose constant coefficient is a unit in O=m. t u In particular, we may apply this lemma to D Gal.Fq `1 =Fq / with being the image of the Frobenius F 2 Gal.FN q =Fq /. We then have elements L.F ; 1 /;

LŠ .F ; 1 / 2 . / S:

for every flat O-sheaf F on X . Assume that ` ¤ p and let W Gal.FN q =Fq / ! Z ` denote the cyclotomic character. For any Artin character W 1eK t .X; / ! GLk .OE / the elements L.M ./; 1 / satisfy the interpolation property L.M . n /; 1/ D LA .; n/: for n 2 Z. (This holds also for the leading terms.) For ` D p one should not expect to obtain an analogous result, for in the p-adic world, the Tate twist of the motive defined by does not correspond to a continuous character , but takes on a very different shape. However, we may still view L.F ; 1 / as an interpolation of the leading terms of L.M ./ ˝Z` F ; t/ at t D 1.

1.5 The Main Conjecture We are now ready to formulate the non-commutative main conjecture of Iwasawa theory in the geometric case. For any profinite group G, let C.G/ denote the set of continuous representations W G ! GLk .O / for the valuation ring O of a finite field extension of Qp . Likewise, we let A.G/ denote the set of all Artin representations W G ! GLk .O /, i.e. representations with finite image. Theorem 1.4 ([Wit3]). Fix any prime `. Let f W Y ! X be an admissible `-adic Lie extension of a geometrically connected scheme X of finite type over Fq . Set G D Gal.Y =X /. For any flat Z` -sheaf F , the cohomology of C.Y =X; F / is S -torsion and there exists an element Š .F / 2 K1 .Z` .G/S / such that 1. @Š .F / D ŒC.Y =X; F / in K0 .Z` .G/; Z` .G/S /, 2. ˚ .Š .F // D LŠ .M ./ ˝Z` F ; 1 / in O . / S for any continuous representation 2 C.G/.

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Recall that for ` ¤ p, we have LŠ .F ; t/ D L.F ; t/ in the commutative situation. Thus, in this case, the above result is indeed a full analogue of Theorem 1.1. Note that we do not need to assume a D 0-conjecture. We can directly prove the much stronger S -torsion property. In fact, the result in [Wit3] is more general than the one stated above. By replacing the fundamental group by an arbitrary principal covering, one may also deal with nonconnected schemes. Moreover, the theorem still holds if G is no longer an `-adic Lie group, but topologically finitely generated and virtually pro-`. This applies for example to the maximal pro-`-quotient of the fundamental group 1eK t .X; /. We may also replace Z` by more general coefficient rings, for example, the valuation ring of any finite field extension of Q` . For ` D p, Burns proves the following result: Theorem 1.5 ([Bur2]). Let p ¤ 2. Let f W Y ! X be an admissible p-adic Lie extension of a geometrically connected scheme X of finite type over Fq . Set G D Gal.Y =X /. For any flat Zp -sheaf F , there exists an element .F / 2 K1 .Zp .G/S / such that 1. @.F / D ŒC.Y =X; F / in K0 .Zp .G/; Zp .G/S /, 2. ˚ ..F // D L.M ./ ˝Zp F ; 1 / in O . / S for any Artin representation 2 A.G/. Again, the restriction to geometrically connected schemes and to `-adic Lie groups may easily be lifted. However, it is not clear that .F / satisfies the interpolation property with respect to all continuous representations. The exclusion of the prime 2 in the statement is a purely technical restriction. There is no reason to expect any odd behaviour for the even prime. Also for technical reasons, we may not replace Zp by the valuation ring of any finite extension of Qp . At present, we can only deal with unramified extensions. Remark 1.2. Note that both [Wit3] and [Bur2] use the inverse sign convention for @.

2 Sketch of Proofs We will give a fairly detailed sketch of the proofs of Theorems 1.4 and 1.5 below. In some details we will deviate from the original proofs in [Wit3] and [Bur2], but the general ideas remain the same.

