Algebraic modular forms - Dartmouth College

ISRAEL JOURNAL OF MATHEMATICS 113 (1999), 61-93

ALGEBRAIC MODULAR FORMS

BY

BENEDICT H. GROSS

Department of Mathematics, Harvard University

Cambridge, MA 02138-2901, USA

e-mail: [email protected]

ABSTRACT

In this paper, we develop an algebraic theory of modular forms, for con

nected, reductive groups G over Q with the property that every arithmetic

subgroup r of G(Q) is finite. This theory includes our previous work [15]

on semi-simple groups G with G(R) compact, as well as the theory of alge

braic Heeke characters for Serre tori [20]. The theory of algebraic modular

forms leads to a workable theory of modular forms (mod p), which we hope

can be used to parameterize odd modular Galois representations.

The theory developed here was inspired by a letter of Serre to Tate in

1987, which has appeared recently [21]. I want to thank Serre for sending

me a copy of this letter, and for many helpful discussions on the topic.

TABLE OF CoNTENTS

0. Notation Chapter I: Groups

1. Groups ...

2. The root datum of G . . . . .

3. Irreducible representations of G Chapter II: Algebraic Modular Forms

4. Modular forms . . .

5. Serre tori . . . . . . . . . .

6. Operators and an inner product

7. Simple submodules . . .

8. Completions and lattices

9. Reduction (mod p) ... 10. Algebraic Heeke characters

11. The trivial representation .

12. The Steinberg space

Chapter III: £-Group Parameters

13. The dual group, and the co-character '7/

14. Conjugacy classes in Lc . . . . . .

Received October 17, 1997

61

62

62

65 67

68 69 71

74 75

77

78

80 80

82

84

62 B. H. GROSS

15. Archimedean parameters . . .

16. Unramified parameters . . . .

Chapter IV: Galois Representations 17. A global conjecture . . . . .

18. Steinberg parameters . . . .

19. The local representation at f. .

References

0. Notation

Isr. J. Math.

85 86

88

89 90

92

Let Z = I~ ZjnZ and Q = Z ®z Q. Then Q is the Q-algebra of finite adeles, n

and A= R x Q is the ring of adeles of Q.

Let G be a connected, reductive group over Q. We write G(Q) for the Q

rational points of G, and G(A) = G(R) x G(Q) for the group of adelic points.

An element g in G(A) has components g00 in G(R) and g in G(Q). We let G(R)+ be the connected component of the identity in the Lie group G(R).

Let V be an irreducible representation of the algebraic group Gover Q. We let D = Endc(V) be the endomorphism ring of V, which is a division algebra of

finite dimension over Q, and let F be the center of D.

Let Q be a fixed algebraic closure of Q. Each embedding Q -t C defines a

complex conjugation T in Gal(Q/Q), and these involutions form a single con

jugacy class in the Galois group. Let 6. be the closed subgroup of Gal(Q/Q)

generated by products T1 T2 of two complex conjugations, and let Q t:. be the fixed

field. Then complex conjugation defines a non-trivial central involution T of Qe:,. We call a finite extension E of Q which embeds into Qe:, a CM field, and put

E+ ={a E E: T(a) =a}. If E = E+, we say E is totally real.

Chapter I: Groups

1. Groups

Let G be a connected reductive group over Q, let S be the maximal split torus

in the center of G, and letS' be the maximal quotient of G which is a split torus.

The composite mapS -t G -t S' is an isogeny of tori; put n = dim(S) = dim(S').

Unicity of prime factorization gives a decomposition:

(1.1) Gm(A) =A*= Q* x R~ x z•. We let N(a) in Q* be the first factor in this decomposition of a, and llall in R+ be the second factor in this decomposition.

Vol. 113, 1999 ALGEBRAIC MODULAR FORMS 63

Once we fix an isomorphismS'~ G~, we get a continuous homomorphism

(1.2) G(A)---+ S'(A) ~ Gm(A)n ---t(R* )n. 1111 +

The kernel G(Ah of this map is independent of the isomorphism chosen, and

contains G(Q) by (1.1). A basic result, due to Borel and Harish-Chandra [5], is

the following.

PROPOSITION 1.3: The subgroup G(Q) is discrete in the locally compact group

G(A)I, and the quotient space G(Q)\G(A)I has finite Haar measure.

The quotient space is compact if and only if S is a maximal split torus in G

over Q.

Our main result in this section gives a number of equivalent conditions on G, which insure that every arithmetic subgroup r of G(Q) is finite. Some of these

conditions appear in [19, Ch. VI].

PROPOSITION 1.4: The following conditions are all equivalent: (1) Every arithmetic subgroupf ofG(Q) is finite.

(2) r = {e} is an arithmetic subgroup ofG(Q).

(3) G(Q) is a discrete subgroup of the locally compact group G(Q).

· (4) G(Q) is a discrete subgroup of the locally compact group G(Q), and the

quotient space G(Q)\G(Q) is compact.

(5) S is a maximal split torus in Gover R. (6) The Lie group G(R)I = G(R) n G(A)I is a maximal compact subgroup of

G(R).

(7) For every irreducible representation V of G there is a character p,: G ---+ Gm

and a positive definite symmetric bilinear form (,): V x V ---+ Q which satisfy

(rv,1v') = p,(r)(v,v'),

for all 1 in G(Q) and v, v' in V.

Note 1.5: We will see that the character p, of Gin part (7) is determined by the

representation V. The bilinear form is not; however, if V ® R is an irreducible

representation of G(R), the forni (,) is unique up to scaling by Q+.

Proof: Since S---+ S' is an isogeny over Q, the induced map S(R)+ ---+ S'(R)+

is an isomorphism of real Lie groups. Hence we obtain product decompositions

(1.6) { G(A) = S(R)+ x G(A)I, G(R) = S(R)+ x G(R)I.

64 B. H. GROSS Isr. J. Math.

Write g00 = 8 00 x h00 in the second decomposition, and define a homomorphism

G(A) --+ G(Rh x G(Q) taking g = g00 x g to h00 x g. The restriction of this

map to the subgroup G(Ah gives an isomorphism of locally compact topological

groups

(1.7) G(A)l ~ G(R)l x G(Q).

We view G(Q) as a discrete subgroup of the product G(Rh x G(Q) via (1.7);

the quotient has finite Haar measure by Proposition 1.3.

By (1.6), the conditions (5) and (6) are easily seen to be equivalent. We now

show that they are equivalent to (3) and (4). If Sis a maximal split torus over

R, it is certainly a maximal split torus over Q. By Proposition 1.3, the quotient

space G(Q)\G(Rh x G(Q) is compact. Let K C G(Q) be an open compact

subgroup. Since G(Rh is assumed compact, the intersection

f = G(Q) n (G(R)l x K)

in G(Rh x G(Q) is both discrete and compact, so is finite. Shrinking K, we may

assume r = 1, so G(Q) is discrete in G(Q). The quotient space G(Q)\G(Q) is

the image of a compact space under a continuous projection map, so is compact.

Conversely, assume (3), so G(Q) is discrete in G(Q), and choose an open,

compact subgroup K C G(Q) with G(Q) n K = 1. Then G(Rh x K is an open and closed subgroup of G(Rh x G(Q), and G(Q) n (G(Rh x K) = 1. Since the

quotient G(Q)\G(Rh x G(Q) has finite Haar measure, the group G(Rh x K has

finite Haar measure. Hence G(Rh is compact; it is a maximal compact subgroup

of G(R) by (1.6). Then Sis a maximal split torus over Q, which shows that the

quotient G(Q)\G(Q) is actually compact, by Proposition 1.3.

We now show the equivalence of (1), (2) and (6). Clearly (1) implies (2), as a

subgroup of finite index in an arithmetic subgroup is arithmetic. Also (2) implies

(6), as the quotient f\G(Rh = G(Rh has finite Haar measure [3]. Finally, if

G(Rh is compact, then any arithmetic r is discrete in G(R)l, so is finite.

