Tony Feng (Stanford/IAS)
On a conjecture of Tate concerning the Brauer group of a surface

Abstract: Artin and Tate discovered a dictionary relating the Birch and Swinnerton-Dyer Conjecture over function fields to the Tate Conjecture for surfaces over finite fields. Under this dictionary the Cassels-Tate pairing on Sha corresponds to (what I call) the Artin-Tate pairing on the Brauer group. An old question of Tate asks if this pairing is alternating. In this talk I will present the answer to Tate's question. The key new ingredient is a circle of ideas originating in algebraic topology, centered around the Steenrod operations, that are imported to algebraic geometry on the ships of etale homotopy theory. The talk will advertise these new tools (while assuming minimal background in algebraic topology).