Symétries spectrales des fonctions zêtas
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.

Received:
Published online:
DOI: 10.5802/jtnb.697
Frédéric Paugam 1

1 Université paris 6 Institut de mathématiques de Jussieu 175, rue du Chevaleret 75012 Paris
@article{JTNB_2009__21_3_713_0,
     author = {Fr\'ed\'eric Paugam},
     title = {Sym\'etries spectrales des fonctions z\^etas},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {713--720},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     doi = {10.5802/jtnb.697},
     zbl = {1214.11095},
     mrnumber = {2605542},
     language = {fr},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.697/}
}
TY  - JOUR
TI  - Symétries spectrales des fonctions zêtas
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2009
DA  - 2009///
SP  - 713
EP  - 720
VL  - 21
IS  - 3
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.697/
UR  - https://zbmath.org/?q=an%3A1214.11095
UR  - https://www.ams.org/mathscinet-getitem?mr=2605542
UR  - https://doi.org/10.5802/jtnb.697
DO  - 10.5802/jtnb.697
LA  - fr
ID  - JTNB_2009__21_3_713_0
ER  - 
%0 Journal Article
%T Symétries spectrales des fonctions zêtas
%J Journal de Théorie des Nombres de Bordeaux
%D 2009
%P 713-720
%V 21
%N 3
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.697
%R 10.5802/jtnb.697
%G fr
%F JTNB_2009__21_3_713_0
Frédéric Paugam. Symétries spectrales des fonctions zêtas. Journal de Théorie des Nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 713-720. doi : 10.5802/jtnb.697. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.697/

[Arm72] J. V. Armitage, Zeta functions with a zero at s=1 2. Invent. Math. 15 (1972), 199–205. | MR | Zbl

[BCDT01] Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, On the modularity of elliptic curves over Q : wild 3-adic exercises. J. Amer. Math. Soc. 14(4) (2001), 843–939 (electronic). | MR | Zbl

[CCM07] Alain Connes, Caterina Consani, and Matilde Marcolli, The weil proof and the geometry of the adeles class space. ArXiv, (math.NT/0703392) (2007). | MR

[Con99] Alain Connes, Trace formula in noncommutative geometry and the zeros of the Riemann zeta function. Selecta Math. (N.S.) 5(1) (1999), 29–106. | MR | Zbl

[Den94] Christopher Deninger, Motivic L-functions and regularized determinants. In Motives (Seattle, WA, 1991), volume 55 of Proc. Sympos. Pure Math., pages 707–743. Amer. Math. Soc., Providence, RI, 1994. | MR | Zbl

[Den98] Christopher Deninger,Some analogies between number theory and dynamical systems on foliated spaces. In Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), number Extra Vol. I, pages 163–186 (electronic), 1998. | MR | Zbl

[GJ72] Roger Godement and Hervé Jacquet, Zeta functions of simple algebras. Lecture Notes in Mathematics, Vol. 260. Springer-Verlag, Berlin, 1972. | MR | Zbl

[KS99] Nicholas M. Katz and Peter Sarnak, Zeroes of zeta functions and symmetry. Bull. Amer. Math. Soc. (N.S.) 36(1) (1999), 1–26. | MR | Zbl

[Man95] Yuri Manin, Lectures on zeta functions and motives (according to Deninger and Kurokawa). Astérisque 228 :4 (1995), 121–163. Columbia University Number Theory Seminar (New York, 1992). | MR | Zbl

[Mey05] Ralf Meyer, On a representation of the idele class group related to primes and zeros of L-functions. Duke Math. J., 127(3) :519–595, 2005. | MR | Zbl

[Mic02] Philippe Michel, Répartition des zéros des fonctions L et matrices aléatoires. Astérisque 282, Exp. No. 887, viii, 211–248, 2002. Séminaire Bourbaki, Vol. 2000/2001. | Numdam | MR | Zbl

[Osb75] M. Scott Osborne, On the Schwartz-Bruhat space and the Paley-Wiener theorem for locally compact abelian groups. J. Functional Analysis 19 (1975), 40–49. | MR | Zbl

[Ram05] Niranjan Ramachandran, Values of zeta functions at s=1/2. Int. Math. Res. Not. 25 (2005), 1519–1541. | MR | Zbl

[Sar01] Sarnak, Dear Enrico. Letter to Bombieri, pages 1–7, 2001.

[Sou99] Christophe Soulé, Sur les zéros des fonctions L automorphes. C. R. Acad. Sci. Paris Sér. I Math. 328(11) (1999), 955–958. | MR | Zbl

[Wil95] Andrew Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2) 141(3) (1995), 443–551. | MR | Zbl

Cited by Sources: