Number Theory Seminar: A relative Oda’s criterion

SEMINARS, NUMBER THEORY

Speaker:

Alex Betts - Harvard University

The Neron--Ogg--Shafarevich criterion asserts that an abelian variety over ℚp has good reduction if and only if the Galois action on its ℤℓ-linear Tate module is unramified (for ℓ 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-ℓ 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-ℓ 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.