Higher regularizations of zeros of cuspidal automorphic L -functions of GL d

Masato Wakayama[1]; Yoshinori Yamasaki[2]

  • [1] Faculty of Mathematics Kyushu University Motooka, Nishiku, Fukuoka 819-0395, JAPAN
  • [2] Graduate School of Science and Engineering Ehime University Bunkyo-cho, Matsuyama 790-8577, JAPAN

Journal de Théorie des Nombres de Bordeaux (2011)

  • Volume: 23, Issue: 3, page 751-767
  • ISSN: 1246-7405

Abstract

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We establish “higher depth” analogues of regularized determinants due to Milnor for zeros of cuspidal automorphic L -functions of 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.

How to cite

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Wakayama, Masato, and Yamasaki, Yoshinori. "Higher regularizations of zeros of cuspidal automorphic $L$-functions of ${\rm GL}_d$." Journal de Théorie des Nombres de Bordeaux 23.3 (2011): 751-767. <http://eudml.org/doc/219842>.

@article{Wakayama2011,
abstract = {We establish “higher depth” analogues of regularized determinants due to Milnor for zeros of cuspidal automorphic $L$-functions of $\{\rm GL\}_d$ 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.},
affiliation = {Faculty of Mathematics Kyushu University Motooka, Nishiku, Fukuoka 819-0395, JAPAN; Graduate School of Science and Engineering Ehime University Bunkyo-cho, Matsuyama 790-8577, JAPAN},
author = {Wakayama, Masato, Yamasaki, Yoshinori},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {cuspidal automorphic $L$-functions; regularized products (determinants); explicit formulas; grand Riemann hypothesis; cuspidal automorphic -functions},
language = {eng},
month = {11},
number = {3},
pages = {751-767},
publisher = {Société Arithmétique de Bordeaux},
title = {Higher regularizations of zeros of cuspidal automorphic $L$-functions of $\{\rm GL\}_d$},
url = {http://eudml.org/doc/219842},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Wakayama, Masato
AU - Yamasaki, Yoshinori
TI - Higher regularizations of zeros of cuspidal automorphic $L$-functions of ${\rm GL}_d$
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/11//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 3
SP - 751
EP - 767
AB - We establish “higher depth” analogues of regularized determinants due to Milnor for zeros of cuspidal automorphic $L$-functions of ${\rm GL}_d$ 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.
LA - eng
KW - cuspidal automorphic $L$-functions; regularized products (determinants); explicit formulas; grand Riemann hypothesis; cuspidal automorphic -functions
UR - http://eudml.org/doc/219842
ER -

References

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  9. N. Kurokawa, M. Wakayama and Y. Yamasaki, Milnor-Selberg zeta functions and zeta regularizations. Preprint, 2011. arXiv:1011.3093. Zbl06138305
  10. W. Luo, Z. Rudnick and P. Sarnak, On the generalized Ramanujan conjecture for GL ( n ) . In Automorphic forms, automorphic representations, and arithmetic, Proc. Sympos. Pure Math. 66, Part 2 (1999), 301–310. Zbl0965.11023MR1703764
  11. J. Milnor, On polylogarithms, Hurwitz zeta functions, and the Kubert identities. Enseignement Mathématique. 29 (1983), 281–322. Zbl0557.10031MR719313
  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. Zbl1075.11056MR1975180
  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. Zbl0809.11045MR1265548
  14. J. Steuding, Value-distribution of L -functions. Lecture Notes in Mathematics, Vol. 1877. Springer, Berlin, 2007. Zbl1130.11044MR2330696
  15. Y. Yamasaki, Factorization formulas for higher depth determinants of the Laplacian on the n -sphere. Preprint, 2010. arXiv:1011.3095. 

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