1.3. THE MAIN RESULT 3

[FC90]). As an example, the vanishing of some N(k1, · · · , kn) translates to a con-

jectural non-existence theorem about Galois representations or motives. A famous

result in this style is the proof by Abrashkin and Fontaine that there are no abelian

scheme over Z (hence no projective smooth curve over Z of nonzero genus), which

had been conjectured by Shaffarevich (see [Fon85],[Fon93]). The corresponding van-

ishing statement about cuspidal automorphic forms had been previously checked

by Mestre and Serre (see [Mes86]). See also Khare’s paper [Kha07] for other conjec-

tures in this spirit as well as a discussion about the applications to the generalized

Serre’s conjecture.

A second motivation, which is perhaps more exotic, is the well-known problem

of finding an integer n ≥ 1 such that the cuspidal cohomology Hcusp(SLn(Z),

∗

Q)

does not vanish. It would be enough to find an integer n ≥ 1 such that

N(n − 1, · · · , 2, 1, 0) = 0. Results of Mestre [Mes86], Fermigier [Fer96] and

Miller [Mil02] ensure that such an n has to be ≥ 27 (although those works do

not assume the self-duality condition). We shall go back to these questions at the

end of this introduction.

Last but not least, it follows from Arthur’s endoscopic classification [Art11] that

the dimensions of various spaces of modular forms for classical reductive groups

over Z have a ”simple” expression in terms of these numbers. Part of this paper is

actually devoted to explain this relation in very precise and concrete terms. This

includes vector valued holomorphic Siegel modular forms for Sp2g(Z) and level 1

algebraic automorphic forms for the Z-forms of SOp,q(R) which are semisimple over

Z (such group schemes exist when p − q ≡ 0, ±1 mod 8). It can be used in both

ways: either to deduce the dimensions of these spaces of modular forms from the

knowledge of the integers N(−), or also to compute these last numbers from known

dimension formulas. We will say much more about this in what follows as this is

the main theme of this paper (see Chapter 3).

1.3. The main result

We will now state our main theorem. As many results that we prove in this

paper, it depends on the fabulous work of Arthur in [Art11]. As explained loc. cit.,

Arthur’s results are still conditional to the stabilization of the twisted trace formula

at the moment. All the results below depending on this assumption will be marked

by a simple star ∗. We shall also need to use certain results concerning inner forms

of classical groups which have been announced by Arthur (see [Art11, Chap. 9])

but which are not yet available or even precisely stated. We have thus formulated

the precise general results that we expect in two explicit Conjectures 3.18 and 3.20.

Those conjectures include actually a bit more than what has been announced by

Arthur in [Art11], namely also the standard expectation that for Adams-Johnson

archimedean Arthur parameters, there is an identification between Arthur’s packets

in [Art11] and the ones of Adams and Johnson in [AJ87]. The precise special cases

that we need are detailed in §3.10. We state in particular Arthur’s multiplicity

formula in a completely explicit way, in a generality that might be useful to arith-

metic geometers. We will go back to the shape of this formula in §1.8.2. All the

results below depending on the assumptions of [Art11] as well as on the assump-

tions 3.18 and 3.20 will be marked by a double star

∗∗.

Of course, the tremendous

recent progresses in this area allow some optimism about the future of all these