Spectral symmetries of zeta functions

TL;DR

This paper introduces a spectral symplectic pairing for the zeros of the Riemann zeta function, linking spectral interpretations with classical proofs of the functional equation and exploring similar structures for automorphic L-functions.

Contribution

It defines a spectral symplectic pairing answering Sarnak's question, connecting spectral interpretations with classical and automorphic L-function theories.

Findings

Spectral symplectic pairing for Riemann zeta zeros

Connection to classical proofs of the functional equation

Construction of an orthogonal pairing for automorphic L-functions

Abstract

We define, answering a question of Sarnak in his letter to Bombieri, a symplectic pairing on the spectral interpretation (due to Connes and Meyer) of the zeroes of Riemann's zeta function. This pairing gives a purely spectral formulation of the proof of the functional equation due to Tate, Weil and Iwasawa, which, in the case of a curve over a finite field, corresponds to the usual geometric proof by the use of the Frobenius-equivariant Poincar\'e duality pairing in etale cohomology. We give another example of a similar construction in the case of the spectral interpretation of the zeroes of a cuspidal automorphic $L$-function, but this time of an orthogonal nature. These constructions are in adequation with Deninger's conjectural program and the arithmetic theory of random matrices.