Effective Methods for Shafarevich Problems
David Urbanik (Toronto)
Abstract: Given a smooth projective family $f : X \to S$ 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 $f$.
number theory
Audience: researchers in the topic
Comments: This week's talk is in APM 7321 rather than APM 6402.
pre-talk at 1:20 pm
Series comments: Most talks are preceded by a pre-talk for graduate students and postdocs. The pre-talks start 40 minutes prior to the posted time (usually at 1:20pm Pacific) and last about 30 minutes.
Organizers: | Kiran Kedlaya*, Alina Bucur, Aaron Pollack, Cristian Popescu, Claus Sorensen |
*contact for this listing |