Extension of Estermann’s theorem to Euler products associated to a multivariate polynomial

Ludovic Delabarre

Bulletin de la Société Mathématique de France (2013)

  • Volume: 141, Issue: 2, page 225-265
  • ISSN: 0037-9484

Abstract

top
Given a multivariate polynomial h X 1 , ⋯ , X n with integral coefficients verifying an hypothesis of analytic regularity (and satisfying h ( 0 ) = 1 ), we determine the maximal domain of meromorphy of the Euler product ∏ p prime h p - s 1 , ⋯ , p - s n and the natural boundary is precisely described when it exists. In this way we extend a well known result for one variable polynomials due to Estermann from 1928. As an application, we calculate the natural boundary of the multivariate Euler products associated to a family of toric varieties.

How to cite

top

Delabarre, Ludovic. "Extension of Estermann’s theorem to Euler products associated to a multivariate polynomial." Bulletin de la Société Mathématique de France 141.2 (2013): 225-265. <http://eudml.org/doc/272653>.

@article{Delabarre2013,
abstract = {Given a multivariate polynomial $h\left(X_1,\dots ,X_n\right)$ with integral coefficients verifying an hypothesis of analytic regularity (and satisfying $h(\textbf \{0\})=1$), we determine the maximal domain of meromorphy of the Euler product $\prod _\{p \ \textrm \{prime\}\}h\left(p^\{-s_1\},\dots ,p^\{-s_n\}\right)$ and the natural boundary is precisely described when it exists. In this way we extend a well known result for one variable polynomials due to Estermann from 1928. As an application, we calculate the natural boundary of the multivariate Euler products associated to a family of toric varieties.},
author = {Delabarre, Ludovic},
journal = {Bulletin de la Société Mathématique de France},
keywords = {multivariables Euler products; meromorphic continuation; natural boundary; cyclotomic polynomial; rational point on a toric variety},
language = {eng},
number = {2},
pages = {225-265},
publisher = {Société mathématique de France},
title = {Extension of Estermann’s theorem to Euler products associated to a multivariate polynomial},
url = {http://eudml.org/doc/272653},
volume = {141},
year = {2013},
}

TY - JOUR
AU - Delabarre, Ludovic
TI - Extension of Estermann’s theorem to Euler products associated to a multivariate polynomial
JO - Bulletin de la Société Mathématique de France
PY - 2013
PB - Société mathématique de France
VL - 141
IS - 2
SP - 225
EP - 265
AB - Given a multivariate polynomial $h\left(X_1,\dots ,X_n\right)$ with integral coefficients verifying an hypothesis of analytic regularity (and satisfying $h(\textbf {0})=1$), we determine the maximal domain of meromorphy of the Euler product $\prod _{p \ \textrm {prime}}h\left(p^{-s_1},\dots ,p^{-s_n}\right)$ and the natural boundary is precisely described when it exists. In this way we extend a well known result for one variable polynomials due to Estermann from 1928. As an application, we calculate the natural boundary of the multivariate Euler products associated to a family of toric varieties.
LA - eng
KW - multivariables Euler products; meromorphic continuation; natural boundary; cyclotomic polynomial; rational point on a toric variety
UR - http://eudml.org/doc/272653
ER -

References

top
  1. [1] S. S. Abhyankar – Local analytic geometry, Pure and Applied Mathematics, Vol. XIV, Academic Press. MR175897
  2. [2] G. Bhowmik, D. Essouabri & B. Lichtin – « Meromorphic continuation of multivariable euler products », Forum Math. 19, p. 1111–1139. Zbl1181.11057MR2367957
  3. [3] R. de la Bretèche – « Sur le nombre de points de hauteur bornée d’une certaine surface cubique singulière », Astérisque 251, p. 51–77, Nombre et répartition de points de hauteur bornée (Paris, 1996). MR1679839
  4. [4] A. Chenciner – Courbes algébriques planes, Publications Mathématiques de l’Université Paris VII, vol. 4, Université de Paris VII U.E.R. de Mathématiques. 
  5. [5] G. Dahlquist – « On the analytic continuation of eulerian products », Ark. Mat. 1, p. 533–554. Zbl0046.27205MR49298
  6. [6] L. Delabarre – « On the domain of meromorphy of a multivariate Euler product of Igusa type », Forum Mathematicum, doi://10.1515/forum-2011-0092. 
  7. [7] T. Estermann – « On certain functions represented by dirichlet series », Proc. London Math. Soc. 27, p. 435–448. Zbl54.0366.03MR1575403
  8. [8] N. Kurokawa – « On the meromorphy of euler products. i », Proc. London Math. Soc. 53, p. 1–47. Zbl0595.10031MR842154
  9. [9] —, « On the meromorphy of euler products. ii », Proc. London Math. Soc. 53, p. 209–236. Zbl0609.10020MR850219
  10. [10] N. Kurokawa & H. Ochiai – « A multivariable euler product of igusa type and its applications », J. Number Theory 129, p. 1919–1930. Zbl1176.11065MR2522714
  11. [11] C. Laurent-Thiébaut – Théorie des fonctions holomorphes de plusieurs variables, Savoirs Actuels, InterÉditions. 
  12. [12] B. Z. Moroz – « On a class of dirichlet series associated to the ring of representations of a weil group », Proc. London Math. Soc. 56, p. 209–228. Zbl0608.12014MR922653
  13. [13] R. M. Range – « Extension phenomena in multidimensional complex analysis: correction of the historical record », Math. Intelligencer 24, p. 4–12. MR1907191
  14. [14] M. du Sautoy & F. Grunewald – « Zeta functions of groups: zeros and friendly ghosts », Amer. J. Math. 124, p. 1–48. Zbl1020.11058MR1878998
  15. [15] M. du Sautoy & L. Woodward – Zeta functions of groups and rings, Lecture Notes in Math., vol. 1925, Springer. MR2371185
  16. [16] P. Swinnerton-Dyer & R. de la Bretèche – « Fonction zêta des hauteurs associée à une certaine surface cubique », Bull. Soc. Math. France 135, p. 65–92. Zbl1207.11068

NotesEmbed ?

top

You must be logged in to post comments.