Mihran Papikian : Optimal quotients of Mumford curves and component groups

Posted by Alina Carmen Cojocaru , part of the Number Theory Seminar.

At

March 5, 2013, 3 p.m.

In

SEO 636

Abstract

Let $X$ be a Mumford curve. We say that an elliptic curve is an optimal quotient of $X$ is there is a finite morphism $X\to E$ such that the homomorphism $\pi: Jac(X)\to E$ induced by the Albanese functoriality has connected and reduced kernel. We consider the functorially induced map $\pi_\ast: \Phi_X\to \Phi_E$ on component groups of the Neron models of $Jac(X)$ and $E$. We show that in general this map need not be surjective, which answers negatively a question of Ribet and Takahashi. Using rigid-analytic techniques, we give some conditions under which $\pi_\ast$ is surjective, and discuss arithmetic applications to modular curves. This is a joint work with Joe Rabinoff.