# Explicit Hecke series for symplectic group of genus 4

• [1] Université de Franche-Comté Laboratoire de mathématiques de Besançon UFR Sciences et techniques 16 route de Gray 25030 Besançon, France
• Volume: 23, Issue: 1, page 279-298
• ISSN: 1246-7405

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## Abstract

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Shimura conjectured the rationality of the generating series for Hecke operators for the symplectic group of genus $n$. This conjecture was proved by Andrianov for arbitrary genus $n$, but the explicit expression was out of reach for genus higher than 3. For genus $n=4$, we explicitly compute the rational fraction in this conjecture. Using formulas for images of double cosets under the Satake spherical map, we first compute the sum of the generating series, which is a rational fraction with polynomial coefficients. Then we recover the coefficients of this fraction as elements of the Hecke algebra using polynomial representation of basis Hecke operators under the spherical map. Numerical examples of these fractions for special choice of Satake parameters are given.

## How to cite

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Vankov, Kirill. "Explicit Hecke series for symplectic group of genus 4." Journal de Théorie des Nombres de Bordeaux 23.1 (2011): 279-298. <http://eudml.org/doc/219742>.

@article{Vankov2011,
abstract = {Shimura conjectured the rationality of the generating series for Hecke operators for the symplectic group of genus $n$. This conjecture was proved by Andrianov for arbitrary genus $n$, but the explicit expression was out of reach for genus higher than 3. For genus $n=4$, we explicitly compute the rational fraction in this conjecture. Using formulas for images of double cosets under the Satake spherical map, we first compute the sum of the generating series, which is a rational fraction with polynomial coefficients. Then we recover the coefficients of this fraction as elements of the Hecke algebra using polynomial representation of basis Hecke operators under the spherical map. Numerical examples of these fractions for special choice of Satake parameters are given.},
affiliation = {Université de Franche-Comté Laboratoire de mathématiques de Besançon UFR Sciences et techniques 16 route de Gray 25030 Besançon, France},
author = {Vankov, Kirill},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {3},
number = {1},
pages = {279-298},
publisher = {Société Arithmétique de Bordeaux},
title = {Explicit Hecke series for symplectic group of genus 4},
url = {http://eudml.org/doc/219742},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Vankov, Kirill
TI - Explicit Hecke series for symplectic group of genus 4
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/3//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 1
SP - 279
EP - 298
AB - Shimura conjectured the rationality of the generating series for Hecke operators for the symplectic group of genus $n$. This conjecture was proved by Andrianov for arbitrary genus $n$, but the explicit expression was out of reach for genus higher than 3. For genus $n=4$, we explicitly compute the rational fraction in this conjecture. Using formulas for images of double cosets under the Satake spherical map, we first compute the sum of the generating series, which is a rational fraction with polynomial coefficients. Then we recover the coefficients of this fraction as elements of the Hecke algebra using polynomial representation of basis Hecke operators under the spherical map. Numerical examples of these fractions for special choice of Satake parameters are given.
LA - eng
UR - http://eudml.org/doc/219742
ER -

## References

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13. A. Panchishkin, K. Vankov, Explicit formulas for Hecke operators and Rankin’s lemma in higher genus. In Algebra, Arithmetic and Geometry. In Honor of Yu.I. Manin, volume 269–270 of Progress in Mathematics. Birkhäuser Boston Inc., Boston, MA, 2010. Zbl1246.11105
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17. K. Vankov, Hecke algebras, generating series and applications, Thèse de Doctorat de l’Université Joseph Fourier. oai:tel.archives-ouvertes.fr:tel-00349767_v1, November 2008.

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