Spectral symmetries of zeta functions

Details Spectral symmetries of zeta functions Spectral symmetries of zeta functions

2008-03-03

arxiv.org/abs/0803.0199v2

Mathematics

Number Theory

6 pages. Minor modification due to a problem with real zeroes of some general Dedekind zetas

Scientific paper

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.

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