Finally, we will show that (6) is equivalent to (7). Let V be an irreducible

representation of G. Assume that G(Rh is compact. Then the usual averaging

argument gives a positive-definite symmetric bilinear form

(, )R: (V 0 R) 0 (V 0 R) --+ R

which is G(R)l-invariant. We can even find such a form (, ): V 0 V--+ Q, as the

positive-definite forms form an open cone in Sym2 (V* 0 R) = Sym2 (V*) 0 R.

Vol. 113, 1999 ALGEBRAIC MODULAR FORMS

Let x: S --t Gm be the central character of V. By (1.6) we have

{goov,goov') = x(soo) 2. {v, v')

for all 9oo in G(R) and v, v' E V 0 R. Hence

65

for all1 in G(Q) and v, v' in V, where JL: G --t Gm is the unique character of G whose restriction to S is equal to x2

. Conversely, assume that (7) holds. Then

the subgroup G(R)Jl = {goo E G(R): JL(9oo) = 1} maps to a closed subgroup

of the orthogonal group of V 0 R. Hence the quotient of G(R)Jl by the kernel

of the action of G(R) on V is compact. Since this is true for all irreducible

representations V of G, one concludes easily that G(R)r is compact.

2. The root datum of G

We henceforth assume that G is a group satisfying the equivalent conditions of

Proposition 1.4. Let Q be a fixed algebraic closure of Q, and define the based root

datum 1/J = (X•, ~ •, X.,~.) of Gas in [4] or [16]. The Galois group Gal(Q/Q)

acts on 1/J. Let k be the finite extension of Q which is fixed by the kernel of this

action. We call k the splitting field of G, as the quasi-split inner form G0 of G is

split over k. The abelian group x• can be viewed as the character group (over

k) of a Levi factor To of a Borel subgroup B 0 C G0 over Q.

The Weyl group W of G over Q is a Coxeter group with generators so: indexed

by the roots a in~ •. It admits an action of Gal(k/Q) permuting the generators,

and the semi-direct product W ><l Gal(k/Q) acts on the free Z-modules X. and

x•. The maps S --t G --t S' induce injections

(2.1) {

x.(S) --t x.

x•(s') --t x·

with image the subgroups fixed by W ><l Gal(k/Q).

PROPOSITION 2.2: The splitting field k of G is contained in Qb., so is a CM

field.

Let r be complex conjugation in Gal(k/Q) and let w be the longest element

of W. Then e = w X T is a central involution in w )<l Gal(k/Q), and the maps (2.1) induce isomorphisms:

X.(S) = (X.)Ii=l,

x•(s') = (x•)li=l.


Proof: The splitting field for '1/J is the same as the splitting field of the root datum

'lj;* for G* = GIS. Similarly, the Weyl group W of G is the same as the Weyl

group of G*. Since the subgroups X.(S) and x•(S') are fixed by W ><1 Gal(kiQ),

they are certainly fixed by e. Hence it suffices to prove the result for G*, where

S* = 1. Here we must show that B* = -1 on the character and co-character

groups of 'lj;*.

When G = G*, the real group G(R) is compact. Fix a complex embedding

Q ~ C, letT C G be a maximal torus over R, and let B be a Borel subgroup

containing T over C. We may identify x• with x• (T) such that ~ • maps to

~ • (T, B); this identification does not respect the Galois actions.

Since B n B = T, B = w(B) is the opposite Borel subgroup with roots-~·.

Hence e = w X Ton x· gives the action of Gal(CIR) on x•(T). Since T(R) is

compact, Gal(CIR) acts as -1 on x•(T), so e acts as -1 on x•.

Since all complex conjugations in Gal(kiQ) act as -won x•, k is a CM field.

To determine when k is totally real, we let M = ffid2_l Md(1- d) be the Artin

Tate motive of G, defined in [12]. We recall that Md is the Gal(kiQ)-module

of primitive W-invariants of degree din the symmetric algebra on x• 0 Q. We

have M 1 = qn EB Mt, with n = dim(S) and Mt the rational character group of

the maximal anisotropic torus in the center of G. Similarly, M = qn EB M*, with

M* the motive of G* = GIS.

PROPOSITION 2.3: The splitting field k is totally real if and only if Mt = 0 and

Md = 0 for all odd d 2: 3.

The value L(M) = L(M, 0) of the Artin L-function of M at s = 0 is non-zero

and rational.

Proof: Since L(M) = ((O)n L(M*) and ((0) = -112, we are reduced to the case

when G = G*. Then Tin Gal(kiQ) acts as (-1)d on Md, and k is totally real if

and only if the primitive invariants all have even degree.

Since L(M) = [Jd2.l L(Md, 1 -d), the rationality and nonvanishing follows

from the basic results of Siegel [23].

PROPOSITION 2.4: The Tate cohomology group (X•)O=l 1(1 + B)X• is an ele

mentary abelian 2-group of rank:::; n = dim(S). It is isomorphic to the dual of

1r0 (G(R)) = G(R)IG(R)+·

Proof: By a result of Borel and Tits [6, §14], the maps S ~ G ~ S' induce an

isomorphism

1r0 (G(R)) = Image(7ro(S(R)) ~ 7ro(S'(R))).


The dual of 1r0 (S'(R)) is the 2-group x•(S')/2X.(S') = (X•)O=l /2(X•)O=l

of rank n, and the annihilator of the image of 1r0 (S(R)) is the subgroup

(1 + O)X• /2(X•)0=1.

COROLLARY 2.5: Let J.L: G--+ Gm be a character ofG over Q. Then the following

are equivalent:

(1) p,: G(Q)--+ Q* takes values in Q+. (2) J.L: G(R)--+ R* takes values in R+. (3) J.Llies in the subgroup (1 + ())X• of (X•)0= 1 = x•(S').

This is clear. We note that the character Jl defined by an irreducible represen

tation V of G in part (7) of Proposition 1.4 satisfies the equivalent conditions of

Corollary 2.5.

3. Irreducible representations of G

In this section, we recall the parameterization of irreducible representations V

of G over Q, and show that the division algebra D = Enda(V) has a positive

anti-involution. The center F of D is a subfield of k, and the anti-involution

induces complex conjugation on F.

Let p+ C x• be the Weyl chamber of weights x which satisfy (x, av) 2: 0 for

all av in L1 •. The group Gal(k/Q) permutes the elements of~., so stabilizes p+ in x·.

PROPOSITION 3.1 (cf. [28]): The irreducible representations V of G are parameterized by the orbits ofGal(k/Q) on p+_

If { a(x)} is an orbit, and F C k is the subfield fixed by the stabilizer of x, then

aEHom(F,Q)

where Wa(x) is the irreducible representation of G over Q with highest weight

a(x). The field F is the center of D = Enda(V), and c2 = [D: F].

This result is true for any reductive group over Q, and Tits gives a recipe for

the class of D in the Brauer group ofF [28]. For example, if V has trivial central

character, then D = F and e = 1.

We now establish some results which are particular to the class of groups G

satisfying Proposition 1.4. Let Tr: End(V) --+ Q be the trace map.


PROPOSITION 3.2: The division algebra D = Enda(V) has an anti-involution

d 1--t d', which satisfies Tr( dd') > 0 for all d f. 0. The restriction of this anti

involution to the center F is equal to complex conjugation T.

Proof: We fix an inner product (,) on V satisfying (7) of Proposition 1.4, and

define an anti-involution of the ring End(V) by the adjoint: (Tv, v') = (v, T'v').

Then TT' is self-adjoint and positive, so its eigenvalues >. satisfy .A ~ 0. Hence

Tr(TT') ~ 0, with equality if and only if TT' = 0.

If 1 is in G(Q), then 1' = f.L(/) ·,-I in End(V). Hence the anti-involution

stabilizes D = Enda(V). Since d f. 0 implies that dd' f. 0, we have Tr(dd') > 0.

Since the division algebras over Q with positive involution have been classified by

Albert (cf. [18, p. 193ff]), it follows directly that F is a CM field with conjugation

the involution a 1--t a'.

PROPOSITION 3.3: Assume that V corresponds to the orbit of X in p+, and let

w G--+ Gm be the character determined by V in Proposition 1.4. Then

where 0 is the central involution defined in Proposition 2.2.