2.1 On the Proof of Theorem 1.4 We begin by assuming that we already know that C.Y =X; F / is S -torsion. The proof of the existence of the non-commutative L-function Š .F / is very different from the number field case: We are in the happy position to be able to give an explicit construction of a hot candidate for Š .F /. The proof then boils down to verifying that this candidate does indeed satisfy the required interpolation property.


197

Let X be the base change of X to the algebraic closure FN q of Fq and set C .Y =X; F / D Rc .X ; FG /: The geometric Frobenius F 2 Gal.FN q =Fq / acts on this complex and from the Hochschild-Serre spectral sequence it follows that there exists a distinguished triangle idF

C .Y =X; F / ! C .Y =X; F / ! C.Y =X; F /Œ1: Since C.Y =X; F / is S -torsion we see by Theorem 1.2 that Œid F is a class in K1 ..G/S /. We take the inverse of it as our definition of Š .F /. Since C.Y =X; F / is the cone of id F shifted by 1 it follows again by Theorem 1.2 that @Š .F / D ŒC.Y =X; F /. The construction of Œid F is compatible with taking derived tensor products. In particular, we have ˚ .Š .F // D Š .M ./ ˝Z` F / 2 K1 .. /S / for any continuous representation 2 C.G/. It remains to verify that Š .F / D LŠ .F ; 1 / for any flat Z` -sheaf F and the cyclotomic Z` -extension X cyc =X . This follows from the commutativity of the diagram

. / ˝LZ` Rc .X ; F /

1 ˝F

Rc .X ; F /

F

/ . / ˝L Rc .X ; F / Z`

(1)

/ Rc .X; F /

in the derived category of complexes of . /-modules, with denoting the canonical quasi-isomorphism. We will now give a sketch of the proof of the S -torsion property. As before, let H D ker.G ! /. The general case is easily reduced to the case that H is finite by considering quotients by open subgroups of H which are normal in G. From now on, we assume that H is finite. In particular, Y may be viewed as a scheme of finite type over Fq `1 . Moreover, in this situation, the cohomology groups of C.Y =X; F / are S -torsion precisely if they are finitely generated as Z` -modules. Hence, the S torsion property follows from Proposition 2.1. Assume that H is finite. For each integer i, Gal.FN q =Fq `1 / : Hi C1 .C.Y =X; F // Š Hic .Y; F / Š Hic .Y Fq`1 FN q ; F /

In particular, the cohomology groups of C.Y =X; F / are finitely generated as Z` -modules.

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Proof. We first note that Hi .C.Y =X; F // Š lim Hic .V; F / V

where the limit goes over the finite Galois subextensions of Y =X . From this, we see that we may replace X by an appropriate V and enlarge Fq if necessary such that we may assume that Y is the cyclotomic Z` -extension of X and that Y Fq`1 FN q D X : The cohomology groups Hic .X; F / are known to be finitely generated Z` -modules, even in the case ` D p [Del, p. 84 Sect. 2.10] (or use Proposition 1.1 for G D 1). Since Gal.FN q =Fq `1 / is of order prime to ` the Hochschild-Serre spectral sequence gives us Hic .X; F /

Gal.FN q =Fq `1 /

D Hic .Y; F /;

which is still finitely generated over Z` . Let Xn denote the subextension of degree `n in Y =X . From the Hochschild-Serre spectral sequence we obtain a short exact sequence 0 ! Hic .Y; F / `n ! HicC1 .Xn ; F / ! HicC1 .Y; F /

`n

!0

As HicC1 .Y; F / is finitely generated over the noetherian ring Z` , the increasing `n family of submodules HicC1 .Y; F / becomes stationary. This means in turn that the inverse limit with respect to the norm maps vanishes. Hence, Hi C1 .C.Y =X; F // Š lim Hic .Y; F / `n n is the compactification of the . /-module Hic .Y; F /. However, this module, being finitely generated over Z` , is already compact and therefore, lim Hic .Y; F / `n D Hic .Y; F /: n

t u

2.2 On the Proof of Theorem 1.5 In the view of Theorem 1.4 we are reduced to showing Proposition 2.2. Let Y =X be an admissible p-adic Lie extension with Galois group G. For any flat Zp -sheaf F on X , there exists a unique .F / 2 K10 .Zp .G// such that


