Representations and classical automorphic forms Bill Casselman

Essays on representations of real groups

Representations and classical automorphic forms Bill Casselman

University of British Columbia [email protected]

This essay will display the realization of discrete series representations of SL

(R) on spaces of holomor- phic functions and relate them to automorphic forms for certain arithmetic groups. Then I shall relate these in turn to representations of ad`ele groups.

By now the realization of discrete series in terms of holomorphic functions on the upper half plane H is so well known that explaining how this remarkable construction arises naturally is rarely done. I’ll not explore the history of this construction, but short circuit the development by following a suggestion to be found in [Serre:1970].

The classical theory of automorphic forms, in spite of initial appearances, is about the group GL

, not SL

. The classical theory is concerned with functions on the upper half plane, which is acted on by fractional linear transformations in GL

(R), and it happens that the intersection of GL

(R) with GL

(Z) is SL

(Z). This accident causes at first mild confusion.

Contents

1. Representations on holomorphic functions 2. Identification with discrete series

3. Automorphic forms 4. Strong approximation 5. Introducing the ad`ele group 6. Hecke operators

7. Remarks 8. References

1. Representations on holomorphic functions

Let Z be the space of complex pairs (z, w) with

(z/w) > 0 (that is to say, z/w is real, with positive imaginary part). The map (z, w) 7→ z/w specifies it as a fibre space over the upper half plane H with fibre equal to C

. This fibring is split by the inverse map z 7→ (z, 1), and H may be identified with its image in Z.

The group GL

(C) acts on C

in the standard way:

g =

 a b c d

 : ω =

 z 1



7−→ gω =

 az + bw cz + dw

 . if g is real, we have

az + bw

cz + dw = a(z/w) + b c(z/w) + d

= aζ + b

cζ + d (ζ = z/w)

IM

 aζ + b cζ + d



= 1 2i

 aζ + b

cζ + d − aζ + b cζ + d



= ad − bc

|cζ + d|

(ζ) .

Therefore the group GL

(R) takes Z into itself. Since it commutes with scalar multiplication it preserves the fibring over H. The traditional action of G on H is derived from this, and the automorphy factor j(g, z) measures the extent to which the splitting fails to be G-covariant:

g

 z w



= (cz + d)

 (az + b)/(cz + d) 1



= j(g, z)

 g(z) 1



if g =

 a b c d

 .

That j(g, z) measures an obstruction explains why the equation j(gh, z) = j g, h(z)

j(h, z) is reminiscent of a cocycle relation.

The group G acts also on the space of smooth functions on Z:

F 7−→ F

, F

(ω) = F (gω)

is the associated right action. The space Z is an open subset of C × C, hence possesses an inherited complex structure. Since G preserves this complex structure, it takes the subspace of holomorphic functions into itself. Since G commutes with scalar multiplication it leaves stable the subspace C

(Z) of smooth functions of weight m—those F such that

F (λz, λw) = λ

F (z, w)

as well as its subspace C

(Z) of holomorphic functions. In effect, the functions in C

(Z) are smooth sections of a holomorphic bundle over H.

The group G acts transitively on Z, which is in fact a principal homogeneous space over G. That is to say that it is G-isomorphic to G given a choice of base point. The stabilizer of any fibre of the projection onto H is a copy of C

in G. In particular, the fibre over i in H may be identified with the copy of C

given by the embedding of C into M

(R)

ι: a + ib 7−→

 a −b

b a

 ,

since 

a −b

b a

  i 1



=

 ai − b bi + a



= (a + bi)

 (ai − b)/(a + bi) 1



= (a + bi)

 i 1

 .

Choosing a base point in Z allows us to identify functions on Z with functions on G. For each smooth F on Z, therefore, let Φ

be the corresponding function on G:

Φ

(g) = F

 g

 i 1



.

It is easy to see:

Lemma 1.1. The map F 7→ Φ

identifies C

(Z) with the space of smooth Φ satisfying the condition

[lifting-K]

Φ g · ι(λ)

= λ

Φ(g)

for λ in C

.

