Symétries spectrales des fonctions zêtas

Paugam, Frédéric

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

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.

      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.

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