199

˚ . .F // D v.M ./ ˝Zp F ; 1 / in O . / for every Artin representation of G. We extend the definition of v.F ; t/ to the non-commutative world by setting K1 .AŒŒt/ 3 v.F ; t/ D Œid F tW AŒŒt ˝A Rc .X ; F / X Œid Fx t deg x W AŒŒt ˝A Fx x2X 0

for any compact noetherian ring A (not necessarily commutative) and any flat A-sheaf F . Note that the sum converges in the profinite topology of K1 .AŒŒt/ because there are only finitely many closed points of X of a given degree. Note further that the definition of v.F ; t/ is compatible with derived tensor products in the following sense: If B is another compact noetherian ring and P a B-A-bimodule which is finitely generated and projective as B-module, then the image of v.F ; t/ under the map P ˝L A

K1 .AŒŒt/ ! K1 .BŒŒt/ is v.P ˝A F ; t/. If A is commutative, then v.F ; t/ 2 Ahti AŒŒt D K1 .AŒŒt/ by Proposition 1.3. In particular, we may consider its image v.F ; 1/ 2 A under t 7!1

the homomorphism Ahti ! A. Proposition 2.3. Assume that Y =X is an admissible p-adic Lie extension with G D Gal.Y =X / abelian. For any compact commutative noetherian Zp -algebra O and any flat O-sheaf F , the element .F / D v.FG ; 1/ validates the interpolation property of Proposition 2.2. Proof. Choose A D .G/, B D O . /. By the compatibility with derived tensor products we conclude that ˚ . .F // D .M ./ ˝Zp F / D v..M ./ ˝Zp F / ; 1/: One then checks using the diagram (1) on page 197 that v..M ./ ˝Zp F / ; t/ D v.M ./ ˝Zp F ; 1 t/: t u Unfortunately, it is a priori not clear that we can evaluate the element v.FG ; t/ in 1 if G is not commutative. Even if we knew that v.FG ; t/ was in the image of K1 ..G/hti/ ! K1 ..G/ŒŒt/, the map could still be so far from injective that

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the evaluation of an element in the preimage of v.FG ; t/ in t D 1 depends on the particular choice of it. However, we will prove below that the evaluation is possible for a p-adic Lie group G under the hypotheses (H1) G contains an open central subgroup Z, (H2) G is a pro-p-group, (H3) There exists a system of representatives R G for the cosets in G=Z which contains 1 and consists of full G-orbits, (H4) O is the valuation ring of a finite unramified extension of Qp and p ¤ 2 introduced in [SV]. Recall the notation S .G; Z/ D fU W Z U Gg and the homomorphism ZG W K1 ..G// !

Y

.U ab /

U 2S .G;Z/

from [SV, Sect. 4]. We also recall from loc. cit., Theorem 4.1, that under the above hypotheses the kernel of ZG is the group SK1 ..G// and that its image consists of precisely those elements satisfying the congruence conditions .M1/–.M 4/ given in loc. cit., before Lemma 4.5. Note that if G satisfies .H1/–.H 3/ then so does G Zp , with Z replaced by Z Zp . Moreover, we may canonically identify S .G Zp ; Z Zp / Š S .G; Z/ via the projection map. Using the isomorphism .G Zp / Š .G/ŒŒt that maps GZ 1 2 Zp to the power series 1 t and the map ZZpp we obtain a homomorphism t W K1 ..G/ŒŒt/ !