Next, I’ll show how to characterize the image of C

(Z). with respect to Φ. Before I state the main result, I’ll recall some elementary facts of complex analysis. A smooth C-valued function f = u(x, y) + iv(x, y) on an open subset of C is holomorphic if and only if the real Jacobian matrix of f



 

∂u

∂x

∂u

∂y

∂v

∂x

∂v

∂y



 

considered as a map from R

to itself lies in the image of C in M

(R). (Since this image generically coincides with the group of orientation-preserving similitudes, this means precisely that it is conformal.) This condition is equivalent to the Cauchy-Riemann equations

∂u

∂x = ∂v

∂y , ∂u

∂y = − ∂v

∂x . Holomorphicity may also be expressed by the single equation

∂f

∂z = 0 where

∂

∂z = 1 2

 ∂

∂x + i ∂

∂y

 . When f is holomorphic, its complex derivative is

∂f

∂z = 1 2

 ∂

∂x − i ∂

∂y

 . The notation is designed so that for an arbitrary smooth function

df = ∂f

∂z dz + ∂f

∂z dz where dz = dx + i dy.

Define these elements of the Lie algebra of G:

ζ =

 1 0 0 1



α =

 1 0 0 −1



η =

 1 0 0 0



= (1/2)(α + ζ) κ =

 0 −1

1 0



ν

=

 0 1 0 0



x

=

 1 −i

−i −1



=

 1 0 0 −1



− i

 0 1 1 0



= α − i(2ν

+ κ)

= (α + ζ) − ζ − i(2ν

+ κ)

= (2η − ζ) − i(2ν

+ κ) .

The significance of x

and its conjugate x

is that

[κ, x

] = ±2ix

.

Now for F in C

(Z), define a function f on the upper half-plane H to be its restriction to its image in Z:

f (z) = F (z, 1) . Lemma 1.2. For F in C

(Z) we have

[lifting-hol]

R

Φ

(p) = −4iy ∂f (z)

∂z if

p =

 y x 0 1



, p(i) = z = x + iy .

Proof. Since κ and ζ are in the Lie algebra of ι(C

):

R

F = −miF, R

F = −mF . But then

R

F (p) = (R

− 2iR

− iR

)F (p)

= (2R

− R

− 2iR

− iR

)F (p)

= (2R

− 2iR

+ m − i(−mi))F (p)

= (2R

− 2iR

)F (p) Now I apply the basic formula R

f (g) = L

f (g) to get

(2R

− 2iR

)F (p) = (2L

− 2iL

)F (p) . But

L

= x ∂

∂x + y ∂

∂y and

pηp

= yη − xν

so

(2L

− 2iL

)F (p) = 2y ∂f

∂y − 2iy ∂f

∂x

= −2iy

 ∂f

∂x + i ∂f

∂y



= −4iy ∂f

∂z . This has as consequence:

Proposition 1.3. The image of C

(Z) under the map F 7→ Φ

is the space C

(G) of all smooth

[lift-hola]

functions Φ in C

(G) such that

(a) Φ g · ι(λ)

= λ

Φ(g) for all λ in C

; (b) R

Φ = 0.

A function F in C

(Z) is determined by its restriction f to H:

F (z, w) = w

F (z/w, 1) = w

f (z/w) .

How does the action of G on F translate to an action of G on f ? If F in C

(Z) restricts to f then F

(z, 1) = F (az + b, cz + d) where g =

 a b c d



= (cz + d)

F (az + b)/(cz + d), 1

= (cz + d)

f

 az + b cz + d



= j(g, z)

f g(z)
.

For z in H and g in G, let  f 

 (g)



(z) = j(g, z)

f g(z)
.

Here is a summary of the situation. The group G = GL

(R) acts on Z and consequently also on the space C

(Z) of holomorphic functions F on Z such that F (λω) = λ

F (ω). On the one hand, this space may be identified with the space of smooth functions Φ on G such that (a) R

Φ = λ

Φ and (b) R

Φ = 0. This identification is compatible with the left regular action of G on the space of such Φ.

