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Register a new userFormes modulaires de Hilbert modulo p et valeurs dextensions galoisiennes
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Let F be a totally real field, v an unramified place of F dividing p and rho a continuous irreducible two-dimensional mod p representation of G_F such that the restriction of rho to G_{F_v} is reducible and sufficiently generic. If rho is modular (and satisfies some weak technical assumptions), we show how to recover the corresponding extension between the two characters of G_{F_v} in terms of the action of GL_2(F_v) on the cohomology mod p.
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This is the memoir of my habilitation thesis, defended on March 29 th, 2013 (Universite Paris XI).
Let $X$ be a real-analytic manifold and $gcolon Xto{mathbf R}^n$ a proper triangulable subanalytic map. Given a subanalytic $r$-form $omega$ on $X$ whose pull-back to every non singular fiber of $g$ is exact, we show tha $omega$ has a relative primitive: there is a subanalytic $(r-1)$-form $Omega$ such that $dgLambda (omega-dOmega)=0$. The proof uses a subanalytic triangulation to translate the problem in terms of relative Whitney forms associated to prisms. Using the combinatorics of Whitney forms, we show that the result ultimately follows from the subanaliticity of solutions of a special linear partial differential equation. The work was inspired by a question of Franc{c}ois Treves.
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We describe a conjectural construction (in the spirit of Hilberts 12th problem) of units in abelian extensions of certain base fields which are neither totally real nor CM. These base fields are quadratic extensions with exactly one complex place of a totally real number field F, and are referred to as Almost Totally real (ATR) extensions. Our construction involves certain null-homologous topological cycles on the Hilbert modular variety attached to F. The special units are the images of these cycles under a map defined by integration of weight two Eisenstein series on GL_2(F). This map is formally analogous to the higher Abel-Jacobi maps that arise in the theory of algebraic cycles. We show that our conjecture is compatible with Starks conjecture for ATR extensions; it is, however, a genuine strengthening of Starks conjecture in this context since it gives an analytic formula for the arguments of the Stark units and not just for their absolute values. The last section provides numerical evidences for our conjecture.
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