Symétries spectrales des fonctions zêtas

Paugam, Frédéric

Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 713-720.

      Spectral symmetries of zeta functions

We define, answering a question of Sarnak in his letter to Bombieri [Sar01], 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.

On définit, en réponse à une question de Sarnak dans sa lettre a Bombieri [Sar01], un accouplement symplectique sur l’interprétation spectrale (due à Connes et Meyer) des zéros de la fonction zêta. Cet accouplement donne une formulation purement spectrale de la démonstration de l’équation fonctionnelle due à Tate, Weil et Iwasawa, qui, dans le cas d’une courbe sur un corps fini, correspond à la démonstration géométrique usuelle par utilisation de l’accouplement de dualité de Poincaré Frobenius-équivariant en cohomologie étale. On donne un autre exemple d’accouplement similaire dans le cas de l’interprétation spectrale des zéros de la fonction $L$ d’une forme automorphe cuspidale, mais cette fois-ci de nature orthogonale. Ces constructions sont en adéquation avec les prévisions du programme conjectural de Deninger et de la théorie arithmétique des matrices aléatoires.

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