## Arithmetic Geometry and Number Theory RTG Seminar: Arithmetic representations of fundamental groups

Seminar | January 29 | 3:10-5 p.m. | 748 Evans Hall

Daniel Litt, Columbia University

Department of Mathematics

Let $X$ be an algebraic variety over a field $k$. Which representations of $\pi _1(X)$ arise from geometry, e.g. as monodromy representations on the cohomology of a family of varieties over $X$? We study this question by analyzing the action of the Galois group of $k$ on the fundamental group of $X$, and prove several fundamental structural results about this action.

As a sample application of our techniques, I show that if $X$ is a normal variety over a field of characteristic zero, and $p$ is a prime, then there exists an integer $N=N(X,p)$ such that any non-trivial $p$-adic representation of the fundamental group of $X$, which arises from geometry, is non-trivial mod $p^N$.

Seminar Format: The seminar consists of two 50-minute talks, a pre-talk (3:10-4:00) and an advanced talk (4:10-5:00), with a 10-minute break (4:00-4:10) between them. The advanced talk is a regular formal presentation about recent research results to general audiences in arithmetic geometry and number theory; the pre-talk (3:10-4:00) is to introduce some prerequisites or background for the advanced talk to audiences consisting of graduate students.

yxy@berkeley.edu