Proof: This follows from an application of the identity

('yv, 1v') = f.Lb)(v, v')

to a highest weight vector v in Wx C V 0 Q, where v' is a lowest weight vector

in Wr(x)· Then (v,v') i 0. If/ lies in a maximal torus, /V = x(T)v and

1(v') = Ox(T)v. Hence f.L(T) = x(T) ·Ox(!), so f.L =X+ Ox.

Chapter II: Algebraic Modular Forms

4. Modular forms

We assume that G satisfies the equivalent conditions of Proposition 1.4, and fix

an irreducible representation V of Gover Q. Let D = Enda(V), and let F be

the center of D. Let f.L: G--+ Gm be the character of G determined by V.

Our aim is to study the Q-vector space of modular forms on G with coefficients

in V. This is the space of functions

(4.1) { f is locally constant on G(A) and}

M(V) = f: G(A) --+ V: . f(Tg) = lf(g) for 1 in G(Q)


This is a left D-vector space; we will show it affords an admissible representation

of 1r0 (G(R)) x G(Q), by right translation. The elements of Z(Q) act on M(V)

by the central character of V.

Since each f in M(V) is locally constant, it is constant on the cosets of an

open subgroup of the form G(R)+ x K, where K is an open compact subgroup

of G(Q). Hence M(V) is the direct limit of the subspaces

( 4.2)

M(V, K) = {!: G(A)/(G(R)+ x K)-+ V: f(rg) = if(g) for 1 in G(Q)}.

PROPOSITION 4.3: The double coset space

Z:K = G(Q)\G(A)/(G(R)+ x K)

is finite, and the D-vector space M(V, K) is finite-dimensional.

Proof: Since G(Q) is discrete and co-compact in G(Q), G(Q)\G(Q)/ K is finite.

Hence Z:K is finite. Since f in M(V, K) is determined by its values on Z:K,

dimM(V, K) :S Card(Z:K) · dim(V).

More precisely, fix representatives {ga} for the classes in Z:K. For each o:,

define

( 4.4)

This is an arithmetic subgroup of G(Q), so is finite.

PROPOSITION 4.5: The function fin M(V, K) is completely determined by the

values f(ga), which lie in vr", the subspace off a-invariants.

The map f f---t ( ... , f (ga), ... ) gives a D-linear isomorphism:

5. Serre tori

We now consider the spaces M(V, K) when G is a torus, of the type studied by

Serre (cf. [20]).

Let L be a CM field of degree 2g over Q, which is not totally real and whose

Galois closure in Q is the field k. Let

(5.1) J = Hom(L,Q) = Hom(L,k),

which is a set of cardinality 2g with an action of Gal(k/Q). The conjugation 7

acts freely on J.


The torus ResL/Q (Gm) has character group ZJ, and is split by k. Let G be

the quotient torus, whose character group is the subgroup

(5.2) x•(G) = { x = ~ n(a)a: n(a) + n(ar) = n(x) is constant}.

PROPOSITION 5.3: The torus G is defined over Q, and has dimension g + 1.

The maximal split torus S -+ G has dimension 1, and G* = G / S is anisotropic

over R.

The splitting field of G is the CM field k, and the motive of G is M =

x•(G) 0 Q.

Proof: G is defined over Q, as x•(G) is stable under Gal(k/Q). It has

dimension = rank x• (G) = g + 1.

The map x• (G) -t Z taking x to n(x) is surjective. Its kernel is the character

group of a quotient torus, on which T acts by -1. Hence the quotient torus is

compact over R, and Z = x•(S). Since Gal(k/Q) acts faithfully on x•(G), k is the splitting field. The motive

of a torus is the rational vector space of characters.

COROLLARY 5.4: x•(S') is the subgroup of x•(G) generated by l:J a. The

isogeny S -+ S' has degree 2, and G(R) ~ R+. x (S1 ) 9 is connected.

Proof: The last statement follows from the fact that 7ro ( G (R)) is the image of

7ro(S(R)) in 7ro(S'(R)).

The irreducible representations V of G over Q correspond to the orbits of

Gal(k/Q) on x•(G). If V corresponds to the orbit of x, then D = Endc(V) = F is the subfield of k which is fixed by the stabilizer of X in Gal(k/Q), and e = 1.

The dimension of V over F is equal to 1. Furthermore, the character f.L is given

by X+ Bx =X+ TX = n(x) · l:J a. Let K C G(Q) be open and compact, and let

(5.5)

(5.6)

h = Card(G(Q)\G(Q)/K),

r =G(Q) n K.

Since G(R) = G(R)+, h is the order of I:K defined in Proposition 4.3. Since

G is commutative, r, = r for all a. By Proposition 4.5, we have an F-linear

isomorphism

(5.7) M(V,K) ~ h · vr.


Since V has dimension 1 over F, M(V, K) has dimension h over F if r acts

trivially on V, and has dimension 0 otherwise.

For a specific example, let '1.'._0 be the connected component of the Neron-o ' 0 Raynaud model ofT over Z [7, Ch. X], and put K ='I'._ (Z). Then r ='I'._ (Z)

lies in an exact sequence

1---+ (±1)---+ r---+ UL/Uu ---+ 1,

where U L are the units of the CM field L, and U L+ are the units of L +. Also,

h = hL/hu is the quotient of the respective class numbers. If r acts trivially

on V, M has dimension

L( E, 0) h = hL/h£+ = Card(r). ~'

where L(E, s) is the Artin £-function of the quadratic character E of £+

corresponding to the extension L / L +.

6. Operators and an inner product

We first define an inner product on the finite-dimensional Q-vector space M =

M(V, K). This depends on the choice of an inner product (, ): V x V ---+ Q,

satisfying (1v,1v') = p,(r)(v,v') for 1 in G(Q). We fix such an inner product on

v. By Corollary 2.5 and Proposition 3.3, the character p,: G ---+ Gm takes positive

values on G(R). Defining N: A*---+ Q* as the projection onto the first factor of

(1.1), we see that the composite homomorphism

(6.1) ~-tA: G(A)----+ A*----+ Q* I" N

takes values in Q"i-.

Fix representatives {ga} for I:K as in §4, and define the arithmetic subgroups

fa by (4.4). For f, f' in M, we define the inner product (!, f')M in Q by the formula

(6.2) (!, J') M = L Card(r a~· J-LA(9a) (f(ga), J' (ga)). a

This is clearly a positive definite, symmetric bilinear form (, ) M: M x M --+ Q,

by Proposition 4.5. It is also independent of the choice of coset representatives;

if ha = 1 · 9a · k, then f(ha) c:::' f(go:) and

(f(ha), J'(ha)) = p,(r)(f(ga), J'(ga)),

/-tA(ha) = p,(r)p,A(9o:).


Furthermore, if K' C K then M(V, K) C M(V, K'), and for J, f' in M(V, K) we

find

(6.3) ( I 1 ') J,f)M(V,K) = [K: K'](f,J M(V,K')·

Using the inner product (,) M' we define an anti-involution of EndQ(M) by the

adjoint: (T J, f') = (!, T' f'). On the sub-algebra D = En de (V) of EndQ ( M), this is clearly the positive anti-involution d r-+ d' studied in Proposition 3.2. We

now define some linear maps T: M-+ M, and compute their adjoints T'.

If Yeo E G (R), then the map

(6.4)

defines an automorphism of M, which depends only on the image of Yeo in

G(R)/G(R)+. This gives an action of the 2-group 1r0 (G(R)) on M, via self

adjoint operators:

(6.5)

A more interesting family of operators comes from the Heeke algebra H K of

all locally constant, compactly supported functions F: G(Q) -+ Q which are

K-bi-invariant. We write G(Q)/ / K for the double coset space K\G(Q)/ K. The

product in HK is by convolution, using the Haar measure dxK on G(Q) giving

the open compact subgroup K volume 1:

F. F'(g) = r A F(x). F'(x- 1g)dxK lc(Q)

= {. F(gy- 1 )·F'(y)dYK· lc(Q)

A Q-basis of HK is given by the characteristic functions of the double cosets

KgK, and each characteristic function defines a linear operator T(g) of M as

follows.

