Square roots of symplectic L-functions and Reidemeister torsion

Abstract

We study the existence of a square root of a symplectic L-function. In joint work with A. Venkatesh we conjecture a global cohomological formula for the central value of a symplectic L-function up to squares, which in particular gives a criterion for the existence of a square root. Using ideas from arithmetic topology we formulate an analogous conjecture about the Reidemeister torsion of 3-manifolds up to squares. We give a proof of the topological statement in full generality and of the arithmetic statement under some assumptions. We note that the topological theorem is central in the proof of the arithmetic statement.