4 1. INTRODUCTION

assumptions!5

Besides Arthur’s work, let us mention the following results which

play a crucial role here: the proof by Chaudouard, Laumon, Ngˆ o and Waldspurger

of Langlands’ fundamental lemma ([Wald09],[Ngˆ o10],[CLau12]), the works of Shel-

stad [She08] and Mezo [Mezb] on endoscopy for real groups, and the recent works

of Labesse and Waldspurger on the twisted trace formula [LW13].

Theorem∗∗ 1.2. Assume n ≤ 8 and n = 7. There is an explicit, computable,

formula for N(k1, · · · , kn).

Although our formulas are explicit, one cannot write them down here as they

are much too big: see §1.8.1 for a discussion of the formula. Nevertheless, we im-

plemented them on a computer and have a program which takes (k1,k2,...,kn) as

input and returns N(k1,...,kn). When k1 − kn ≤ 100, the computation takes less

than ten minutes on our machine6: see the website [CR] for some data and for our

computer programs. We also have some partial results concerning N(k1,...,k7).

This includes an explicit upper bounds for these numbers as well as their values

modulo 2, which is enough to actually determine them in quite a few cases (for

instance whenever k1 − k7 ≤ 26). On the other hand, as we shall see in Propo-

sition 1.10 below, these numbers are also closely related to the dimensions of the

spaces of vector valued Siegel modular forms for Sp6(Z). In a remarkable recent

work, Bergstr¨ om, Faber and van der Geer [BFG11] actually found a conjectural

explicit formula for those dimensions. Their method is completely different from

ours: they study the number of points over finite fields of M3,n and of certain bun-

dles over the moduli space of principally polarized abelian varieties of dimension

3. Fortunately, in the few hundreds of cases where our work allow to compute this

dimension as well, it fits the results found by the formula of these authors! Even

better, if we assume their formula we obtain in turn a conjectural explicit formula

for N(k1, · · · , k7).

1.4. Langlands-Sato-Tate groups

We not only determine N(k1, · · · , kn) for n ≤ 8 (with the caveat above for

n = 7) but we give as well the conjectural number of π of weights k1 · · · kn

having any possible Langlands-Sato-Tate group. We refer to the appendix B for a

brief introduction to this conjectural notion (see also [Ser68, Ch. 1, appendix]).

Here are certain of its properties.

First, a representation π as above being given, the Langlands-Sato-Tate group

of π (or, for short, its Sato-Tate group) is a compact Lie group Lπ ⊂ GLn(C),

which is well-defined up to GLn(C)-conjugacy. It is ”defined” as the image of

the conjectural Langlands group LZ of Z, that we view as a topological following

Kottwitz, under the hypothetical morphism LZ → GLn(C) attached to π ⊗ | · |

w(π)

2

([Lan79],[Kot88],[Art02]). The natural representation of Lπ on

Cn

is irreducible

and self-dual.

The group Lπ is equipped with a collection of conjugacy classes

Frobp ⊂ Lπ

5Note

added in proof: in 2014, Moeglin and Waldspurger published on the arXiv a series

of preprints culminating in a proof of the stabilization of the twisted trace formula, thus making

unconditional the results of [Art11] hence the results of this paper that are marked with a simple

star.

6Four processors Northwood Pentium 4, 2.80 GHz, 5570.56 BogoMIPS.