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
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