Holly Green

The Parity Conjecture: from characteristic p to characteristic 0.

The parity conjecture predicts that the parity of the Mordell—Weil rank of an abelian variety is determined by its global root number, a product of local root numbers. A common strategy for proving it is to introduce a local invariant that controls the parity of the rank, and then relate this invariant to the corresponding local root number.

In this talk, I will explain a new way to prove these local identities. The method avoids computing local data explicitly and instead exploits the fact that it can be transported between fields of characteristic 0 and characteristic p. I will also present some new results on the parity conjecture for hyperelliptic curves and their quadratic twists that follow from this method.