Number Theory Seminar

Speaker: David Urbanik (University of Toronto)

Title: Effective Methods for Shafarevich Problems

Abstract: Given a smooth projective family f : X --> S defined over the ring of integers of a number field, the Shafarevich problem is to describe those fibres of f which have everywhere good reduction. This can be interpreted as asking for the dimension of the Zariski closure of the set of integral points of S, and is ultimately a difficult diophantine question about which little is known as soon as the dimension of S is at least 2. Recently, Lawrence and Venkatesh gave a general strategy for addressing such problems which requires as input lower bounds on the monodromy of f over essentially arbitrary closed subvarieties of S. In this talk we review their ideas, and describe recent work which gives a fully effective method for computing these lower bounds. This gives a fully effective strategy for solving Shafarevich-type problems for essentially arbitrary families f.

The talk will be exclusively on Zoom. To receive the Zoom link, please contact one of the organizers if you are not on the Number Theory Seminar mailing list.