Write

as a disjoint union of a finite number of single cosets; the number is finite as

KgK is compact and each g;K is open. Now define, for fin M,

(6.6) T(g)f(y) = :2.:: f(g[JiK).


Then T(g)f is still right K-invariant, as left multiplication by K permutes the

right cosets giK. Hence T(g)f lies in M; extending linearly to HK gives a map

(6.7)

which is painfully checked to be a homomorphism of Q-algebras. If F in H K

takes the value F(giK) on the coset ?JiK, then

(6.8) T(F)f(g) = L F(giK). f(ggiK)

where the sum on the right is finite (as F is compactly supported).

PROPOSITION 6.9: The adjoint ofT(.q) is the operatOTf.LA(g) ·T(?J-l) in End(M).

Proof: This is also a painful computation (cf. [24, Prop. 3.39]), which is one of

the reasons for the inclusion of the terms Card(f a) in the definition of the inner

product (,) M in (6.2). Another is to obtain formula (6.3).

Let A be the Q-sub-algebra of End(M) spanned by the operators in D = Endc(V), as well as the automorphisms T(g00 ), and the endomorphisms T(g)

coming from H K. One easily checks that these 3 types of linear transformations

commute with each other, so we have a surjective homomorphism of Q-algebras

(6.10) D Q9 Q[Iro(G(R))] Q9 HK--+ A~ End(M).

PROPOSITION 6.11: The anti-involution T H T' of End(M) stabilizes A. The

A-module M is semi-simple.

Proof: The fact that A is stable under the adjoint follows from (6.5) and Propo

sition 6.9. Let N C M be an A-submodule, and let N j_ be the orthogonal comple

ment of N with respect to (, ) M. Since the form is positive definite, M = NEB N j_

as a rational vector space. It suffices to show N j_ is an A-module. Let f be in

Nj_, and f' be inN. For any Tin A,

Since T' is in A, and N is an A-submodule, T' f' lies inN. Hence(!, T' f') M = 0.

Since this holds for any f', T f is in N j_.


7. Simple submodules

Let M be equipped with the inner product (,) M of the previous section, and let

A C End(M) be the Q-algebra defined by (6.10). We have seen that M is a semi

simple A-module. In this section, we study invariants of simple A-submodules

NcM. Let

(7.1)

(7.2)

C = EndA(N),

E = center of C.

Then C is a division algebra, of finite dimension over Q, and E is a number field

containing F, the center of D = Endc(V). Indeed, if m is the dimension of N

over C and C0 is the opposite algebra, the bi-commutant theorem shows that

the image of A in End(N) is the simple algebra Endc(N) = Mat(m,C0 ). Hence

the center of A maps to the center of Endc(N), which are the scalar matrices

with entries in E. But we have, from (6.10), a homomorphism of commutative

Q-algebras

(7.3) F ~ Q[7ro(G(R))] ~ Z(HK) --+ Z(A)

where Z(HK) and Z(A) denote the centers of HK and A, respectively. This

embeds F as a subfield of E, and gives two characters:

(7.4)

(7.5)

<fJoo: 7ro(G(R))--+ ( ± 1) c E*,

rp : Z(HK) --+ E.

We remark that, by results of Bernstein, HK is a finite module over its center.

Since M = NEBN.l, we may view elements ofC = EndA(N) as endomorphisms

of M, which are zero on N .l and c9mmute with A. The following is an analog

of Proposition 3.2.

PROPOSITION 7.6: The anti-involution T--+ T' ofEnd(M) which is defined by

(,) M stabilizes the sub-algebra C, and induces a positive anti-involution of C.

Proof: Since the anti-involution given by the adjoint stabilizes A, it stabilizes

the sub-algebra of End(M) commuting with A. Since C is the sub-algebra of

the corr:muting algebra which satisfies T(N) C Nand T(N.l) = 0, it suffices to

check that these conditions are closed under adjoint.

Since T(N) c N, T'(N.l) c Nl.. Since T(N.l) = 0, T'(N.l) c N. Hence

T'(N.l) c N n N.l = 0. Similarly, T'(N) C N. Therefore Cis stable under

adjoint. Since Tr(cc') 2: 0 with equality if and only if c = 0, the division algebra

C has a positive anti-involution.


COROLLARY 7. 7: The center E of C is a CM field, and the anti-involution given

by the adjoint induces complex conjugation on E.

Proof: This follows from Albert's classification of division algebras with a

positive anti-involution ( cf. [18, p. 193ff]).

CoROLLARY 7.8: Assume that the characteristic function of KgK lies in the

center of H K. If

rp: Z(HK) -+ E = center of End A (N)

is the character associated to a simple A submodule N c M, then

rp(T(g)) = !1A(9) · tp(T(g- 1)) in E

where o: f-t a denotes complex conjugation on E.

Proof: This follows from Proposition 6.9 and Corollary 7.7. The former also

shows that T(g- 1) is in the center of A.

As an example of the use of Corollary 7.8, assume that G is semi-simple, and

that -1 is in the Weyl group. The splitting field k of '1/J( G) is then totally real,

by Proposition 2.2. Hence the center F of Endc(V) = D is totally real, as, by

Proposition 3.1, F is a subfield of k. By Albert's classification, either D = F, or

D is a quaternion algebra over F (which is totally indefinite or totally definite

over F 0 R). Now assume further that K c G(Q) has the following property: KgK =

Kg- 1 K for all g. This is true, for example, if Kp is hyperspecial for all primes p

[14]. Then HK is commutative; since f1 = 1 we conclude from Corollary 7.8 that,

for each simple submodule N C M = M(V,K), the center E of C = EndA(N) is also totally real. Hence C has degree 1 or 4 over E.

8. Completions and lattices

In this section, we consider the spaces M0R and M0Qp, where M = M(V, K). We relate the vector space M 0 R to objects in the usual theory of automorphic

forms, and define certain Zp-lattices in M 0 Qp·

Since V is an algebraic representation of G over Q, the Lie group G(R) acts

on V 0 R and the p-adic Lie group G(Qp) acts on V 0 Qp. Iff: G(A) -+ V is

an element of M, we define

(8.1)

(8.2)

Foo(g) = g;;} f(g) in V 0 R,

Fp(g) = g;1 f(g) in V 0 Qp.


PROPOSITION 8.3: The map f e-+ F= identifies the space M 0 R with the real vector space of functions

F: G(Q)\G(A)/ K -tV 0 R

which satisfy F(gg=) = g-;;,1 F(g) for all 9= in G(R)+. This is an isomorphism

of H K 0 R-modules.

Now assume that G(R) is compact, so G(Q) is discrete and co-compact in

G(A), and G(R) is connected. Let L2 (G(Q)\G(A)) be the real vector space of

square-integrable functions on the quotient space, with respect to Haar measure

on G(A). This is a unitary representation of G(A), by right translation. Fix

an invariant inner product (,) on V. Then to each F in the vector space of

Proposition 8.3 we can associate a linear map CF: V 0 R -7 L2 (G(Q)\G(A)), defined by

(8.4) £ F ( v) (g) = ( F (g), v).

The functions in the image of eF are actually analytic on G(R), and locally

constant on G(Q).

PROPOSITION 8.5: The map F e-+ eF defined in (8.4) is an HK 0 R-linear

isomorphism:

M 0 R ~ Homa(R)xK(V 0 R, L2 (G(Q)\G(A)).

The proofs of Propositions 8.3 and 8.5 are simple computations. If V is

absolutely irreducible, Proposition 8.5 identifies the simple H K-submodules

Nc C M 0 C with the irreducible automorphic representations 1r = 1r= 0 1T of G(A) which satisfy 1r= ~ V 0 C and 7rK =j:. 0.

We now consider the space M 0 Qp, under the hypothesis that K has the form

Kp x K', where K' is open and compact in G(Q') = Ul,ep G(Qe). The following

is an analog of Proposition 8.3.

PROPOSITION 8.6: The map f e-+ Fp identifies the space M 0 Qp with the

Qp-vector space of functions

F: G(Q)\G(A)/(G(R)+ x K') -7 V 0 Qp

which satisfy F(gkp) = k;1 F(g) for kp in Kp. This is an isomorphism of

HK' 0 Qp-modules, and of representations of 7ro(G(R)).

