331 research outputs found

    Spectre automorphe des vari\'et\'es hyperboliques et applications topologiques

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    This book is made of two parts. The first is concerned with the differential form spectrum of congruence hyperbolic manifolds. We prove Selberg type theorems on the first eigenvalue of the laplacian on differential forms. The method of proof is representation theoritic, we hope the different chapters may as well serve as an introduction to the modern theory of automorphic forms and its application to spectral questions. The second part of the book is of a more differential geometric flavor, a new kind of lifting of cohomology classes is proved.Comment: 237 pages, book (in french

    Even Galois Representations and the Fontaine--Mazur conjecture II

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    We prove, under mild hypotheses, that there are no irreducible two-dimensional_even_ Galois representations of \Gal(\Qbar/\Q) which are de Rham with distinct Hodge--Tate weights. This removes the "ordinary" hypothesis required in previous work of the author. We construct examples of irreducible two-dimensional residual representations that have no characteristic zero geometric (= de Rham) deformations.Comment: Updated to take into account suggestions of the referee; the main theorems remain unchange

    On the canonical degrees of curves in varieties of general type

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    A widely believed conjecture predicts that curves of bounded geometric genus lying on a variety of general type form a bounded family. One may even ask whether the canonical degree of a curve CC in a variety of general type is bounded from above by some expression aχ(C)+ba\chi(C)+b, where aa and bb are positive constants, with the possible exceptions corresponding to curves lying in a strict closed subset (depending on aa and bb). A theorem of Miyaoka proves this for smooth curves in minimal surfaces, with a>3/2a>3/2. A conjecture of Vojta claims in essence that any constant a>1a>1 is possible provided one restricts oneself to curves of bounded gonality. We show by explicit examples coming from the theory of Shimura varieties that in general, the constant aa has to be at least equal to the dimension of the ambient variety. We also prove the desired inequality in the case of compact Shimura varieties.Comment: 10 pages, to appear in Geometric and Functional Analysi

    Oddness of residually reducible Galois representations

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    We show that suitable congruences between polarized automorphic forms over a CM field always produce elements in the Selmer group for exactly the ±-Asai (aka tensor induction) representation that is critical in the sense of Deligne. For this we relate the oddness of the associated polarized Galois representations (in the sense of the Bella ̈ıche-Chenevier sign being +1) to the parity condition for criticality. Under an assumption similar to Vandiver’s conjecture this also provides evidence for the Fontaine-Mazur conjecture for polarized Galois representations of any even dimension
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