Let 1 G be a profinite group which is topologically finitely generated 2 , p a prime number and d ≥ 1 an integer. We show that the functor from rigid analytic spaces over Q p to sets, which associates to a rigid space Y the set of continuous d-dimensional pseudocharacters G −→ O(Y ), is representable by a quasi-Stein rigid analytic space X, and we study its general properties.Our main tool is a theory of determinants extending the one of pseudocharacters but which works over an arbitrary base ring ; an independent aim of this paper is to expose the main facts of this theory. The moduli space X is constructed as the generic fiber of the moduli formal scheme of continuous formal determinants on G of dimension d.As an application to number theory, this provides a framework to study rigid analytic families of Galois representations (e.g. eigenvarieties) and generic fibers of pseudodeformation spaces (especially in the "residually reducible" case, including when p ≤ d).σ∈S n+1 ε(σ)T σ (g 1 , g 2 , . . . , g n+1 ) = 0, 1 The author is supported by the C.N.R.S., as well as by the A.N.R. project ANR-10-BLAN 0114. 2 Actually, we only assume that for any normal open subgroup H ⊂ G, there are only finitely many continuous group homomorphisms H −→ Z/pZ.3 This expression has the following important interpretation, due to Kostant. Assume that T : GL m (A) → A is the trace map, if g 1 , . . . , g n ∈ GL m (A) and if σ ∈ S n , then T σ (g 1 , . . . , g n ) coincides with the trace of the element (g 1 , . . . , g n )σ acting on V ⊗An , where V := A m . 1 2 GAËTAN CHENEVIERwhere ε(σ) denotes the signature of the permutation σ. We say that T is a d-dimensionalThe main interest of pseudocharacters lies in the close relations they share with traces of representations : by an old result of Frobenius [Fr, p. 50], the trace of a representation G −→ GL d (A) is a d-dimensional pseudocharacter 4 , and it is known that the converse holds when A is an algebraically closed field with d! ∈ A * (Procesi [P3], Taylor [T] for Q-algebras, [Rou] in general 5 ) as well as in various other situations (see below). In particular, we obtain this way an interesting parametrization of the isomorphism classes of semisimple representations of G over such algebraically closed fields. As the covariant functor from the category of Z[1/d!]−commutative algebras with unit to the category Ens of sets, which associates to A the set of d-dimensional pseudocharacters G → A, is obviously representable 6 , it turned out to be an interesting substitute for the quotient functor Hom(G, GL d (−))/PGL d (−) of isomorphism classes of d-dimensional representations of G. Indeed, since they have been introduced in number theory by Wiles [W] (when d = 2), and by Taylor [T] under the form above (sometimes under the name of pseudorepresentations), they have proved to be a successful tool, first to actually construct some (Galois) representations, and then to study Galois representations and Hecke-algebras.Over Q-algebras, most of the basic properties of pseudocharacters follow ac...
Let K be a CM number field and G K its absolute Galois group. A representation of G K is said to be polarized if it is isomorphic to the contragredient of its outer complex conjugate, up to a twist by a power of the cyclotomic character. Absolutely irreducible polarized representations of G K have a sign ±1, generalizing the fact that a self-dual absolutely irreducible representation is either symplectic or orthogonal. If Π is a regular algebraic, polarized, cuspidal automorphic representation of GL n (A K ), and if ρ is a p-adic Galois representation attached to Π, then ρ is polarized and we show that all of its polarized irreducible constituents have sign +1. In particular, we determine the orthogonal/symplectic alternative for the Galois representations associated to the regular algebraic, essentially self-dual, cuspidal automorphic representations of GL n (A F ) when F is a totally real number field.
In this paper, we give definitions for p-adic automorphic forms on any twisted form of GL n =Q compact at infinity, and we construct the ''eigenvariety'' of finite slope eigenforms of wild level G 0 ðpÞ, at a split place p. Besides some generalisation of Coleman-Mazur theory, the main ingredients are the very existence of the ''orthonormal'' p-adic analytic family of principal series of G 0 ðpÞ and its most basic properties. We give analogues of Coleman's ''small slope forms are classical'' and of Wan's bounds for explicit radii for the families. As an application, we can construct n-dimensional p-adic families of non ordinary, n-dimensional, refined Galois representations coming from Shimura varieties of some unitary groups.
We determine the number of level 1, polarized, algebraic regular, cuspidal automorphic representations of GL n over Q of any given infinitesimal character, for essentially all n ≤ 8. For this, we compute the dimensions of spaces of level 1 automorphic forms for certain semisimple Z-forms of the compact groups SO 7 , SO 8 , SO 9 (and G 2 ) and determine Arthur's endoscopic partition of these spaces in all cases. We also give applications to the 121 even lattices of rank 25 and determinant 2 found by Borcherds, to level one self-dual automorphic representations of GL n with trivial infinitesimal character, and to vector valued Siegel modular forms of genus 3. A part of our results are conditional to certain expected results in the theory of twisted endoscopy.1 An important finiteness result of Harish-Chandra ([HC68, Thm. 1.1]) asserts that this number is indeed finite, even if we omit assumption (a). As far as we know, those numbers have been previously computed only for n ≤ 3. For n = 1, the structure of the idèles of Q shows that if π satisfies (a), (b) and (c) then w(π) = 2 k 1 is even and π = |.| −k 1 . By considering the central character of π, this also shows the relation n w(π) = 2 n i=1 k i for general n. More interestingly, classical arguments show that N(k − 1, 0) coincides with the dimension of the space of cuspidal modular forms of weight k for SL 2 (Z), whose dimension is well-known (see e.g. [Ser70]) and is about 2 [k/12]. Observe that up to twisting π by | · | kn , there is no loss of generality in assuming that k n = 0 in the above problem. Moreover, condition (a) implies for i = 1, · · · , n the relation k i +k n+1−i = w(π).1.3. Motivations. There are several motivations for this problem. A first one is the deep conjectural 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(k 1 , k 2 , · · · , k n ) share with arithmetic geometry and pure motives 3 over Q. More precisely, consider the three following type of objects :
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