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We give a survey of results on the Galois group of polynomials obtained by truncation of power series, the main example being the exponential series. We also present some evidence of a new phenomena: Galois groups of Pade approximation polynomials seem to have special properties as well.
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Let G be a connected reductive group defined over Q_p. The set of crystals contained in a given G-isocrystal is viewed from a Bruhat-Tits building-theoretic vantage point as a kind of tubular neighborhood of a skeleton characterized by a minimality property arising from metric space theory.
This is the memoir of my habilitation thesis, defended on March 29 th, 2013 (Universite Paris XI).
We describe the behaviour of the rank of the Mordell-Weil group of the Picard variety of the generic fibre of a fibration in terms of local contributions given by averaging traces of Frobenius acting on the fibres. The results give a reinterpretation of Tates conjecture (for divisors) and generalises previous results of Nagao, Rosen-Silverman and the authors.
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.
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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