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 $\Omega\subset\mathbbR^N$ from a complex vector space $E$ to a complex vector space $F$, then the Hardy-Sobolev inequality

$$
\int_\mathbbR^N\frac|D^\nu-\ellu(x)||x-x_0|^\ell\,dx\leq C \int_\mathbbR^N|A(x,D)u|dx, \quad u \in C_c^\infty(B ;E),
$$

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

$$
\bigcap_\xi\in\mathbbR^N\setminus{0}a_\nu(x_0,\xi)[E]={0}.
$$

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