Y

.U ab /ŒŒt :

U 2S .G;Z/

We let K1˘ ..G/ŒŒt/ denote the preimage of Y

.U ab /hti

U 2S .G;Z/

under the homomorphism t . Proposition 2.4. Assume .H1/–.H 4/. There exists a unique homomorphism " filling the commutative diagram


201

t

K1˘ ..G/ŒŒt/

Q /

.U ab /hti

U 2S .G;Z/ t 7!1

"

K10 ..G//

ZG

.U ab / :

Q /

U 2S .G;Z/

Proof. Since the homomorphism ZG in the diagram is injective, it suffices to show that the evaluation in t D 1 of an element in the image of t lies in the image of ZG . As noted above, the images of t and ZG may be described by a list of explicit congruences .M1/–.M 4/. Let f .t/ D .fU .t//U 2S .G;Z/ 2 t .K1˘ ..G/ŒŒt//: We will exemplarily check that f .1/ satisfies .M 3/

verVU .fV .1// fU .1/ D UV .x/

for some x 2 .U ab / if ŒV W U D p:

Here, UV W .U ab / ! .U ab /;

x 7!

X

gxg 1 ;

g2V =U

and verVU W .V ab / ! .U ab / is the unique continuous ring homomorphism which coincides with the transfer map on V ab and with the absolute Frobenius automorphism on the coefficient ring O. Fix U V 2 S .G; Z/ with ŒV W U D p and let .g; a/ 2 V ab Zp . Then V Z

verU Zpp .g; a/ D .verVU .g/; ŒV W U a/; in particular,

V Z

verU Zpp .t/ D 1 .1 t/p V Z

for the indeterminate t. We conclude that the power series verU Zpp .fV .t// lies in .U ab /hti and that its evaluation in 1 agrees with verVU .fV .1//. By assumption, there exist a power series x.t/ D

1 X i D0

xi t i 2 .U ab /ŒŒt

202

M. Witte

with V Z

V Z

verU Zpp .fV .t// fU .t/ D U Zpp .x.t// D

X

gx.t/g 1 2 .U ab /hti:

g2V =U

Hence, for every n, UV .xi / 2 Jac..U ab //n for almost all i . Since .U ab / is compact, the image of the continuous map UV is closed and therefore, there exists an x 2 .U ab / with UV .x/ D

1 X

UV .xi / D verVU .fV .1// fU .1/:

i D0

Similarly, one shows that f .1/ also satisfies .M1/, .M 2/, and .M 4/. In each case, the main step is to verify that each of the maps involved in the formulation of the respective congruence relations are compatible with the evaluation in 1. This can be achieved by a quick inspection of the corresponding definitions given in [SV]. u t Proof (Proposition 2.2). Using the same reduction arguments as in [Suj, Sect. 4] and in [Kak2, Proposition 2.1] we see that it suffices to consider the cases 1. G is abelian, 2. H is a finite p-group, The first case has been settled in Proposition 2.3. Therefore, assume that H is a finite p-group. Using Proposition 2.4, we may set

.F / D ".v.FG ; t//: The interpolation property is then verified as in Proposition 2.3. This concludes the proof of Proposition 2.2 and hence, also the proof of Theorem 1.5. t u Note that our t differs from the one used in [Bur2]. Instead of using the identification .G/ŒŒt Š .G Zp / Burns considers .G/ŒŒt as an Iwasawa algebra with coefficient ring OŒŒt and transfers the logarithm techniques of Oliver and Taylor to power series rings. This also yields slightly different congruence relations, but the difference vanishes if one evaluates in t D 1. We also remark that ker t D SK1 ..G Zp // D SK1 ..G// by Witte [Wit2, Proposition 5.3].


