Symplectic Reidemeister torsion and symplectic $L$-functions

Abstract: Many of the quantities appearing in the conjecture of Birch and Swinnerton-Dyer look suspiciously like squares. Motivated by this and related examples, we may ask if the central value of an $L$-function "of symplectic type" admits a preferred square root.

The answer is no: there's an interesting cohomological obstruction. More formally, in the everywhere unramified situation over a function field, I will describe an explicit cohomological formula for the $L$-function modulo squares. This is based on a purely topological result about $3$-manifolds. If time permits I'll speculate on generalizations. This is based on joint work with Amina Abdurrahman.

number theory

Audience: researchers in the topic

Harvard number theory seminar