Steenrod operations and Tate's Conjecture on the Brauer group of a surface

Diophantine Analysis working group seminar
Topic:Steenrod operations and Tate's Conjecture on the Brauer group of a surface
Speaker:Tony Feng
Affiliation:Stanford University
Date:Monday, April 9
Time/Room:4:45pm - 6:00pm/Fine Hall 1201, Princeton University

There is a canonical pairing on the Brauer group of a surface over a finite field, which is the analogue of the Cassels-Tate pairing on the Tate-Shafarevich group of a Jacobian variety. An old conjecture of Tate predicts that this pairing is alternating. In this talk I will present a proof of Tate’s conjecture. The key new ingredient is a circle of ideas originating in algebraic topology, centered around the Steenrod operations, that is imported to algebraic geometry on the ships of eÄ›tale homotopy theory. The talk will advertise these new tools (while assuming minimal background in algebraic topology).