2 1. INTRODUCTION

1.2. Motivations

There are several motivations for this problem. A first one is the deep conjec-

tural relations, due on the one hand to Langlands [Lan79], in the lead of Shimura,

Taniyama, and Weil, and on the other hand to Fontaine and Mazur [FM95], that

those numbers N(k1,k2, · · · , kn) share with arithmetic geometry and pure

motives3

over Q. More precisely, consider the three following type of objects:

(I) Pure motives M over Q, of weight w and rank n, with coeﬃcients in Q,

which are: simple, of conductor 1, such that M

∨

M(w), and whose

Hodge numbers satisfy

hp,q(M)

= 1 if (p, q) is of the form (ki,w − ki) and

0 otherwise.

(II) Continuous irreducible representations ρ : Gal(Q/Q) → GLn(Q ) which

are unramified outside , crystalline at with Hodge-Tate numbers k1

· · · kn, and such

that4 ρ∨

ρ ⊗

ωw.

(III) Cuspidal automorphic representations π of GLn over Q satisfying (a), (b)

and (c) above, of weights k1 k2 · · · kn,

Here is a fixed prime, and Q and Q are fixed algebraic closures of Q and Q .

To discuss the aforementioned conjectures we need to fix a pair of fields embeddings

ι∞ : Q → C and ι : Q → Q . According to Fontaine and Mazur, Grothendieck’s

-adic ´ etale cohomology, viewed with Q coeﬃcients via ι , should induce a bijection

between isomorphism classes of motives of type (I) and isomorphism classes of Ga-

lois representations of type (II) . Moreover, according to Langlands, the L-function

of the -adic realizations of a motive of type (I), which makes sense via ι∞ and

ι , should be the standard L-function of a unique π of type (III), and vice-versa.

These conjectural bijections are actually expected to exist in greater generality (any

conductor, any weights, not necessarily polarized), but we focus on this case as it

is the one we really consider in this paper. In particular, N(k1, · · · , kn) is also the

conjectural number of isomorphism classes of objects of type (I) or (II) for any .

Let us mention that there has been recently important progresses toward those con-

jectural bijections. First of all, by the works of many authors (including Deligne,

Langlands, Kottwitz, Clozel, Harris, Taylor, Labesse, Shin, Ngˆ o and Waldspurger,

see [GRFA11],[Shi11] and [CH13]), if π is of type (III) then there is a unique as-

sociated semisimple representation ρπ : Gal(Q/Q) → GLn(Q ) of type (II) with

the same L-function as π (via ι∞,ι ), up to the fact that ρπ,ι is only known to be

irreducible when n ≤ 5 (see [CG11]). Second, the advances in modularity results

in the lead of Wiles and Taylor, such as the proof of Serre’s conjecture by Khare

and Wintenberger (see e.g. [Kha06]), or the recent results [BGGT], contain striking

results toward the converse statement.

An important source of objects of type (I) or (II) comes from the cohomology

of proper smooth schemes (or stacks) over Z, about which solving problem 1.1 would

thus shed interesting lights. This applies in particular to the moduli spaces Mg,n

of stable curves of genus g with n-marked points and to certain spaces attached to

the moduli spaces of principally polarized abelian varieties (see e.g. [BFG11] and

3The reader is free here to choose his favorite definition of a pure motive [Mot94].

4Here

ω denotes the -adic cyclotomic character of Gal(Q/Q), and our convention is that its

Hodge-Tate number is −1.