On the other hand, we may restrict F to the copy of H in Z, and the action of G is compatible with the action on holomorphic functions f on H taking f to f 

(g

)

. The restriction is an isomorphism. The action of g on C

(Z) is compatible with the map f 7→ f | (g)

. In short, we have so far a diagram of G-spaces

C

(Z)

C

(G)

C

(H)

images/cm-diagram.eps

in which the arrows are linear isomorphisms, and the action of G on the lower space is via f 7→  f 

 (g)

 . It will be convenient in the rest of this lecture to modify things slightly. Let

Φ

= Φ

(g) · det

(g) .

The functions Φ

are now characterized by the conditions (a) Φ g · ι(λ)

= (λ/|λ|)

Φ(g) and (b) R

Φ = 0. In particular, Φ is fixed by the connected component of the center of G. To go with this modification, redefine 

f   [g]

 (z) = det

(g)(cz + d)

f g(z)
.

Here, the connected component of the scalars also acts trivially. The left regular action of G on the functions Φ and this new one on functions on H are compatible. This normalization is just one of several in the literature. Each has its own charms, but for us invariance under the positive real scalars turns out to be most convenient. Keep in mind that the embedding of SL

(R) into GL

(R) induces an isomorphism of it with the quotient GL

(R)/R

.

The explicit relation between f and Φ

is

Φ

(g) = f g(i)

det

(g) j(g, i)

.

2. Identification with discrete series

Let π

be the left representation of G = GL

(R) on the space of holomorphic functions on H taking f to f 

 [g

]

. This is trivial on the connected component of the scalar matrices, hence may be identified with a representation of SL

(R). In this section we’ll see that for m > 1 it is an incarnation of the discrete series D

and for m = 1 one of the so-called ‘limit’ of discrete series.

Let K be the maximal compact subgroup SO

of G. Recall the character

ε:

 c −s

s c



7−→ c + is .

The restriction of π

to K will decompose into a sum of one-diemnsional eigenspaces with respect to K.

Proposition 2.1. The eigenfunctions of K in π

are the functions

[k-finite-holomorphic]

f

(z) =

 z − i z + i



 1 z + i



on H with p ≥ 0. For k in K

π

(k)f

= ε

(k)f

.

Proof. The action of G on Z is inherited from a linear one. It maps linear functions to linear functions, polynomials to polynomials. Linear functions correspond to row vectors, since

pz + qw = [ p q ]

 z w

 .

So the matrix g takes the linear function pz + qw to that with row vector [ p q ] 7→ [ p q ] g

.

The vectors [1, ±i] are eigenfunctions of K in this representation, since

[1, ±i]

 c −s

s c



= (c ± is)[1, ±i] .

Equivalently, the functions z ± iw are eigenfunctions of K, and more precisely k takes z ± iw to ε(k)

(z ± iw). Hence when acting on polynomials it takes

(z − iw)

(z + iw)

7−→ ε

(k)(z − iw)

(z + iw)

.

Since 1/(z − iw) has a singularity on Z, the function (z − iw)

(z + iw)

will lie in C

(Z) if and only if

−m = p + q and p ≥ 0. Restricting to the image of H in Z concludes.

Suppose m > 0. The restriction of π

to K is the direct sum of characters ε

for ℓ = 0, 1, . . . It is an interesting exercise to verify explicitly that when m ≤ 0 the representation π

contains the unique irreducible representation of dimension |m| + 1.

From now on in this essay, assume m > 0.

We know already that the measure dx dy/y

is G-invariant on H. Here is a useful generalization of this:

Proposition 2.2. The measure y

dx dy is G-invariant for π

.

[H-measure]

Proof. What this means is that for f in C

(H) and f

= π

(g)f we have Z

H

 f

(z) 

y

dx dy y

=

Z

H

 f(z) 

y

dx dy y

. We may as well assume that g is in SL

(R). The first integrand is

 f g(z)
 

j(g, z) 

y

(z) = 

f g(z)
 

j(g, z) 

y

g(z)
 j(g, z) 

= 

f g(z)


y

g(z)
, so the result follows from the invariance of dx dy/y

.

