School of Mathematics Colloquium
The Tate conjecture for K3 surfaces over fields of odd characteristic
by Prof. Keerthi Madapusi (Harvard University, USA)
Thursday, September 20, 2012
from
to
(Asia/Kolkata)
at Colaba Campus ( AG-69 )
at Colaba Campus ( AG-69 )
Description |
Using the theory of integral canonical models of Shimura varieties (due to Faltings-Kisin-Vasiu), we extend the classical Kuga-Satake construction for K3 surfaces over fields of odd characteristic. This construction attaches to every polarized K3 surface X an abelian variety A, and allows us (always when p>3; in certain cases when p=3) to identify the Picard group of X with a certain space of endomorphisms (called 'special endomorphisms') of A. Using new results of Kisin towards a proof of the Langlands-Rapoport conjecture, we can now reduce the Tate conjecture for X to the Tate conjecture for endomorphisms of A, which is already known due to Tate and Zarhin. Over finite fields of characteristic at least 5, the Tate conjecture for K3 surfaces is already known by work of Nygaard-Ogus, Maulik and Charles, but our proof is uniform and works also over infinite, finitely generated fields. |