# Seminars Calendar

## Thursday October 18, 2018 - room 2AB45 - time 14:20 - Yukako Kezuka (Regensburg)

** The p-part of Birch-Swinnerton-Dyer conjecture for the Gross family of elliptic curves **

Abstract: Take $q$ to be any prime congruent to 7 modulo 8, and let $K= Q(\sqrt{-q})$. B. Gross proved the existence of an elliptic curve $A$ defined over the Hilbert class field $H$ of $K$ with complex multiplication by the ring of integers of $K$ and minimal discriminant $-q^3$. We study a large family of quadratic twists $E$ of $A$ for which the complex L-series of $E/H$ does not vanish at 1. We discuss the $p$-part of Birch-Swinnerton-Dyer conjecture for $E/H$ for any prime $p$ which splits in $K$, including $p=2$. This is a joint work in progress with J. Coates, Y.Li and Y. Tian.

## Thursday October 18, 2018 - room 2AB45 - time 15:30 - Takashi Suzuki (Chuo University)

** Duality for cohomology of local fields and curves with coefficients in abelian varieties **

Abstract: I will explain a duality for cohomology of local fields and curves over perfect base (or residue) fields of positive characteristic with coefficients in abelian varieties. The cohomology mentioned here is equipped with a structure of a sheaf over a Grothendieck site called the "rational etale site" of the base field, and we consider a sheaf-theoretic relative duality. With this duality, we are able solve Grothendieck's duality conjecture in SGA 7 on special fibers of abelian varieties. Also, Tate-Shafarevich groups, as sheaves, are represented by unipotent algebraic groups. Cassels-Tate pairings are generalized with these geometric structures. I will then try to express my naive hope that, in this duality, abelian varieties and unipotent groups should be generalized to motives and curves should be generalized to morphisms between varieties. The big picture here is that there should be a relative duality and six operations formalism for mixed etale motives with $p$-torsion, a program that has yet to be developed.

# Working Seminars

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