The function (z − i)

/(z + i)

is bounded on H and 1/(z + i)

is square-integrable with respect to the measure y

dx dy for m > 1. It follows that the representation π

is square-integrable for m > 1—i.e.

is in fact a discrete summand of L

SL

(R)
. 3. Automorphic forms

A congruence subgroup Γ of GL

(R) is a subgroup of

SL

(Z) = GL

(Z) ∩ GL

(R) containing some principal congruence subgroup

Γ(N ) = 

γ ∈ SL

(Z) 

 γ ≡ I (mod N)} . The region

D = 

z = x + iy 

 |x| ≤ 1/2, |z| ≥ 1

is a fundamental domain of SL

(Z) in H, and some finite number of transforms of this region will be a fundamental domain for Γ. A function on D is said to be of moderate growth if it is O(y

) for some M > 0, and rapidly decreasing if O(y

) as y → ∞. Similarly for a fundamental domain of Γ.

An automorphic form of weight m for Γ is a holomorphic function f (z) on H satisfying a condition of moderate growth on every fundamental domain such that:

f 

 [γ]

= f or f γ(z)

= j(g, z)

f (z)

for all γ in Γ. Following the previous section, define the corresponding function on G = GL

(R) Φ

(g) = f g(i)

det

(g) j(g, i)

.

It is also invariant under Γ and again in some sense of moderate growth. From the results of the previous section:

Theorem 3.1. The space of holomorphic forms of weight m for Γ is isomorphic to the space of smooth

[classical]

(in fact, necessarily real analytic) functions Φ on Γ\G of moderate growth such that (a) Φ(gk) = ε

(k)Φ(g) for k in SO

;

(b) R

Φ = 0;

(c) Φ(gz) = Φ(g) for z in the connected component of the center of G.

The existence of an automorphic form of weight m says something about the occurrence of a discrete series representation of G in C

(Γ\G), but not quite what might be naively expected. Conditions (a)–(c) mean that the representation of G generated by Φ is a copy of the dual discrete series b π

. Formally, this is related to Frobenius reciprocity, since a Γ-invariant vector in π

lies in

Hom

(b π

, C)

which according to a kind of Frobenius reciprocity is formally the same as Hom

π

, C

(Γ\G)

.

This formal analysis is not legal, because the Γ-invariant function is not in π

but in some larger space.

However, this heuristic idea can be justified with some modest amount of trouble.

4. Strong approximation

We start with an elementary result:

Proposition 4.1. The canonical projection from SL

(Z) to SL

(Z/N ) is surjective.

[sa-ZN]

Proof. The Chinese Remainder Theorem says that the ring Z/N is isomorphic to the product of rings Z/p

if N = Q

p

. Therefore it suffices to prove the theorem in the case N is a prime power. It is simple to proceed from p

to p

, so the only critical case is N = p. In this case one can use the Bruhat decomposition to reduce to the case of a diagonal matrix in SL

(Z/p). So we are given an integral matrix

 a 0 0 d



with ad ≡ 1 modulo p. Choose α, δ inverse to each other modulo p

, congruent to a, d. Thus αδ − βp

= 1

for some β, and the matrix 

α p

βp δ



is what we are looking for.

The previous result can be formulated as saying that if one is given for each of a finite number of primes p an integral matrix A

with det(A

) congruent to 1 modulo p

then one can find an integral matrix A of determinant 1 simultaneously congruent to each A

modulo p

. This can be generalized.

Proposition 4.2. (Strong approximation for SL

(Q)) Suppose that for each of a finite set S of primes p

[strong-approx]

we are given g

in SL

(Q

) and n

≥ 0. There exists γ in SL

(Q) such that (a) the matrix entries of γ lie in Z

for p not in S;

(b) for each p in S, γ

g

lies in SL

(Z

) and is congruent to I modulo p

.

Proof. By the previous result, it suffices to deal with the case where all n

= 0. What has to be shown now is that there exists γ satisfying (a) with γ

g

in SL

(Z

) for all p in S. An induction argument reduces this to the case where S has just one prime.

