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Register a new userProgr`es recents sur la conjecture de Zagier et le programme de Goncharov [dapr`es Goncharov, Rudenko, Gangl, ...]
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Authors
Clement Dupont
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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 motives. We survey recent progress on the conjecture and in particular the proof of the $n=4$ case of the conjecture by Goncharov and Rudenko.
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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.
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 the generic fibre (modulo the constant part) in terms of average of the traces of Frobenius on the fibers of $f$. The results also give a reinterpretation of the Tate conjecture for the surface $mathcal{X}$ and generalizes results of Nagao, Rosen-Silverman and Wazir.
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.
This article reviews the results obtained by the Rosetta space probe on comet 67P/Churyumov-Gerasimenko and the expectations of future space exploration of comets. -------------- Cet article presente une synth`ese des resultats obtenus par la sonde Rosetta sur la com`ete 67P/Churyumov-Gerasimenko et envisage le futur de lexploration spatiale des com`etes.
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.
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