Paugam: Symétries spectrales des fonctions zêtas

Table of contents for this issue | Previous article | Next article

Frédéric Paugam
Symétries spectrales des fonctions zêtas
Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), p. 713-720
Article: subscription required

Résumé - Abstract

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.

Bibliography

[Arm72] J. V. Armitage, Zeta functions with a zero at $s = \frac{1}{2}$ . Invent. Math. 15 (1972), 199–205. MR 291122 | Zbl 0233.12006
[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 1839918 | Zbl 0982.11033
[CCM07] Alain Connes, Caterina Consani, and Matilde Marcolli, The weil proof and the geometry of the adeles class space. ArXiv, (math.NT/0703392) (2007).
arXiv | MR 2349719
[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 1694895 | Zbl 0945.11015
[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 1265547 | Zbl 0816.14010
[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 1648030 | Zbl 0899.14001
[GJ72] Roger Godement and Hervé Jacquet, Zeta functions of simple algebras. Lecture Notes in Mathematics, Vol. 260. Springer-Verlag, Berlin, 1972. MR 342495 | Zbl 0244.12011
[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 1640151 | Zbl 0921.11047
[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 1330931 | Zbl 0840.14001
[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.
Article | MR 2132868 | Zbl 1079.11044
[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 1975180 | Zbl 1075.11056
[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 407534 | Zbl 0295.43008
[Ram05] Niranjan Ramachandran, Values of zeta functions at $s = 1 / 2$ . Int. Math. Res. Not. 25 (2005), 1519–1541. MR 2152893 | Zbl 1089.11034
[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 1696186 | Zbl 0943.11030
[Wil95] Andrew Wiles, Modular elliptic curves and Fermat’s last theorem. Ann. of Math. (2) 141(3) (1995), 443–551. MR 1333035 | Zbl 0823.11029