Let Q

be the ring of fractions whose denominators are powers of p. The image of an element of Q

in Q

for q 6= p lies in Z

. We now want to show that given g

in SL

(Q

) there exists γ in SL

(Q

) such that γ

g

lies in SL

(Z

).

By the elementary divisors theorem for M

(Q

), we can find matrices γ

and γ

such that γ

gγ

is

a diagonal matrix d whose diagonal entries are powers p

of p. Let m = sup |m

− m

|. Then for any

matrix M in M

(Z

) the conjugate d

p

M d will have entries in Z

. Choose γ

, γ

congruent to γ

, γ

modulo p

. If γ = γ

dγ

, then γ lies in SL

(Q

) and γ

g

lies in SL

(Z

).

5. Introducing the ad `ele group

The ring A of ad`eles over Q is the subring of all a

, (a

)

in the direct product R × Q

Q

with a

in Z

for all but a finite number of primes p. It is a topological ring in which a basis of neighbourhoods of 0 are subsets I × Q

U

where I is a neighbourhood of 0 in R, and each U

is a subgroup p

Z

with all but a finite number of n

= 0.

The group GL

(A) consists of the subgroup of elements of GL

(R) × Q

GL

(Q

) with all but a finite number of g

in GL

(Z

). As we’ll see in a moment, the field Q is a discrete subring of A. Consequently, the group GL

(Q) is also a discrete subgroup of GL

(A). In the modern theory of automorphic forms, automorphic forms are specified to be functions on GL

(Q)\GL

(A) rather than on arithmetic quotients Γ\H or even Γ\GL

(R). There are many reasons for this change. One good one is that in many problems involving automorphic forms somewhat complicated questions involving number theory are replaced by simpler questions about analysis on the local groups GL

(R) and GL

(Q

). This is especially true of Hecke operators. Another good reason is that the discrete group GL

(Q) is in most ways much simpler than arithmetic subgroups of SL

(Z). In applying the trace formula, for example, the conjugacy classes of GL

(Q) are much simpler to understand than those of Γ(N ). In any case, what I want to do here is very limited—to provide a simple introduction to the ad`eles and to explain what is involved in transferring classical automorphic forms to functions on GL

(Q)\GL

(A). I’ll not prove everything in detail.

The fields R and Q

are all the reasonable completions of Q, and are usefully considered uniformly.

To emphasize this and make notation more efficient I set Q

= R. In this notation A is a subring of Q

p≤∞

Q

.

Proposition 5.1. The field Q is a discrete additive subgroup of A. We have

[discrete-adeles]

A = Q + 

[0, 1) × Y

p

Z

 .

In fact the region [0, 1)× Q

p

Z

is a fundamental domain for the discrete subgroup Q, and the embedding of R into A induces an isomorphism of Z\R with Q\A/ Q

Z

.

Proof. To see that Q is discrete in A we just need to verify that the open subset

 |x| < 1

× Y

Z

contains no rational numbers.

As for the second claim, what it means is that if we are given an ad`ele a = (a

) there exists a rational number α such that a − α lies in I × Z

. Choosing an integer n such that a

− n lies in I, we are reduced to the claim that given a finite set S of primes and for each p in S a p-adic number a

, there exists a rational number α lying in Z

for p not in S with a

− α lying in Z

for p in S. This is a variant of the Chinese Remainder Theorem.

Proposition 5.2. We have

[adele-factors]

A

= Q

· 

R

× Y Z

 .

Proof. Suppose a unit ad`ele (a

) to be given, with no a

= 0 and all but a finite number of a

in Z

, say for p not in the finite set S. We want to find α > 0 in Q

with α in Z

for p in S and a

α

in Z

for p in S. This problem reduces to unique factorization for integers.

The strong approximation theorem for SL

(Q) is equivalent to the following result about SL

(A):

Theorem 5.3. Suppose that for each prime p we are given a compact open subgroup K

⊆ SL

(Z

) such

[strong-approx]

that all but a finite number of K

= SL

(Z

). Then SL

(A) = SL

(Q) · 

SL

(R) × Y K

 .

