CHAPTER 1

Introduction

1.1. A counting problem

Let n ≥ 1 be an integer. Consider the cuspidal automorphic representations π

of GLn over Q (see [GGPS66, Ch. 3],[BJ79, §4],[Cog04]) such that:

(a) (polarization)

π∨

π ⊗ | ·

|w

for some w ∈ Z,

(b) (conductor 1) πp is unramified for each prime p,

(c) (algebraicity) π∞ is algebraic and regular.

Our main aim in this paper is to give for small values of n, namely for n ≤ 8,

the number of such representations as a function of π∞. Recall that by the Harish-

Chandra isomorphism, the infinitesimal character of π∞ may be viewed following

Langlands as a semisimple conjugacy class in Mn(C) (see §3.4, §3.6). Condition

(c)

means1

that the eigenvalues of this conjugacy class are distinct integers. The

opposite of these integers will be called the weights of π and we shall denote them

by k1 k2 · · · kn. When n ≡ 0 mod 4, we will eventually allow that kn/2 =

kn/2+1 but to simplify we omit this case in the discussion for the moment. If π

satisfies (a), the necessarily unique integer w ∈ Z such that

π∨

π ⊗ | ·

|w

will be

called the motivic weight of π, and denoted w(π).

Problem 1.1. For any n ≥ 1, determine the number N(k1,k2, · · · , kn) of cus-

pidal automorphic representations π of GLn satisfying (a), (b) and (c) above, and

of weights k1 k2 · · · kn.

An important finiteness result of Harish-Chandra ([HC68, Thm. 1.1]) asserts

that this number is indeed finite, even if we omit assumption (a). As far as we

know, those numbers have been previously computed only for n ≤ 2. For n = 1,

the structure of the id` eles of Q shows that if π satisfies (a), (b) and (c) then

w(π) = 2 k1 is even and π = | ·

|−k1

. By considering the central character of π,

this also shows the relation n w(π) = 2

∑

n

i=1

ki for general n. More interestingly,

classical arguments show that N(k − 1, 0) coincides with the dimension of the space

of cuspidal modular forms of weight k for SL2(Z), whose dimension is well-known

(see e.g. [Ser70]) and is

about2

[k/12]. Observe that up to twisting π by

|·|kn

, there

is no loss of generality in assuming that kn = 0 in the above problem. Moreover,

condition (a) implies for i = 1, · · · , n the relation ki + kn+1−i = w(π).

1The

term algebraic here is in the sense of Borel [Bor77, §18.2], and is reminiscent to Weil’s

notion of Hecke characters of type A0: see §3.6. Langlands also uses the term of type Hodge, e.g.

in [Lan96, §5]. See also [BG], who would employ here the term L-algebraic, for a discussion of

other notions of algebraicity, as the one used by Clozel in [Clo90].

2We

denote by [x] the floor of the real number x.

1