Higher regularizations of zeros of cuspidal automorphic $L$-functions of ${\mathrm{GL}}_d$

Wakayama, Masato; Yamasaki, Yoshinori

Journal de théorie des nombres de Bordeaux, Tome 23 (2011) no. 3, pp. 751-767.

Nous établissons un analogue de la “plus haute profondeur” des déterminants régularisés due à Milnor pour les zéros des fonctions $L$ automorphes cuspidales de ${\mathrm{GL}}_d$ sur un corps de nombres général. C’est une généralisation du résultat de Deninger au sujet du déterminant régularisé pour les zéros de la fonction zêta de Riemann.

We establish “higher depth” analogues of regularized determinants due to Milnor for zeros of cuspidal automorphic $L$-functions of ${\mathrm{GL}}_d$ over a general number field. This is a generalization of the result of Deninger about the regularized determinant for zeros of the Riemann zeta function.

[1] K. Barner, On A. Weil’s explicit formula. J. Reine Angew. Math. 323 (1981), 139–152. | EuDML | MR | Zbl

[2] D. Bump, Automorphic forms and representations. Cambridge Studies in Advanced Mathematics 55. Cambridge University Press, Cambridge, 1997. | MR | Zbl

[3] C. Deninger, Local $L$-factors of motives and regularized determinants. Invent. Math. 107 (1992), 135–150. | EuDML | MR | Zbl

[4] R. Godement and H. Jacquet, Zeta functions of simple algebras. Lecture Notes in Mathematics, Vol. 260. Springer-Verlag, Berlin-New York, 1972. | MR | Zbl

[5] T. Ibukiyama and H. Saito, On zeta functions associated to symmetric matrices. I. An explicit form of zeta functions. Amer. J. Math. 117 (1995), 1097–1155. | MR | Zbl

[6] H. Iwaniec and P. Sarnak, Perspectives on the analytic theory of $L$-functions. Geom. Funct. Anal. Special Volume (2000) Part II, 705–741. | MR | Zbl

[7] N. Kurokawa, H. Ochiai and M. Wakayama, Milnor’s multiple gamma functions. J. Ramanujan Math. Soc. 21 (2006), 153–167. | MR | Zbl

[8] N. Kurokawa and M. Wakayama, Analyticity of polylogarithmic Euler products. Rend. Circ. Mat. di Palermo. 200 (2003), 382–388. | MR | Zbl

[9] N. Kurokawa, M. Wakayama and Y. Yamasaki, Milnor-Selberg zeta functions and zeta regularizations. Preprint, 2011. arXiv:1011.3093. | MR

[10] W. Luo, Z. Rudnick and P. Sarnak, On the generalized Ramanujan conjecture for $\mathrm{GL}\left(n\right)$. In Automorphic forms, automorphic representations, and arithmetic, Proc. Sympos. Pure Math. 66, Part 2 (1999), 301–310. | MR | Zbl

[11] J. Milnor, On polylogarithms, Hurwitz zeta functions, and the Kubert identities. Enseignement Mathématique. 29 (1983), 281–322. | MR | Zbl

[12] P. Michel, Répartition des zéros des fonctions $L$ et matrices aléatoires. Seminaire Bourbaki, Vol. 2000/2001, Astérisque. 282 (2002), Exp. No. 887, viii, 211–248. | Numdam | MR | Zbl

[13] M. Schröter and C. Soulé, On a result of Deninger concerning Riemann’s zeta function. In Motives, Proc. Sympos. Pure Math. 55, Part 1 (1994), 745–747. | MR | Zbl

[14] J. Steuding, Value-distribution of $L$-functions. Lecture Notes in Mathematics, Vol. 1877. Springer, Berlin, 2007. | MR | Zbl

[15] Y. Yamasaki, Factorization formulas for higher depth determinants of the Laplacian on the $n$-sphere. Preprint, 2010. arXiv:1011.3095.