In other words, SL

(Q) · SL

(R is dense in SL

(A).

Corollary 5.4. Suppose that for each prime p we are given a compact open subgroup K

⊆ GL

(Z

)

[strong-approx]

such that (a) all but a finite number of K

= GL

(Z

); (b) det(K

) = Z

for all p. Then GL

(A) = GL

(Q) · 

GL

(R) × Y K

 .

More precisely,

GL

(Q)\GL

(A)/SO

× Y K

may be identified with Γ\H if

Γ = GL

(Q) ∩ 

GL

(R) × Y K

 .

Now I explain how functions on Γ\GL

(R) become functions on GL

(Q)\GL

(A). Suppose we are given for each p a compact open subgroup K

of GL

(Z

), with almost all K

= GL

(Z

and with all det K

= Z

. Let K = Q

K

. Then

U

= GL

(R) × Y K

is an open subgroup of GL

(A). The group

Γ

= Γ ∩ U

will be a congruence subgroup. If K

= GL

(Z

) for all p, for example, then Γ = SL

(Z).

Every congruence subgroup arises in this fashion. For a given Γ, the intersection K

∩ SL

(Z

) is determined by Γ (it is its closure in SL

(Z

)) but there will be several choices possible for K

itself. If Γ = Γ(N ) the standard choice is the group of all k of the form

 ∗ 0 0 1



modulo N .

Lemma 5.5. In this situation, the injection

[k-gamma]

Γ

\GL

(R) −→ GL

(Q)\GL

(A)/K is a bijection.

This means among other things that automorphic forms of weight m on Γ\H lift uniquely to functions

on GL

(Q)\GL

(A) fixed by K. It is important to realize that this lifting depends not only on Γ but

also on the choice of K. This phenomenon occurs also in the classical theory, in the definition of Hecke

operators for general congruence groups.

6. Hecke operators

The starting point of the classical theory of Hecke operators is the observation that if g lies in GL

(Q) and f is an automorphic form for the congruence subgroup Γ then the function f

: z 7→ f (gz) is an automorphic form for g

Γg, since

f

(g

γg z) = f (gg

γ gz) = f

(z) . This should be combined with another simple observation:

Proposition 6.1. If Γ is a congruence subgroup of SL

(Z) and g lies in GL

(Q) then g

Γg ∩ Γ is also

[congruence-intersection]

a congruence group.

Proof. We may assume, by the elementary divisors theorem, that g is diagonal and Γ = Γ(M ). We must show that g

Γ(M )g ∩ Γ(M ) contains some Γ(N ). This is easy.

Suppose g in M

(Z) with positive determinant. The double coset ΓgΓ will be a union of left Γ-cosets Γg

. Each g

can be chosen to be of the form gγ

with γ

in Γ. The map

γ 7−→ Γgγ

is therefore surjective onto ΓgΓ. The isotropy subgroup of g is Γ ∩ g

Γg, so that the associated map (Γ ∩ g

Γg)\Γ −→ Γ\ΓgΓ

is a bijection, and as a consequence of the previous result the right hand side is finite. Corresponding to the double coset ΓgΓ = S

Γg

we can therefore define the Hecke operator T [g]

f 7−→ f 

 [ΓgΓ]

= X f 

 [g

]

on the space of automorphic forms of weight m for Γ. This definition differs from the classical definition by a power of det(g). It can be extended consistently to operators T [g] for any rational matrix, since scalar matrices act trivially. If we associate to f the function

Φ(g) = f   [g]

on Γ\G, then f 

 [ΓgΓ]

corresponds to P

Φ(g

g) (where ΓgΓ = S Γg

).

Cusp forms on Γ\H are square-integrable, and give rise to square-integrable functions on Γ\G. The representation of G on L

(Γ\G) is unitary, which means that the adjoint of L

is L

. The Hecke operator T [g] is therefore self-adjoint when T [g] = T [g

].

The classical theory of Hecke operators applies to all congruence groups, but there are complications in the general case that I wish to avoid (related to the problem of choosing K

for a given Γ), so I’ll assume for a while that Γ = SL

(Z).