Now let Lp c V 0 Qp be a Zp-lattice which is stable under the action of the

compact group Kw Then the space of functions

(8.7) F: G(Q)\G(A)/(G(R)+ x K') -7 Lp


which satisfy F(gkp) = k; 1 F(g) for kp in Kp gives a Zp-lattice

(8.8)

which is stable under Zp-linear combinations of the operators T(g'), for f/ in

G(Q')/ /K'. As an application of the lattices M(Lp), we note the following. Assume that

Ke c G(Qe) is hyperspecial, and that K =TIP Kp. Let 9£ be in G(Qe)/ j Ke and

put

.§e=(l,l, ... ,g£, ... 1,1) inG(Q)//K.

The operator T(ge) is then in the center of A, so acts on a simple module N C M

by the scalar

ae = I{!(T(flt)) in E.

PROPOSITION 8.9: Let OE denote the ring of integers of the CM-field E. Then

ae lies in the subring OE[l/R] of E.

Proof: Since T(gt) stabilizes a lattice M(Lp) in M ® Qp, its eigenvalues are

p-adically integral, for all p =!= e. Finally, in the case when V = Q is the trivial representation of G, we have

F00 = Fp = f in (8.1) and (8.2). In this case, we have the Z lattice

(8.10) M(Z) = {f: G(Q)\G(A)/(G(R)+ x K) -t Z}

inside the Q-vector space M, which is stable under the 2-group n0 (G(R)) and

the integral Heeke algebra HK(Z) of Z-linear combinations of the elements T(g). In this case, the eigenvalues of elements in the center of HK(Z) on a simple

submodule N C Mare elements of OE, the ring of integers in the CM-field E.

9. Reduction (modp)

Let M(Lp) be the Zp-lattice in M(V, K) constructed in (8.8). A function F lies

in the submodule pM(Lp) if and only if it takes values in pLP. Let L = LpjpLp; the map taking Fin M(Lp) to its values (mod pLp) gives an injection of Z/pZ

modules

(9.1) M(Lp)jpM(Lp) -t M

= {F: G(Q)\G(A)/(G(R)+ x K') -t L: F(gkp) = k; 1 F(g)}

for kp in Kp.


PROPOSITION 9.2: If the finite groups f"' of (4.4) have orders prime top, then

the map in (9.1) is an isomorphism.

Proof: Let~"'= g;; 1G(Q)g"' n (G(R)+ x K). This is a finite group isomorphic

to r "'' which acts on Lp through the quotient Kp of K. We have Zp-linear

isomorphisms

Furthermore, the map L~" -+ L/',." has cokernel equal to the p-torsion subgroup

of H1 (~"'' Lp). If~"' has order prime top, H1 (~"'' Lp) = 0.

The representation L = Lp/pLp of Kp may not be irreducible over Z/pZ, and this reducibility gives a decomposition of the space M. More generally, we

will define spaces M(W, K) of modular forms (mod p), associated to irreducible

representations W of the group KP over Z/pZ. Let

M(W,K) = {F: G(Q)\G(A)/(G(R)+ x K')-+ W :F(gkp) = k; 1 F(g)} (9.3)

for kp in Kp.

This Z/pZ-vector space is finite-dimensional, and is a module for the algebra

(9.4) Z/pZ[no(G(R))] Q9 Z/pZ[G(Q')/ / K']0 EndKp (W).

We end with a remark on the irreducible modules W for Kp, when G/Qp is

unramified and Kp is hyperspecial. We then have a reduction homomorphism

KP -+ G(p), whose kernel is a pro-p-group which acts trivially on W. The

quotient G(p) consists of the Z/pZ-points of a quasi-split reductive group over

Z/pZ, which is split over the residue field of a prime p dividing p in k. When

G (and hence G) are simply-connected, the irreducible modules W for G(p) over

Z/pZ are parameterized as in §3 by the Gal(kp/Qp) orbits {x} on p+, which

satisfy the additional condition that (x, a.v) :::; (p- 1) for all a.v in~. [25].

10. Algebraic Heeke characters

We consider the theory of the previous sections in the case when G is a Serre

torus, associated to a CM field L as in §5. We have seen t'hat D = F is commutative, and that V has dimension 1 over

F. Similarly HK is a commutative Q-algebra, isomorphic to the group algebra


of G(Q)/ K. Let N C M be a simple A-module, where A is the image of the

commutative algebra F ® HK in End(M). Let C = EndA(N) and let m be the

dimension of N over C; by the Wedderburn theory the image of A in End(N) is

Endc(N) = Mat(m, C0 ). Since A is commutative, this shows that C = E is a

CM field, and that N has dimension 1 over E. We therefore obtain a character

(10.1)

Equivalently, we obtain a group homomorphism

(10.2) x: G(Q)/K-+ E*.

We may view x as a character

(10.3) x: G(A)-+ E*

which is trivial on G(R) x K and extends the character

(10.4) x: G(Q)-+ F*

defining the irreducible representation V. From Proposition 7.8, we obtain the

formula

(10.5) x(g) · T(x(g)) = /LA(g),

which extends the formula x + TX = JL of §5. In particular, the values of x lie in

the subgroup {a: E E*: a:· T(a:) E Q*} of E*.

The torus G was defined as a quotient of ResL;QGm. Hence there are homo

morphisms (not necessary surjective)

(10.6) { A~-+ G(A),

L*-+ G(Q).

If we compose x and x with these maps, we obtain Serre's algebraic Heeke

character A~ -+ E*, and its algebraic part L * -+ F* [20]. In this way, we obtain

those algebraic Heeke characters of A~ whose restriction to A~+ factors through

the norm map A~+ -+A*.


11. The trivial representation

We consider the general theory when V = Q is the trivial representation. Then

D = F = Q and f.1 = 1. We take the inner product (v, v') = vv' on V, and fix

coset representatives {ga} for ~K = G(Q)\G(A)/(G(R)+ x K). Since M = {!: ~K -7 Q}, a basis is given by the functions {fa} which satisfy

(11.1)

These functions are orthogonal with respect to (, ) M, and satisfy

(11.2) Wa = Card(f a).

Hence the dual basis is f:;_ = Wa ·fa· The functions fa and f:;_ all lie in the

lattice M(Z) defined in (8.10).

If

then we find

(11.3)

(11.4)

(11.5)

1 (e,e)M = 2:-,

Wa T(g00 )e = e,

T(g)e = degree(KgK) · e,

where degree(KgK) is the number of single cosets contained in the double coset.

In particular, N = Qe is a simple A-submodule of M, with C = E = Q. More interesting A-modules occur in the space

( 11.6)

When G is an inner form of G £ 2 corresponding to a quaternion algebra ramified

at p and oo, and K is maximal, the simple A-submodules of M 0 correspond to

cuspidal eigenforms of weight 2 for f 0 (p) [10].

12. The Steinberg space

Now we consider a special case of the general theory, when G is semi-simple

and simply-connected over Q, and V is absolutely irreducible. Then G(R) is

connected, D = Q, and f.1 = 1. We will also assume that the center Z(Q) of

G(Q) acts trivially on V.


For this section (where there is no central torus) we let S denote a finite,

non-empty set of primes, which includes all p where G is ramified over Qp.

For p E S, we let Kp be an Iwahori subgroup of G(Qp)· We recall that the

Heeke algebra for G(Qp)/ / KP has a presentation with generators ta,p satisfying

(ta,p + 1)(ta,p- qa) = 0, with qa > 0 [8, p. 142]. If G(Qp) is split, qa = p for all

a. The Steinberg character is the unique homomorphism

Stp: HKp --t Q

mapping ta,p f--7 -1 for all a. (12.1)

For p tf. S, the group G is unramified over Qp and we let Kp C G(Qp) be a

hyperspecial maximal compact subgroup, chosen as in [11]. Put K' = I1p¢S Kp

and K = TipES Kp x K'. The Heeke algebra HK' is commutative [14].

