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The aim of this paper is twofold. First, we introduce a new method for evaluating the multiplicity of a given discrete series representation in the space of level 1 automorphic forms of a split classical group $G$ over $\mathbf {Z}$ , and provide numerical applications in absolute rank $\leq 8$ . Second, we prove a classification result for the level one cuspidal algebraic automorphic representations of $\mathrm{GL}_{n}$ over $\mathbf {Q}$ ( $n$ arbitrary) whose motivic weight is $\leq 24$ . In both cases, a key ingredient is a classical method based on the Weil explicit formula, which allows to disprove the existence of certain level one algebraic cusp forms on $\mathrm{GL}_{n}$ , and that we push further on in this paper. We use these vanishing results to obtain an arguably “effortless” computation of the elliptic part of the geometric side of the trace formula of $G$ , for an appropriate test function. Thoses results have consequences for the computation of the dimension of the spaces of (possibly vector-valued) Siegel modular cuspforms for $\mathrm{Sp}_{2g}(\mathbf {Z})$ : we recover all the previously known cases without relying on any, and go further, by a unified and “effortless” method.

We give an explicit construction of global Galois gerbes constructed more abstractly by Kaletha to define global rigid inner forms. This notion is crucial to formulate Arthur's multiplicity formula for inner forms of quasi-split reductive groups. As a corollary, we show that any global rigid inner form is almost everywhere unramified, and we give an algorithm to compute the resulting local rigid inner forms at all places in a given finite set. This makes global rigid inner forms as explicit as global pure inner forms, up to computations in local and global class field theory.

The goal of this paper is to remove the irreducibility hypothesis in a theorem of Richard Taylor describing the image of complex conjugations by $p$-adic Galois representations associated with regular, algebraic, essentially self-dual, cuspidal automorphic representations of $\GL_{2n+1}$ over a totally real number field $F$. We also extend it to the case of representations of $\GL_{2n}/F$ whose multiplicative character is ''odd''. We use a $p$-adic deformation argument, more precisely we prove that on the eigenvarieties for symplectic and even orthogonal groups, there are ''many'' points corresponding to (quasi-)irreducible Galois representations. The recent work of James Arthur describing the automorphic spectrum for these groups is used to define these Galois representations, and also to transfer self-dual automorphic representations of the general linear group to these classical groups.

Real forms of a complex reductive group are classified by Galois cohomology H^1(Gamma,G_ad) where G_ad is the adjoint group. Cartan's classification of real forms in terms of maximal compact subgroups can be stated in terms of H^(Z/2Z,G_ad) where the action is by a (holomorphic) Cartan involution. The main result is that for any complex reductive group, possibly disconnected, there is a canonical isomorphism between H^1(Gamma,G) and H^1(Z/2Z,G). As applications we give short proofs of some well known results, including the Sekiguchi correspondence, Matsuki duality, results on Cartan subgroups, the rational Weyl group, and strong real forms. We also compute H^1(Gamma,G) for all simple, simply connected real groups., This paper replaces Galois Cohomology of Real Groups, arXiv:1310.7917 (by the first author). The proof here is more direct and does not require that the complex group G be connected

Let G be a special orthogonal group or an inner form of a symplectic group over a number field F such that there exists a non-empty set S of real places of F at which G has discrete series and outside of which G is quasi-split. We prove Arthur's multiplicity formula for automorphic representations of G having algebraic regular infinitesimal character at all places in S., Comment: 33 pages, comments welcome