61,558 research outputs found

    Residual irreducibility of compatible systems

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    We show that if {ρ}\{\rho_{\ell}\} is a compatible system of absolutely irreducible Galois representations of a number field then the residual representation ρ\overline{\rho}_{\ell} is absolutely irreducible for \ell in a density 1 set of primes. The key technical result is the following theorem: the image of ρ\rho_{\ell} is an open subgroup of a hyperspecial maximal compact subgroup of its Zariski closure with bounded index (as \ell varies). This result combines a theorem of Larsen on the semi-simple part of the image with an analogous result for the central torus that was recently proved by Barnet-Lamb, Gee, Geraghty, and Taylor, and for which we give a new proof.Comment: 11 page

    On weakly maximal representations of surface groups

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    We introduce and study a new class of representations of surface groups into Lie groups of Hermitian type, called {\em weakly maximal} representations. We prove that weakly maximal representations are discrete and injective and we describe the structure of the Zariski closure of their image. Furthermore we prove that the set of weakly maximal representations is a closed subset of the representation variety and describe its relation to other geometrically significant subsets of the representation variety.Comment: In this version the paper has been split in two parts. The part that has been removed appears now as http://arxiv.org/abs/1601.02232. The current version of the paper will appear in the Journal of Differential Geometr

    Curtis homomorphisms and the integral Bernstein center for GL_n

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    We describe two conjectures, one strictly stronger than the other, that give descriptions of the integral Bernstein center for GL_n(F) (that is, the center of the category of smooth W(k)[GL_n(F)]-modules, for F a p-adic field and k an algebraically closed field of characteristic l different from p) in terms of Galois theory. Moreover, we show that the weak version of the conjecture (for m at most n) implies the strong version of the conjecture. In a companion paper [HM] we show that the strong conjecture for n-1 implies the weak conjecture for n; thus the two papers together give an inductive proof of both conjectures. The upshot is a description of the integral Bernstein center for GL_n in purely Galois-theoretic terms; previous work of the author shows that such a description implies the conjectural "local Langlands correspondence in families" of Emerton and the author.Comment: Final accepted version. 36 pages. Note that the published version of Helm-Moss, "Converse theorems and the local Langlands correspondence in families" references an earlier version of this paper, and the section numbering has changed; in particular sections 9,10, and 11 of the referenced version correspond to sections 8,9, and 10 of the current version, respectivel

    Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras

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    CC^*-algebraic Weyl quantization is extended by allowing also degenerate pre-symplectic forms for the Weyl relations with infinitely many degrees of freedom, and by starting out from enlarged classical Poisson algebras. A powerful tool is found in the construction of Poisson algebras and non-commutative twisted Banach-*-algebras on the stage of measures on the not locally compact test function space. Already within this frame strict deformation quantization is obtained, but in terms of Banach-*-algebras instead of CC^*-algebras. Fourier transformation and representation theory of the measure Banach-*-algebras are combined with the theory of continuous projective group representations to arrive at the genuine CC^*-algebraic strict deformation quantization in the sense of Rieffel and Landsman. Weyl quantization is recognized to depend in the first step functorially on the (in general) infinite dimensional, pre-symplectic test function space; but in the second step one has to select a family of representations, indexed by the deformation parameter \hbar. The latter ambiguity is in the present investigation connected with the choice of a folium of states, a structure, which does not necessarily require a Hilbert space representation.Comment: This is a contribution to the Special Issue on Deformation Quantization, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA
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