We use the Steinberg character to define a subspace Ms C M(V, K) as follows.

Let

(12.2) Ms = {f EM: T(ta,p)J =-f for all pES, all ta,p}.

The algebra As= Image(HK') acts on Ms, and the simple A-submodules N C

Ms C M correspond to simple As-submodules of Ms. Since As is commutative,

we obtain the following.

PROPOSITION 12.3: If N is a simple A-submodule contained in Ms, then

EndA(N) = E is a CM field and N has dimension 1 over E. The associated

character

restricts to the Steinberg character of HKP, for all pES.

Since we have assumed that V is absolutely irreducible, the simple HK

submodules Nc C M 0 C correspond bijectively to the irreducible automorphic

representations 1r = 7r00 0 -IT of G(A), with 7r00 ~ V 0 C and jK =j:. 0, by

Proposition 8.5. The submodules Nc contained in Ms 0 C correspond to the

automorphic representations with 'lrp the Steinberg (or special) representation

of G(Qp), for all p E S. In particular, for these submodules Nc, dim-ITK = 1.

General conjectures of Arthur on multiplicities in the discrete spectrum [1]lead

one to the following.

CONJECTURE 12.4: The image As of HK' in EndQ(Ms) is a maximal

commutative Q-sub-algebra, and Ms is a free As-module of rank 1.

The dimension of Ms over Q is an interesting invariant, which one can compute

from the stable trace formula. The following estimate, which is the contribution


of the classes in Z(Q), is useful when the integer N = ITpESP is large enough (cf. [15]):

(12.5) dimMs ,....., Card(Z(Q)) ·IL (M)I. dimV 2e 5

In this formula, € is the rank of G(C), and Ls(M) is the partial £-function of

the motive of G (with the factors at oo and primes in S removed) evaluated at

s = 0. The right hand side of (12.5) is asymptotic to cc · Nk as N -+ oo, where

cc is a constant (depending on G) and k 2: 1 is the number of positive roots for

G.

If Conjecture 12.4 holds, and V = Q is the trivial representation, one gets an

interesting commutative ring

(12.6) Rs c Endz(Ms(Z))

generated by the Z-linear combinations of the T(g), which stabilize the lattices

M(Z) and Ms(Z) = Ms n M(Z). This is an order in· As, and is the analog of

Mazur's Heeke ring for forms of weight 2 on r0 (p) [17]. Its completions at maxi

mal ideals may be of interest in the deformation theory of Galois representations.

Chapter III: £-Group Par.ameters

13. The dual group, and the co-character ry

We define the dual group G of G as in [13], but insist that G be a split reductive

group over Z, instead of just over C. The dual group comes with a pinning

(13.1)

over Z, and an isomorphism of root data.

(13.2)

Here 'lj;v is the root datum dual to the datum of Gover Q:

The group Aut('lj;v) = Aut('lj;) acts on the set of isomorphisms (13.2), so acts

as pinned automorphisms of G over Z. Hence Gal(k/Q), which acts faithfully


on '1/J, acts as pinned automorphisms of G. We define the L-group over Z as the

semi-direct product

(13.3) Lc = G ~ Gal(k/Q),

where Gal(k/Q) is viewed as a finite, constant etale group scheme. The multi

plication law in LG is

(g,a) · (g',a') = (g · a(g'),aa')

where g, g' are in G and a, a' are in Gal(k/Q).

Let g = Lie(G) over Z. For all a E ~. = ~·('i',B), define

Then the sum

( 13.4)

is a regular nilpotent element in g, which is fixed by the action of Gal(k/Q).

To define the parameters of eigenforms for G in the L-group, we need an

additional hypothesis on the root datum 'lj;. Consider the following condition:

(13.5) There is an element 'T) in x•, which is fixed by Gal(k/Q) and satisfies

('f), a)= 1 for all a E ~ •.

To see that this is an additional hypothesis on the isogeny class of G, let G be any form of PG L2 over Q which corresponds to a definite quaternion algebra.

Then ~ • = {,8}, ~. = {a}, and x• = Z,B with trivial Galois action. Since (,8, a) = 2, no element 'T) exists satisfying (13.5).

If an element 'T) exists satisfying (13.5), it is unique up to the addition of an

element in the subgroup (X•)Ii=l = Hom(G, Gm). Indeed, if 7)1 is another such

element, the difference 7)1

- 'T) in x• is fixed by the entire group W ~ Gal(k/Q). But the fixed space of this group is the fixed space of the central involution(), by

Proposition 2.2.

Henceforth, we will assume that the condition (13.5) holds for the root datum

'ljJ of G, and will fix the choice of a suitable element 'T) in x• once and for all. Our

parameters for eigenforms will depend on this choice.

In two cases, there is a natural choice for 7). When G is a torus, we will take

'T) = 0. When G is simply-connected, we will take 'T) to be half the sum of the


positive roots (which actually lies in x•). One can also show that (13.5) holds

when the derived group of G is simply-connected, and the center of G is split

[14]. However, it may not be possible to take 'TJ equal to half the sum of the positive roots in this case. For example, when G is an inner form of G £ 2 we have

x• = Ze1 + Ze2 with ~. = { e~ - e¥}. Half the sum of the positive roots is the

element ~(e 1 - e2 ) in ~X- X but the element 'T} = e1 satisfies (13.5).

Via the isomorphism i: '1/J(T, B)':::' '1/Jv of (13.2), we can view 'T} as a co-character

(13.6)

which is fixed by Gal(kiQ) and satisfies

(13. 7) ad'T}(a)(X) =a· X,

where X is the regular nilpotent element defined by (13.4).

14. Conjugacy classes in LG

The local parameters of simple submodules N C M will be points of the variety

of conjugacy classes Cf(LG) of LG with values in E, the center of EndA(N).

Before defining the parameters, we will need to define the affine schemes Cfu, associated to a- E Gal(kiQ).

Let

(14.1) {

Gal(kiQt be the centralizer of a- in Gal(kiQ),

wu be the subgroup fixed by a- in W,

X~ be the sublattice fixed by a- in X •.

The finite group wu >4 Gal(kiQ)u I (a-) acts on X~. We define

(14.2)

(14.3)

Cfu = SpecZ[X:Jw",

Cf(LG) = 11 Cfu.

The relative dimension of Cfu over Spec Z is equal to the rank of x:, and the

group Gal(kiQ)u I (a-) acts on Cfu. Since X. = x•(i'), the scheme Cfu admits an interpretation as a "variety

of semi-simple conjugacy classes." More precisely, the points of Cfu over an

algebraically closed field F correspond to the G(F)-conjugacy classes of elements


(s,a) in LG(F), with s semi-simple in G(F) (cf. [4]). If a= 1, C£1 = C£(G) is the usual scheme of semi-simple conjugacy classes in G [22]:

( 14.4)

If a E Gal(k/Q) satisfies aaa- 1 = a', then conjugation by a gives an

isomorphism of schemes:

(14.5) C£17 , -+ C£a,

( s, a') a = (a - 1 ( s), a).

This defines a right action of Gal(k/Q) on C£(LG).

15. Archimedean parameters

In this section, we define the Archimedean parameter h00 = h00 (N) of a simple

submodule N C M. We have seen that N gives a character (7.4):

(15.1) I.{Joo: Jro(G(R))-+ ( ± 1) c E*.

To every such character, we will associate the class of an involution h00 in C£r(Z),

where Tis complex conjugation in Gal(k/Q).

By Proposition 2.4, the character group of 1r0 (G(R)) is canonically isomorphic

to (X•) 11 = 1 /(1 + fJ)X•. We will view l.fJoo as an element of this quotient group.

If X is any element of (X.) 11= 1 = Hom(LG, Gm), the inner product (l.fJ00 , x) is

well-defined (mod 2).

PROPOSITION 15.2: There is a unique class hoo = h00 (1.fJ00 ) in C£r(Z) which

satisfies:

{

h'?x, = 1 in Lc(Z),

Tr(hool9) = Tr(fJIX• = Lie(T)),

x(hoo) = (-1)(7J+'Poo,x) for all X in Hom(LG, Gm).