Suppose p a prime, and let

g =

 p 0 0 1

 .

In this case ΓgΓ differs from Γg

Γ by a scalar, so T [g] = T [g

] and T [g] is self-adjoint on the space of cusp forms. To get an explicit expression, note that

 p 0 0 1



 a b c d

  p 0 0 1



=

 a p

b

pc d



the intersection Γ ∩ g

Γg is the group of all matrices lying in the Borel subgroup

 ∗ ∗ 0 ∗



modulo p, and the quotient (g

Γg ∩ Γ)\Γ is in bijection with P

(F

), a set of p + 1 elements.

What does the Hecke operator T [g] on Γ\G correspond to on GL

(Q)\GL

(A)? The local Hecke algebra on the p-adic group GL

(Q

) is made up of functions of compact support that are left- and right-invariant under K

= GL

(Z

).

If ϕ is in the p-adic Hecke algebra and Φ a function on the adelic quotient fixed by GL

(Z

), the convolution

R

Φ(g) = Z

GL2(Qp)

ϕ(x)Φ(gx), dx is defined. In particular, the characteristic function ϕ

of the double coset

K

 p 0 0 1

 K

is in this algebra.

Proposition 6.2. If f is an automorphic form for Γ = SL

(Z) and f corresponds to Φ on GL

(Q)\GL

(A)

[hecke-global]

then f | T

corresponds to R

Φ.

How the Hecke operators relate to representations of GL

(Q

) is the next question to take up, but that won’t be done in this essay.

7. Remarks

The transition from classical automorphic forms to functions on the ad`ele quotient depends, as we have seen, on many features of arithmetic of the rational numbers that do not remain valid for other number fields. For them, the correct way to proceed is to start off immediately with functions on the ad`ele quotient, making occasional forays into arithmetic quotients for a few questions of real analysis. One loses, perhaps, a certain amount of intuition, but gains enormously in elegance. One sign that this is the correct approach is that notation and basic definitions expanded rapidly in the period 1920–1965 (starting with Ramanujan, passing through Mordell, Hecke, Siegel, and Weil) but that basic notions have remained stable in the longer period since then. The summer symposium in Boulder in 1965 marks the change from what might be called the classical period to the modern one.

The description of Q\A generalizes nicely to other number fields, but the factorization of A

depends

strongly on the fact that Z is a principal ideal domain. On the other hand, although the proof of the

strong approximation theorem for SL

(Q) depends strongly on factorization in Z, it is remarkable that

the theorem itself continues to remain valid for SL

(F ) where F is any number field. This result is due

to Martin Eichler. The proof is entirely different from the one given here, and and a somewhat simplified

version can be found in [Kneser:1965]. In fact, strong approximation is valid for any simply connected

semi-simple group over a number field. A proof in this generality can be found in [Kneser:1966], on

the assumption of the Hasse principle (which was subsequently verified). A complete if rather dense

discussion along different lines can be found in [Platonov-Rapinchuk:1994].

8. References

1. J. W. S. Cassels and A. Fr¨ohlich, Algebraic number theory , Thompson, 1967.

2. Martin Kneser, ‘Starke approxiamtion in algebraische Gruppen’, Journal f ¨ur die reine and angewandte Mathematik ²¹⁸ (1965), 190–203.

3. ——, ‘Strong approximation’, in Algebraic groups and discontinuous subgroups , Proceedings of Symposia in Pure Mathematics ^IX , 1966.

4. James Milne, Modular functions and modular forms , course notes available at http://www.jmilne.org/math/CourseNotes/math678.html

5. Vladimir Platonov and Andrei Rapinchuk, Algebraic groups and number theory , Academic Press, 1994.

6. Michael Pohst and Hans Zassenhaus, Algorithmic algebraic number theory , Cambridge University Press, 1989.

7. J-P. Serre, Cours d’Arithm ´etique , Presses Universitaires de France 1970.

8. G. Shimura, Arithmetic theory of automorphic functions , Princeton University Press, 1971.