10 octobre 2017

Tiago H. Picon (Université de São Paulo)
Local Hardy-Sobolev inequalities for canceling elliptic differential operators

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Lieu : Salle 113-115 (Bâtiment 425)

Résumé : In this lecture we show that if A(x,D) is a linear differential operator of order \nu with smooth complex coefficients in \Omega\subset\mathbb{R}^N from a complex vector space E to a complex vector space F, then the Hardy-Sobolev inequality
<br class='autobr' />\int_{\mathbb{R}^{N}}\frac{|D^{\nu-\ell}u(x)|}{|x-x_0|^{\ell}}\,dx\leq C \int_{\mathbb{R}^{N}}|A(x,D)u|dx, \quad u \in C_{c}^{\infty}(B;E),<br
class='autobr' />
for \ell \in \left\{ 1,...,\min\left\{\nu,N-1 \right\} \right\} holds locally at any point x_0\in\Omega if and only if A(x,D) is elliptic and the constant coefficients homogeneous operator A_\nu(x_0,D) is canceling in the sense of Van Schaftingen for every x_0\in \Omega, which means that
<br class='autobr' />\bigcap_{\xi\in\mathbb{R}^N\setminus\{0\}}a_\nu(x_0,\xi)[E]=\{0\}.<br
class='autobr' />
Here A_\nu(x,D) is the homogeneous part of order \nu of A(x,D) and a_\nu(x,\xi) is the principal symbol of A(x,D).
This is joint work with Jorge Hounie (UFSCar, Brazil).

Local Hardy-Sobolev inequalities for canceling elliptic differential operators  Version PDF

Yichao Tian (Hausdorff Center for Mathematics, Moringside Center of Mathematics)
The triple product Selmer group of an elliptic curve over a cubic real field

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Lieu : Bât. 425, salle 117-119 - Laboratoire de Mathématiques d’Orsay

Résumé : Let F be a cubic totally real field, and E/F be a modular elliptic curve. We consider its triple product motive M attached to E, which is a 8-dimensional motive over Q. Assume that the functional equation of L(M,s) has sign -1. Under certain technical assumptions, one can construct a cohomology class in the p-adic Bloch-Kato Selmer group of M using Hirzebruch-Zagier cycles. We prove that if this class is non-trivial, then the Bloch-Kato Selmer group of M has dimension 1. A key ingredient in the proof is a congruence formula for the cohomology class at certain unramified level raising primes, whose proof uses the fine geometry of the supersingular locus of a Hilbert modular threefold at an inert prime. This is a joint work with Yifeng Liu.

The triple product Selmer group of an elliptic curve over a cubic real field  Version PDF