The class h00 is fixed by the action ofGal(k/Qt /(T) = Gal(k+ /Q) on C£r·

Proof: Lift l.fJoo to an element ~ in (X•) 11=1 , and view ~ as a co-character ofT

fixed by W )q Gal(k/Q). Define the involution

(15.3) h00 = (17(-l) · ~( -1), T)

86 B. H. GROSS lsr. J. Math.

in LG(Z). This has all the properties stated in the Proposition, and its class

in Ci!T(Z) depends only on r.p00 (not on the lifting ~). For example, when G is

simply-connected, r.p00 = 1 and 1] is half the sum of the positive roots for G. The

proof that

Tr(hoo\9) = Tr(8\X•) = -rank(G)

is given in [13], using Kostant's theory of the principal PGL2 in G. The general

case is similar.

16. Unramified parameters

Let p be a rational prime, which is unramified in k. Let p be a factor of p in k, and let a(p) be the corresponding arithmetic Frobenius element in Gal(k/Q).

Assume that Kp = KnG(Qp) is a hyperspecial maximal compact subgroup of

G(Qp)- Our aim in this section is to define the unramified parameters of simple

A-submodules N C M = M(V,K).

Since Kp is hyperspecial, the group G is quasi-split over QP, and split over

the maximal unramified extension of Qp [27]. Let Hp be the (commutative)

Z[1/p]-algebra of Z[1/p]-valued, compactly supported functions on the double

coset space G(Qp)/ j Kp, under convolution. By (7.5) and Proposition 8.9, each

simple submodule N gives rise to a homomorphism of Z[1/p]-algebras

(16.1)

where E is the center of EndA(N). To such a homomorphism, we will assign a

class

(16.2)

The definition of hp uses the Satake isomorphism over Z[1/p], normalized using

the co-character 1] (cf. [14]).

If p' = a(p) is another factor of pink, with a E Gal(k/Q), then aa(p)a-1 = a(p'), and we will show that

(16.3)

Finally, if G is semi-simple, we will show that


where T is complex conjugation in Gal(k/Q) (which centralizes o-(p) and acts on

the scheme C£a-(p)), and a rl a is complex conjugation on the CM-field E (which

gives an involution of the coefficients of C£a-(p)(E)). We now give the definition of hp. Let G be a model for the group Gover Zp,

with G(Zp) = Kp and good reduction (mod p). LetT_ C B C G be a maximal

torus, contained in a Borel subgroup over Zp, and let W 8 be the Weyl group of

the maximal split torus 'Ls C T_. We have an isomorphism

(16.5) X.(T_)Gal(Qp/Qp) =X.(T_

8)::: T_(Qp)/T_(Zp),

,\ rl ,\(p).

The choice of p dividing pin k gives an embedding k---+ QP, up to the action

of Gal(Qp/Qp)· From this, we get a canonical identification:

(16.6)

(16.7)

x.('LJ::: x~<Pl = x·cty<Pl, W

8 ::: W""(P).

We may then use rJ in x• to define a homomorphism:

(16.8) f/p: T_(Qp)/T_(Zp)---+ Z[1/p]*,

,\(p) rl p-(>-.,1J).

Let U C B be the unipotent radical over Zp, and let du be the unique Haar

measure on U(Qp) which gives the open compact subgroup U(Zp) volume 1. The

Satake transform S, normalized by rJ, is the homomorphism of Z[1/p]-algebras

[8]

(16.9)

which is defined by

Sf(t) = f/p(t) J f(tu)du. U(Qp)

(16.10)

The main result is that the Satake transform gives an isomorphism of Z[l/p]·

algebras [14]:

(16.11)

Since H(T_(Qp)/ /T_(Zp)) = Z[1/p][X.(Ts)] by (16.5), we obtain from (16.6)

(16.7) an isomorphism

(16.12)


Hence an element cpp in Homz[I/pJ(Hp, OE[1/p]) gives a point hp in

C£u(p)(OE[1/p]). This is the definition of the unramified parameter.

The dependence (16.3) of hp on pis easily checked, as this only depends on the

identification of X.([8

) with X~(P) in (16.6). The identity (16.4) is established

as follows. When G is semi-simple, J.L = J.L(V) = 1. There is an involution j of

Hp which satisfies j(KptKp) = Kpt- 1 Kp for all tin I::8(Qp)· By Corollary 7.8,

cppoj = rpp in Hom(Hp,E). Since e = w X T acts as -1 on x·, and w lies in Kp,

the action of j on Hp takes the parameter hp to the parameter hjrp = hr{p)·

To see the dependence of the parameters hp(N) on ry, we will give an explicit

formula for the parameters when k = Q, so G is split over Qp. Let f\ C X. ~ X.(I::) be the elements,\ with (.A, a) 2: 0 for all a E 6_•. For each,\ in P+, let

(16.13)

(16.14)

C>. = char(Kp.A(p)Kp) in Hp,

X>. be the character of the irreducible

representation of G with highest weight .A.

Let hp be the parameter of cpp: Hp --+ OE[1/p] in C£(G)(OE[1/p]). Then we

have [14]

(16.15) X>.(hp) = p<>.,'1- 2P) 2:.: d>.(J.L)(p) · cpp(c,..) in OE[1/p],

1-'9

where p is half the sum of the positive roots for G, and d>. (J.L) (p) is the Kazhdan

Lusztig polynomial (in the variable q = p) associated to a pair of elements J.L ::; ,\ in P+.

Chapter IV: Galois Representations

17. A global conjecture

We fix Gover Q, and an abolutely irreducible representation V over Q with trivial

central character. We assume there is an element 'f) E x• satisfying {13.5), and

fix it once and for all. We let x be the highest weight of V, which we assume is

dominant with respect to 6. •. We will view "1 and x as co-characters off', which

are fixed by Gal(k/Q). Let S denote a finite, non-empty set of primes p, which includes all places

where G is ramified over Qp, and includes at least two places. For p ~ S, we

let Kp be a hyperspecial maximal compact subgroup of G(Qp)· For p E S, we

let Kp be (the connected component of) an Iwahori subgroup of G(Qp)· We let

K = TIP Kp, as in §12.


Let N C M (V, K) be a simple Heeke submodule, with the property that N

gives rise to the Steinberg character of 'Hp for all primes p E S. If G is simply

connected, we haveN C Ms C M(V, K) in the notation of §12. If V = Q and

TipES G(Qp) is compact, we exclude the trivial representation N = Qe, discussed

in §11.

The unramified, abelian Heeke algebra As = @p~S'Hp acts on N, and by

Proposition 12.3 the algebra E = End As (N) is a CM field.

We define the local parameters of N, using the methods of §§15-16, and our

fixed choice of rr

(17.1) { h00 (N) in Cfr(Z), hp(N) in C£cr(p)(OE[1/p]).

CONJECTURE 17.2: Let £ be a rational prime. There is a continuous homo

morphism

p = p(N)e: Gal(Q/Q) ---+ LG(E 0 Qe)

which satisfies:

(a) If S 00 is a complex conjugation in Gal(Q/Q), then p(soo) h00 (N) in

Cfr(E 0 Q,). (b) If p -:J £ and p rf_ S, the representation p is unramified at p. Let sp be

a Frobenius element at pin Gal(Q/Q) which maps to a(p) in Gal(k/Q).

Then p(sp) is a semi-simple elment in LG(E 0 Qt) and p(sp) = hp(N) in

CRcr(pJ(E 0 Q,).

If a representation p satisfying the conditions of Conjecture 17.2 exists, then

the composite homomorphism

Gal(Q/Q)--+ LG(E 0 Qe)--+ Gal(k/Q) p pr

is the standard projection. Indeed, Sp maps to a(p) for almost all primes p of k.

18. Steinberg parameters

Let p be a prime inS, with p -:J £. Let Gmfqz be a Tate elliptic curve over QP.

The Galois action on £-power torsion points gives a continuous homomorphism

The image lies in the subgroup H C GL2 which stabilizes the line V,(Gm) and

acts trivially on the quotient [26]. The algebraic subgroup

(18.1)

90 B. H. GROSS lsr. J. Math.

is defined over Z and isomorphic to the semi-direct product Ga XI Gm, and the

conjugacy class of the resulting homomorphism

(18.2)

is independent of the choice of Tate curve Gm/qz over Qp.

