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