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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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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 V be a closed 3-manifold. In this paper we prove that the homotopy classes of plane fields on V that contain tight contact structures are in finite number and that, if V is atoroidal, the isotopy classes of tight contact structures are also in finite number.
Let X_d be the p-adic analytic space classifying the d-dimensional (semisimple) p-adic Galois representations of the absolute Galois group of Q_p. We show that the crystalline representations are Zarski-dense in many irreducible components of X_d, including the components made of residually irreducible representations. This extends to any dimension d previous results of Colmez and Kisin for d = 2. For this we construct an analogue of the infinite fern of Gouv^ea-Mazur in this context, based on a study of analytic families of trianguline (phi,Gamma)-modules over the Robba ring. We show in particular the existence of a universal family of (framed, regular) trianguline (phi,Gamma)-modules, as well as the density of the crystalline (phi,Gamma)-modules in this family. These results may be viewed as a local analogue of the theory of p-adic families of finite slope automorphic forms, they are new already in dimension 2. The technical heart of the paper is a collection of results about the Fontaine-Herr cohomology of families of trianguline (phi,Gamma)-modules.
We give a new definition, simpler but equivalent, of the abelian category of Banach-Colmez spaces introduced by Colmez, and we explain the precise relationship with the category of coherent sheaves on the Fargues-Fontaine curve. One goes from one category to the other by changing the t-structure on the derived category. Along the way, we obtain a description of the pro-etale cohomology of the open disk and the affine space, of independent interest.
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