203

3 Main Conjectures for Function Fields In this section, we will consider the special case that X is a geometrically connected smooth affine curve over Fq . Let F denote the function field of X , X c the smooth compact curve with function field F and ˙ D X c n X . Every admissible `-adic Lie extension Y =X may then be seen as a Galois extension F1 =F such that 1. the Galois group G of F1 =F is an `-adic Lie group, 2. F1 =F is unramified outside ˙, 3. F1 contains Fq `1 F . Assume that ` ¤ p and let Z` .1/ be the Z` -sheaf on X corresponding to the cyclotomic character Gal.FN q =Fq / ! Z ` . We set C.F1 =F / D C.Y =X; Z` .1//Œ3: Poitou-Tate duality then implies C.F1 =F / D RHom.ReK t .Y; Q` =Z` /; Q` =Z` /; in perfect accordance with the number field case [Kak1, Sect. 2.3]. Since Y is K. ; 1/, one may replace the appearance of e´ tale cohomology of Y by the Galois cohomology of 1eK t .Y; / if one desires. One of the major differences between number fields and function fields is that Spec Z has no sensible compactification in the category of schemes. In particular, the standard construction of the total derived section functor with proper support does not work. However, the above duality statement explains why the complex C.F1 =F / that appears in Kakde’s work is a sensible replacement. As in the number field case, we let X .F1 / denote the Galois group of the maximal abelian `-extension of F1 unramified outside ˙. A quick calculation then shows H1 .C.F1 =F // D X .F1 /; H0 .C.F1 =F // D Z` ; Hi .C.F1 =F // D 0

otherwise.

Beware that the module X .F1 / in itself may have no finite projective resolution and therefore, no well-defined class in K0 ..G/; .G/S / if G has elements of order `. If we additionally assume that such elements do not exist, then ŒC.F1 =F / D ŒZ` ŒX .F1 / in K0 ..G/; .G/S / and Theorem 1.4 gives the precise equivalent of Theorem 1.1 plus the vanishing of the -invariant.

204

M. Witte

A precise equivalent of the non-commutative Iwasawa main conjecture for elliptic curves over number fields [CFKSV] can be deduced from Theorem 1.4 by considering the Z` -sheaf on X given by the `-adic Tate module (` ¤ p) of an elliptic curve or, more generally, any abelian variety over F . More details are given in [Wit4]. The case ` D p looks more difficult and seems very different in nature. Theorem 1.5 may be applied to the constant sheaf Zp on X to deduce a main conjecture for the leading terms of the Artin L-functions LA .; s/ in s D 0. The cohomology of the complex C.Y =X; Zp / is concentrated in degree 2 and its .G/-dual is related to the inverse limit of the p-parts of the class groups of the intermediate fields [Bur1, Proposition 4.1]. However, we cannot apply Burns’ theorem to obtain interpolations of the leading terms in s D n for arbitrary n as the natural constructions of the Tate twists Zp .n/ via Bloch’s cycle complexes do not give Zp -sheaves in our sense. Moreover, the formulas for the leading terms of the zeta functions deduced by Milne [Mil2] and others hint that there is a ‘tangent space’ contribution from the DeRham-complex which should appear in the boundary term of a non-commutative p-adic L-function at non-zero Tate twists. An equivariant generalisation of Milne’s formulas is still missing. Completing the work of Ochiai and Trihan [OT], a noncommutative main conjecture for abelian varieties over F in the case ` D p is formulated in [TV] and there is progress towards a proof, at least under certain hypotheses. The conjectured interpolation property is only for the leading terms of the L-functions of Artin twists of the abelian variety in s D 1. Instead of e´ tale cohomology the authors use flat cohomology as in [KT]. The boundary of the conjectured non-commutative L-function is given by the flat total derived section complex of the (flat) p-adic Tate module plus a tangent space contribution given by the total derived sections of the Lie algebra of the abelian variety. One might be tempted to apply Theorem 1.5 to the Zp -sheaf on X given by the (étale) p-adic Tate module, but the L-function of this sheaf differs from the L-function of the abelian variety and it is not clear how to handle the difference. Fixing a place p of the function field F , one may speculate about yet another, completely different, approach to formulate an analogue of the Iwasawa main conjecture, with the valuation ring Ap of Fp taking over the role of Zp . One can define a characteristic p version of an L-function taking values in the completion of the algebraic closure of Fp [Gos] and one can prove formulas for special values of characteristic p L-functions [Laf]. According to the general philosophy, a possible analogue of the Iwasawa main conjecture should then give information about the limit of these values if F varies in a suitable family of field extensions. Such a suitable family might be given by the extensions obtained from Drinfeld-Hayes modules. These are widely regarded as the right analogues of cyclotomic extensions in this setting.


