The Geometry and Cohomology of Some Simple Shimura Varieties. (Annals of Mathematics Studies) - Softcover

Harris, Michael

Softcover

ISBN 10: 0691090920 ISBN 13: 9780691090924

Verlag: Princeton University Press, 2001

Inhaltsangabe

This book aims first to prove the local Langlands conjecture for GLn over a p-adic field and, second, to identify the action of the decomposition group at a prime of bad reduction on the l-adic cohomology of the "simple" Shimura varieties. These two problems go hand in hand. The results represent a major advance in algebraic number theory, finally proving the conjecture first proposed in Langlands's 1969 Washington lecture as a non-abelian generalization of local class field theory. The local Langlands conjecture for GLn (K), where K is a p-adic field, asserts the existence of a correspondence, with certain formal properties, relating n-dimensional representations of the Galois group of K with the representation theory of the locally compact group GLn (K). This book constructs a candidate for such a local Langlands correspondence on the vanishing cycles attached to the bad reduction over the integer ring of K of a certain family of Shimura varieties. And it proves that this is roughly compatible with the global Galois correspondence realized on the cohomology of the same Shimura varieties. The local Langlands conjecture is obtained as a corollary. Certain techniques developed in this book should extend to more general Shimura varieties, providing new instances of the local Langlands conjecture. Moreover, the geometry of the special fibers is strictly analogous to that of Shimura curves and can be expected to have applications to a variety of questions in number theory.

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The Geometry and Cohomology of Some Simple Shimura Varieties

By Michael Harris, Richard Taylor

PRINCETON UNIVERSITY PRESS

Copyright © 2001 Princeton University Press
All rights reserved.

ISBN: 978-0-691-09092-4

Contents

Introduction, 1, 
Acknowledgements, 15, 
I Preliminaries, 17, 
II Barsotti-Tate groups, 59, 
III Some simple Shimura varieties, 89, 
IV Igusa varieties, 121, 
V Counting Points, 149, 
VI Automorphic forms, 195, 
VII Applications, 217, 
Appendix. A result on vanishing cycles by V. G. Berkovich, 257, 
Bibliography, 261, 
Index, 269, 

CHAPTER 1

Preliminaries

In this chapter we will establish some notation, recall some standard facts and prove a couple of technical lemmas. The reader may like to read only section 1.7 (which establishes some very important assumptions which are particular to this book) and simply refer back to the rest of this chapter as the need arises.

1.1 General notation

In this section we will introduce some notation which we will use throughout this book.

We will let p and l denote rational primes. We will often assume that l ≠ p. In particular from the middle of section III.4 onwards this will always be the case. Let Z(p) denote the ring of elements of Q with denominator coprime to p, valp the p-adic valuation (so that valp (p) = 1) and | "|p the p-adic absolute value (so that |p|p = 1/p).

If X is a scheme and x is a point of X we will let k(x) denote the residue field at x. We will let Ox,x denote the local ring of X at x and we will let O^X,x denote its completion at its maximal ideal. If Y [subset] X is a locally closed subscheme we will let X^Y denote the completion of X along Y. For instance X^x- = Spf O^X,x. If X is a locally noetherian formal scheme then X has a unique largest ideal of definition I. The formal scheme with the same underlying topological space as X and with structure sheaf Ox/I is in fact a scheme which we will refer to as the reduced subscheme of X, and will denote Xred. (See section 10.5 of [EGAI].)

If L is a field we will let Lac denote its algebraic closure. If L'/L is a finite field extension we will let NL'/L denote the norm from L' to L. If X / L' is a scheme we will write RSL'L X for the restriction of scalars of X to L, i.e. (RSL'L X)(S) = X(S XSpec L Spec L') for any L-scheme S.

If k is a field and A/k is an abelian variety we will let T A denote the Tate module of A, i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the limit is over all positive integers N. We will also introduce the eharaeteristic zero version of the Tate module

VA = T A [cross product]Z Q.