We map H to the group 6 over Q by

( 18.3) ( ~ i) r+ exp(yX) · rJ(a)

where ry: Gm -+ Tis our fixed co-character and X = 2:.::: X a is the regular nilpotent

element of g defined in (13.4). This is a homomorphism by (13.7). Since the image

of (18.3) in 6 is fixed by Gal(k/Q), it extends to a homomorphism of algebraic

groups

(18.4) '{J: H x Gal(k/Q) -+ G = 6 ><1 Gal(k/Q).

CONJECTURE 18.5: For p E S with p -:f. £, the restriction of p = p(N)e to a

decomposition group of a prime p dividing p .in Q is conjugate to the homo

morphism

Gal(Qp/Qp)--+ H(E 0 Qe) x Gal(k/Q)--+ LG(E 0 Qe) t ~

by an element of 6(E 0 Qe). Here 'P is defined by" (18.4), and t is the product

of te defined in (18.2) and the projection to the decomposition group of p in

Gal(k/Q) .

In particular, Conjecture 18.5 predicts that the image of p contains regular

unipotent elements of 6(E 0 Qe), so the image is not contained in any proper

parabolic subgroup. The existence of regular unipotent elements, together with

the expectation that the Zariski closure of the image should be a reductive sub

group, also severely restricts the possibilities. For example, if G is of type Eg

the Zariski closure of the image of Gal(Q/Q) should be either 6(E0 Qe), or the

principal subgroup PGL2(E 0 Qe).

19. The local representation at £

We will attempt to predict the restriction of p(N)e to a decomposition group at

£in Gal(Q/Q) in two cases: when N is ordinary at£, and when N is Steinberg

at£ and V = Q.


The ordinary condition depends on the representation V at infinity, with

highest weight X· Assume that I! is not in S, and is split in k. Let >. be a

factor of I! in k, and let h>-(N) be the Satake parameter (defined using TJ) in

Cf!(G)(E) = T jW(E). We say that N is ordinary at I! if there is a lifting h>-(N)

of h>-(N) from

TjW(E0Qe) to T(E0Qe)

such that

(19.1)

in T(E 0 Qe) lies in the maximal compact subgroup of this locally compact,

abelian group.

When G is simply-connected, so G is of adjoint type, the condition that i>- lies

in a compact subgroup is

(19.2)

for all roots a of G and valuations v of (E 0 Qe)*. If N is ordinary at >., it is

ordinary for all factors of I! in k.

Assume that N is ordinary at£. Since x + TJ is a regular co-character ofT, the

lifting h>- of h>- satisfying (19.1) is unique. We let

( 19.3)

be the unramified homomorphism mapping a Frobenius elementS>- to i>-.

Let Ke: Gal(Q>-/Q>-) -+ Aut(Te)(Gm) = Z£ be the cyclotomic character. The

composite of "'>- with x + TJ gives a homomorphism

(19.4)

CoNJECTURE 19.5: Assume that N is ordinary at C. Then tile restriction of

p(N)e to tile decomposition group of a prime.\ dividing>. can be conjugated by

an element of G(E 0 Qe) so that

(1) Res>-p(N)e takes values in B(E 0 Qe), (2) The projection ofRes>-p(N)e to T(E 0 Qe) is equal to tile product homo

morphism (x + TJ) o "'£ · i>-.

A similar result should hold when N is Steinberg at I! and V = Q. We predict

that the image can be conjugated to lie in B(E 0 Qe) ><l Gal(k/Q)>-, and the

projection to T(E 0 Qe) ><l Gal(k/Q)>- should be the product of TJ o "'>- with the

standard projection to Gal(k/Q)A.


References

[1] J. Arthur, Lectures on automorphic £-functions, in £-Functions and Arithmetic,

London Mathematical Society Lecture Notes, Vol. 153, 1991, pp. 1-21.

[2] N. Bourbaki, Groupes et algebras de Lie, Hermann, Paris, 1982.

[3] A. Borel, Reduction theory for arithmetic groups, in Algebraic Groups and

Discontinuous Subgroups, Proceedings of Symposia in Pure Mathematics 9 (1966), 20-25.

[4] A. Borel, Automorphic £-functions, in Automorphic Forms, Representations, and

£-Functions, Proceedings of Symposia in Pure Mathematics 33 (1979), 27-61.

[5] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Annals of Mathematics 75 (1962), 485-535.

[6] A. Borel and J. Tits, Groupes reductifs, Publications Mathematiques de l'Institut

des Hautes Etudes Scientifiques 27 (1965), 55-150.

[7] S. Bosch, W. Liitkebohmert and M. Raynaud, Neron models, Ergebnisse der Mathematik 21, Springer-Verlag, Berlin, 1990.

[8] P. Cartier, Representations of p-adic groups, in Automorphic Forms, Representa

tions, and £-Functions, Proceedings of Symposia in Pure Mathematics 33 (1979), 111-155.

[9] J. de Siebenthal, Sur certains sous-groupes de rang un des groupes de Lie clos, Comptes Rendus de l'Academie des Sciences, Paris 230 (1950), 910-912.

[10] B. Gross, Heights and £-series, Number Theory CMS Conference Series 1 (1985), 115-188.

[11] B. Gross, Groups over Z, lnventiones Mathematicae 124 (1996), 263-279.

[12] B. Gross, On the motive of a reductive group, Inventiones Mathematicae 130

(1997), 287-313.

[13] B. Gross, On the motive of G and the principal homomorphism SL2 ~ G, Asian Journal of Mathematics 1 (1997), 208-213.

[14] B. Gross, On the Satake isomorphism, in Galois Representations in Arithmetic Algebraic Geometry, London Mathematical Society Lecture Notes, Vol. 254, 1998,

pp. 223-237.

[15] B. Gross and G. Savin, Motives with Galois group of type G2: an exceptional theta

correspondence, Compositio Mathematica 114 (1998), 153-217.

[16] R. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Mathematical

Journal 51 (1984), 611-650.

[17] B. Mazur, Modular curves and the Eisenstein ideal, Publications Mathematiques de l'lnstitut des Hautes Etudes Scientifiques 47 (1977), 33-186.

[18] D. Mumford, Abelian varieties, Publications of the Tata Institute 5, 1974.


[19] V. Platonov and A. R.apinchuk, Algebraic Groups and Number Theory, Academic

Press, New York, 1944.

[20] J.-P. Serre, Abelian £-adic representations and elliptic curves, Benjamin, New York,

1968.

[21) J.-P. Serre, Lettre de J-P. Serre a J. Tate, 7 aout 1987, Israel Journal of Mathematics 95 (1996), 282-291.

[22) J.-P. Serre, Proprietes conjecturates des groupes de Galois motiviques et des

representations £-adiques, in Motives, Proceedings of Symposia in Pure Mathe

matics 55 (1991), 377-400.

[23] C. Siegel, Berechnung von Zetafunktionen als ganzziihlige Stellen, Nachricthen der

Akademie der Wissenschaften in Gottingen. II. Mathematisch-Physikalische Klasse

10 (1969), 87-102.

(24] G. Shimura, Arithmetic Theory of Automorphic Functions, Princeton University

Press, 1971.

(25] R.. Steinberg, Representations of algebraic groups, Nagoya Mathematical Journal

22 (1963), 33-56.

[26] J. Tate, A review of non-Archimedean elliptic functions, in Elliptic Curves,

Modular Forms, and Fermat's Last Theorem, International Press Series in Number

Theory 1 (1995), 162-184.

(27] J. Tits, Reductive groups over local fields, in Automorphic Forms, Representations,

and £-Functions, Proceedings of Symposia in Pure Mathematics 33 (1979), 29-69.

[28) J. Tits, Representations lineares irreductibles d 'un groupe reductif sur un corps

quelconque, Journal fUr die reine und angewandte Mathematik 247 (1971), 196-

220.

Documents

Algebraic modular forms - Dartmouth College