École d'été: Valeurs spéciales de fonctions L
ENS Lyon, 2-6 juin 2014

ANR Labex Milyon
English version

Denis-Charles Cisinski
Cohomological descent and rigidity for étale motives.

Olivier Fouquet
Congruences dans l’étude des conjectures sur les valeurs spéciales des fonctions L

À part le cas classique de la formule des classes de Dirichlet, les résultats sur les valeurs spéciales des fonctions L des motifs ont été obtenus en considérant des familles p-adiques de tels objets. En prenant comme fil conducteur le cas de la valeur en 1 de la fonction L d’une courbe elliptique (prédite par la conjecture de Birch et Swinnerton-Dyer), nous montrerons comment l’on peut déduire par congruences certains cas de ces conjectures à partir de résultats en famille.

Jose Ignacio Burgos Gil
The singularities of the invariant metric of the sheaf of Jacobi forms on the universal elliptic curve

A theorem by Mumford implies that every automorphic line bundle on a pure open Shimura variety, provided with an invariant smooth metric, can be uniquely extended as a line bundle on a toroidal compactification of the variety, in such a way that the metric acquires only logarithmic singularities. This result is the key to being able to compute arithmetic intersection numbers from these line bundles. Hence it is natural to ask whether Mumford's result remains valid for line bundles on mixed Shimura varieties.
In this talk we will examine the simplest case, namely the sheaf of Jacobi forms on the universal elliptic curve. We will show that Mumford's result can not be extended to this case and that a new interesting kind of singularities appear that are related to the phenomenon of Height jumping introduced by Hain.
We will discuss some preliminary results. This is joint work with G. Freixas, J. Kramer and U. Kühn.

Jennifer Johnson-Leung
L-functions of almost abelian extensions

I will discuss the equivariant Tamagawa number conjecture for the case of abelian extensions of imaginary quadratic fields. The conjecture is not completely settled in the case. I will explain what is known and some of the difficulties that remain. I will also discuss ongoing work toward analogous results for abelian extensions of CM fields.

Henri Johnston
Hybrid p-adic group rings and the equivariant Tamagawa number conjecture

We discuss applications of the understanding of the structure of group rings of finite groups over p-adic integers to the equivariant Tamagawa number conjecture for Tate motives. This is joint work with Andreas Nickel.

Baptsite Morin
Weil-étale cohomology

Lichtenbaum has suggested the existence of a Weil-étale cohomology in order to describe special values of zeta functions of arithmetic schemes at $s=0$. We give a conditional construction of such cohomology groups for regular proper arithmetic schemes, and state a conjecture for the special value and the vanishing order of the corresponding zeta functions at $s=0$. Then we generalize this construction and give a conjectural description of zeta functions at any integer argument. This is joint work with Matthias Flach.

Andreas Nickel
Hybrid Iwasawa algebras and the equivariant Iwasawa main conjecture

Jörg Wildeshaus
Weight structures : abstract properties and applications to motives