If S is a finite set of rational primes we will let TS A and VS A denote the "away from S" Tate modules, i.e.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the limit is over all positive integers coprime to S, and VS A = TS A [cross product]Z Q. We similarly define the "at S" Tate modules TS A and VS A.

If A and A'/S are abelian schemes then by an isogeny α : A -> A' we shalI mean an invertible element of Horn (A, A') [cross product]Z Q. We will denote Horn (A, A) [cross product]Z Q by End0 (A). By a polarisation [lamba] of A we shall mean a homomorphism λ : A -> Av such that for each geometric point s of S the homomorphism λs is a polarisation in the usual sense. If p is a rational prime then by a prime-to-p-isogeny we shall mean an invertible element of Horn (A, A') [cross product]Z Z(p). By a prime-to-p-polarisation of A we shall mean a polarisation λ : A -> Av which is also a prime-to-p-isogeny.

If R is an Fp-algebra we will let Fr : R -> R denote the Frobenius morphism which takes x [member of] R to xp [member of] R. If X/Fp is a scheme we will let Fr* : X -> X denote the Frobenius morphism indueed by Fr on structure sheaves. If Y -> X is a morphism of schemes over Fp then we will let Y(p)/X (or simply Y(p), if no confusion seems likely to arise) denote the pullback of Y by Fr* : X -> X. We will also let FY/X : Y -> Y/(p)/X (or simply F : Y -> Y(p) when no confusion seems likely to arise) denote the relative Frobenius, i.e. the morphism that arises from Fr : Y -> Y and the universal property of the pull back Y/(p)/X. If Y / X is a finite flat group scheme then we will let V : Y(p) -> Y denote the dual of F : Y[disjunction] -> Y[disjunction],(p) = Y(p),[disjunction], where Y[disjunction] is the Cartier dual of Y. This definition then extends to p-divisible groups Y/X. The morphism V : Y(p) -> Y induces a morphism of quasi-coherent sheaves of Ox-modules

V* : (Fr*)* Lie Y [congruent to] Lie Y(p) -> Lie Y.

Composing this with the natural map Fr : Lie Y -> (Fr*)*Lie Y we get a map, which we will also denote V*, from Lie Y to itself over X, which satisfies

V* (xy) = xpV*(y)

for x a section of Ox and for y a section of Lie Y.

If k/Fp is a finite extension we will let Frobk E Gal (kac/k) denote [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i.e. it will denote a geometric Frobenius element.

Throughout this book K will denote a p-adic field, i.e. a finite extension of Qp. We will let νK : Kx -> Z denote its unique valuation which is normalised to send uniformisers to 1. We will let OK denote its ring of integers, PK the unique maximal ideal of OK and k(vK) = k(PK) = OK/PK its residue field. We will often use [??]K to denote a uniformiser in OK. We will define an absolute value [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] on K by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

for x [member of] Kx. We will let Knr denote the maximal unramified extension of K. We will let [??]ac denote the completion of the algebraic closure of K and [??]nr denote the completion of Knr.

We will let IK [subset] Gal (Kac/K) denote the inertia subgroup, so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We will let WK [subset] Gal (Kac/K) denote the Weil group, i.e. the inverse image in Gal (Kac/K) of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We will write [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and will without comment think of it as an element of WK/IK. If σ [member of] WK then we define vK(σ) by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We will let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. If g is a positive integer we will let DK,g denote the division algebra with...

Details zur Ausgabe

• Verlag: Princeton University Press
• Erscheinungsdatum: 2001
• Sprache: Englisch
• ISBN 10: 0691090920
• ISBN 13: 9780691090924
• Einband: Taschenbuch
• Anzahl der Seiten: 288
• Kontakt zum Hersteller: preigu GmbH & Co. KG; mail@preigu.de; ; Lengericher Landstr. 19; Osnabrück; Niedersachsen; 49078; Deutschland