## EVENT DETAILS AND ABSTRACT

**Number Theory**
**Title:** The Tate conjecture for K3 surfaces in odd characteristic

**Speaker:** Keerthi Madapusi

**Speaker Info:** Harvard

**Brief Description:**

**Special Note**:

**Abstract:**

The Tate conjecture predicts that, given a smooth, projective variety X over a finitely generated field (a number field, finite field, or function field over such), we can deduce a great deal about it by studying its l-adic cohomology, essentially reducing many questions about the variety to problems in linear algebra. In particular, it says that the rank of the Neron-Severi group of X can be computed as the dimension of the largest sub-space of H^2(X) on which the Galois group acts via the l-adic cyclotomic character. When X is an abelian variety over a finite field, Tate proved this last assertion in the 60s, by showing that it has an endomorphism ring of the expected dimension. We will show how his proof (or a reformulation of it) can be leveraged to prove the Tate conjecture for divisors on K3 surfaces. The key input is a miraculous construction (called the Kuga-Satake construction) that attaches to any K3 surface X, an abelian variety A such that divisors on X correspond to certain endomorphisms on A.

**Date:** Monday, April 15, 2013

**Time:** 3:00PM

**Where:** Lunt 107

**Contact Person:** Simon Marshall

**Contact email:** slm@math.northwestern.edu

**Contact Phone:** 847-467-0715

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