205

References [Bru] A. Brumer, Pseudocompact algebras, profinite groups, and class formations. J. Algebra 4, 442–470 (1966) [Bur1] D. Burns, Congruences between derivatives of geometric L-functions, Invent. math 184, 221–256 (2011) [Bur2] D. Burns, On main conjectures of geometric Iwasawa theory and related conjectures (2011). Preprint (Version 6). Available at www.mth.kcl.ac.uk/staff/dj burns/gmcrcvers6.pdf [CFKSV] J. Coates, T. Fukaya, K. Kato, R. Sujatha, O. Venjakob, The GL2 main conjecture for elliptic curves without complex multiplication. Publ. Math. Inst. Hautes Etudes Sci. 101, 163–208 (2005) [CK] J. Coates, D. Kim, Introduction to the work of M. Kakde on the non-commutative main conjectures for totally real fields, in this volume (2012) [Del] P. Deligne, Cohomologie e´ tale (SGA 4 12 ). Lecture Notes in Mathematics, vol. 569 (Springer, Berlin, 1977) [EK] M. Emerton, M. Kisin, Unit L-functions and a conjecture of Katz. Ann. Math. 153(2), 329–354 (2001) [FK] T. Fukaya, K. Kato, A formulation of conjectures on p-adic zeta functions in noncommutative Iwasawa theory, in Proceedings of the St. Petersburg Mathematical Society, (Amer. Math. Soc. Transl. Ser. 2), vol. XII, pp. 1–85 (American Mathematical Society, Providence, 2006) [Gos] D. Goss, Basic structures of function field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 35 (Springer, Berlin, 1996) [Gro] A. Grothendieck, Cohomologie `-adique et fonctions L (SGA 5). Lecture Notes in Mathematics, vol. 589 (Springer, Berlin, 1977) [Jen] C.U. Jensen, Les foncteurs dérivés de lim et leurs applications en théorie des modules. Lecture Notes in Mathematics, vol. 254 (Springer, Berlin, 1972) [Kak1] M. Kakde, The main conjecture of Iwasawa theory for totally real fields (2011). Preprint (Version 3). arXiv:1008.0142v3 [Kak2] M. Kakde, Congruences between abelian p-adic zeta functions, in this volume (2012) [KT] K. Kato, F. Trihan, On the conjectures of Birch and Swinnerton-Dyer in characteristic p > 0. Invent. Math. 153, 537–592 (2003) [KS] M. Kerz, A. Schmidt, Covering data and higher dimensional global class field theory. J. Number Theory 129(10), 2569–2599 (2009) [Laf] V. Lafforgue, Valeurs spéciales des fonctions L en caractéristique p. J. Number Theory 129(10), 2600–2634 (2009) [Mil1] J.S. Milne, Etale Cohomology. Princeton Mathematical Series, vol. 33 (Princeton University Press, New Jersey, 1980) [Mil2] J.S. Milne, Values of zeta functions of varieties over finite fields. Am. J. Math. 108, 297–360 (1986) [Mur] F. Muro, Maltsiniotis’s first conjecture for K1 . Int. Math. Res. Not. 2008, 31 p., rnm153–31 (2008) [MT] F.Muro, A. Tonks, The 1-type of a Waldhausen K-theory spectrum. Adv. Math. 216(1), 178–211 (2007) [OT] T. Ochiai, F. Trihan, On the Selmer groups of abelian varieties over function fields of characteristic p > 0. Math. Proc. Camb. Philos. Soc. 146(1), 23–43 (2009) [SV] P. Schneider, O. Venjakob, K1 of certain Iwasawa algebras, after Kakde, in this volume (2012) [Suj] R. Sujatha, Reductions of the main conjecture, in this volume (2012) [TV] F. Trihan, D. Vauclair, On the Iwasawa main conjecture of abelian varieties over function fields of characteristic p > 0. (2011). In preparation

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