Arithmetic invariants and parity of ranks of abelian surfaces
Heilbronn Number Theory Seminar
18th December 2019, 4:00 pm – 5:00 pm
Fry Building, 2.04
Let A/K be an abelian variety over a number field. If A/K admits an isogeny of a particular type, it is known how to control the parity of its rank. On the other hand, the Birch and Swinnerton-Dyer conjecture (BSD) also provides a way to compute the parity of the rank, but these two expressions for the parity of the rank are not obviously compatible. By proving this compatibility, one proves the so-called parity conjecture. In other words, one proves that BSD correctly predicts the parity of the rank of abelian varieties.
The key to proving the parity conjecture in this setting is to find an expression for the local discrepancy between specific local arithmetic invariants of A/K (including Tamagawa numbers and local root numbers). In this talk, we discuss several properties of this local discrepancy for elliptic curves and some specific abelian varieties, and present a new expression for the local discrepancy in the case of Jacobians of genus 2 curves. This allows us to prove the parity conjecture for principally polarized abelian surfaces under extra conditions.