A relative Oda's criterion

Abstract: The Neron--Ogg--Shafarevich criterion asserts that an abelian variety over $\mathbb{Q}_p$ has good reduction if and only if the Galois action on its $\mathbb{Z}_\ell$-linear Tate module is unramified (for $\ell$ different from $p$). In 1995, Oda formulated and proved an analogue of the Neron--Ogg--Shafarevich criterion for smooth projective curves $X$ of genus at least two: $X$ has good reduction if and only if the outer Galois action on its pro-$\ell$ geometric fundamental group is unramified. In this talk, I will explain a relative version of Oda's criterion, due to myself and Netan Dogra, in which we answer the question of when the Galois action on the pro-$\ell$ torsor of paths between two points $x$ and $y$ is unramified in terms of the relative position of $x$ and $y$ on the reduction of $X$. On the way, we will touch on topics from mapping class groups and the theory of electrical circuits, and, time permitting, will outline some consequences for the Chabauty--Kim method.

number theory

Audience: researchers in the topic

Harvard number theory seminar