Symplectic $L$-functions and symplectic Reidemeister torsion (mod squares)

Abstract

We propose (and prove under some restrictions) that the square class of the central value of the $L$-function of an everywhere unramified symplectic Galois representation is given by a universal cohomological formula. This phenomenon is parallel to the appearance of metaplectic groups in quantization. In the course of the proof we also establish a topological analogue of this statement, concerning Reidemeister torsion of $3$-manifolds.