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The Geometry and Cohomology of Some Simple Shimura Varieties

PRINCETON UNIVERSITY PRESS

Contents

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 centre K and Hasse invariant 1/g. The algebra DK,g has a unique maximal order, which we will denote [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We will let IIK,g denote a uniformiser in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We will let AK,g denote the subgroup of

GLg(K) × D×K,g × WK

consisting of elements (γ, δ, σ) such that the valuation 01 the reduced norm δ equals vK (det γ) - vK(σ). This group will be of particular importance for us.

Local class field theory gives us a canonical isomorphism

ArtK : Kx [??] WabK.

There is a choice of sign in the definition of ArtK. We will choose a normalisation which makes uniformisers and geometric Frobenius elements correspond. If σ [member of] WK we will define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We will let c denote the non-trivial element of Gal (C/R). We take ArtC : C× -> Gal (C/C) to be the trivial homomorphism and we take

ArtR : R× /R×>0 [??] Gal (C/R)

to be the unique isomorphism. We take R. to be the usual absolute value on R and C to be the square of the usual absolute value on C, i.e. |z|C = |z|2 = |zzc|R.

If L is a number field we will let AL denote the adeles of L. If S is a finite set of places of L we decompose AL = ASL × LS where ASL denotes the adeles away from S and where LS = Πx[member of]S LX. Also let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where L' runs over finite extensions of L and where S(L') denotes the set of places of L' above S. The product of the normalised absolute values gives a homomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Global class field theory tells us that the product of the local Artin maps gives an isomorphism

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where (Lx∞)0 denotes the connected component of the identity in Lx∞.

Suppose that ι : Qacl [??] C. If

[??] : A×L /L× -> C×

is a continuous character, we will call ψ algebraic if we can find integers nσ for each embedding σ : L [??] C such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In this case there is a unique continuous character

reclι(ψ) = rec(ψ): Gal (Lac/L) -> (Qacl)×

such that for any finite place × Xl of L we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

More explicitly rel,ι(ψ) = ψ' o Art-1L where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If M is any Gal (Lac / L)-module we will let

ker1 (L,M)

denote the subset of H1 (L, M) consisting of elements which become trivial in H1(Lx, M) for every place x of L.

If L is a CM field we will let c denote complex conjugation on L, i.e. the unique automorphism of L which coincides with complex conjugation on C for any embedding L [??] C.

Let ε [member of] Mn (Z) denote the idempotent

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus for any ring R we have isomorphisms

• Mn(R)ε [congruent to] Rn via the map sending x to its first column;

• εMn(R) [congruent to] (Rn)[disjunction] via the map sending x to its first row;

• the map Mn(R)ε [cross product]R εMn(R) [??] Mn(R) sending xε [cross product] εy to xεy is an isomorphism.

If x [member of] Mn(R) then we will let xt denote its transpose.

1.2 Generalities on representations

If G is a group and g [member of] G we will let

• Z (G) denote the centre of G,

• ZG(g) denote the centraliser of g in G,

• and [g] the conjugacy class of g in G.

If π is a representation of G we will let Wπ denote the vector space on which the image of π acts, and we will let ψπ denote the central character of π (if it has one).

Now suppose that G is a topological group such that every neighbourhood of the identity contains a compact open subgroup. Suppose also that Ω is an algebraicalIy closed field of characteristic 0. Then we will let

C∞c(G)

denote the space of locally constant Ω-valued functions on G with compact support. If [??] : Z(G) -> Ω× is a smooth character of Z(G) then we will let

C∞c(G, ψ)

denote the space of locally constant Ω-valued functions φ on G such that

• φ(zg) = ψ(z)φ(g) for all z [member of] Z(G) and g [member of] G

• and the image of the support of φ in G/Z(G) is compact.

We may choose a (left or right) Haar measure μ on G such that every compact subgroup of G has measure in Q. Then we may speak of an Ω-valued Haar measure meaning a non-zero element of Ωμ. If φ [member of] C∞c(G), if π is an admissible representation G over Ω and if we fix an (Ω-valued) Haar measure on G then we have a well defined endomorphism π(φ) of Wπ. The endomorphism π(φ) has finite rank and so π(φ) makes sense. Similarly if ψ is an admissible character of Z(G), if φ [member of] C∞c(G, ψ-1), if π is an admissible representation G with central character ψ and if we fix Haar measures on G and Z(G) then again we have a welI defined endomorphism π(φ) of Wπ. Again the endomorphism π(φ) has finite rank and so tr π(φ) makes sense. If φ [member of] C∞c(G) or C∞c(G, ψ) and if we fix Haar measures on G and ZG(g) then we will let denote the integral

OGg(φ)

denote the integral

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

if this integral converges. All these notations depend on a choice of Haar measure(s) which we are suppressing. We will try to make clear in the accompanying text which measures we have chosen. If the choice of Haar measure on a particular group enters twice into a particular formula we will always suppose that we make the same choice both times, unless there is an explicit statement to the contrary. In such cases it will often be irrelevant which choice we make, only that we make a consistent choice.

Now suppose that H/K is a reductive algebraic group. Let P [subset] H be a parabolic subgroup with unipotent radical N [subset] P. If π is an admissible representation of (P/N)(K) then we define an admissible representation IndHP (π) of H(K) as follows. The underlying space will be the set of functions f : H(K) ->Wπ such that

• f(hg) = π(h)(f(g)) for all g [member of] H(K) and h [member of] P(K);

• there exists an open subgroup U [subset] H(K) such that f(gu) = f(g) for all u [member of] U and g [member of] H(K).

If f is such a function and g [member of] H(K) then we set

(g(f))(h) = f(hg)

for all h [member of] H(K). Conversely if 7r is an admissible representation of G(K) then the space of N(K)-coinvariants Wπ,N(K) is naturally an admissible representation πN of (P/N)(K).

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Excerpted from The Geometry and Cohomology of Some Simple Shimura Varieties by Michael Harris, Richard Taylor. Copyright © 2001 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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