##### Department of Mathematics,

University of California San Diego

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### Math 209 - Number Theory Seminar

## David Urbanik

#### Toronto

## Effective Methods for Shafarevich Problems

##### 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 $f$.

### March 10, 2022

### 2:00 PM

Pre-talk at 1:20 PM

APM 7321 and Zoom;

See https://www.math.ucsd.edu/~nts

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