2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/156449We 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 Poincaré 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.6 pages. Minor modification due to a problem with real zeroes of some general Dedekind zetasNumber TheoryAlgebraic Geometry11R42; 11F70text