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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.
On the rank of Jacobians over function fields.} Let $f:mathcal{X}to C$ be a projective surface fibered over a curve and defined over a number field $k$. We give an interpretation of the rank of the Mordell-Weil group over $k(C)$ of the jacobian of th
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 cat
Let F be a finite extension of Qp, O_F its ring of integers and E a finite extension of Fp. The natural action of the unit group O_F* on O_F extends in a continuous action on the Iwasawa algebra E[[O_F]]. In this work, we show that non zero ideals of
Let $F$ be a quadratic extension of $mathbb{Q}_p$. We prove that smooth irreducible supersingular representations with central character of $mathrm{GL}_2(F)$ are not of finite presentation.
This survey article is the written version of a talk given at the Bourbaki seminar in April 2021. We give an introduction to Zagiers conjecture on special values of Dedekind zeta functions, and its relation to $K$-theory of fields and the theory of m