Unipotent vector bundles and higher-order non-holomorphic Eisenstein series

Jay Jorgenson[1]; Cormac O’Sullivan[2]

  • [1] Department of Mathematics City College of New York Convent Avenue at 138th Street New York, NY 1003
  • [2] Department of Mathematics Bronx Community College University Avenue and West 181st Street Bronx, NY 1045

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 1, page 131-163
  • ISSN: 1246-7405

Abstract

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Higher-order non-holomorphic Eisenstein series associated to a Fuchsian group Γ are defined by twisting the series expansion for classical non-holomorphic Eisenstein series by powers of modular symbols. Their functional identities include multiplicative and additive factors, making them distinct from classical Eisenstein series. In this article we prove the meromorphic continuation of these series and establish their functional equations which relate values at s and 1 - s . In addition, we construct high rank vector bundles 𝒱 from certain unipotent representations π of Γ and show that higher-order non-holomorphic Eisenstein series can be viewed as components of certain eigensections, 𝔼 , of 𝒱 . With this viewpoint the functional identities of these higher-order series are formally identical to the classical case. Going further, we prove bounds on the Fourier coefficients of the higher-order non-holomorphic Eisenstein series.

How to cite

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Jorgenson, Jay, and O’Sullivan, Cormac. "Unipotent vector bundles and higher-order non-holomorphic Eisenstein series." Journal de Théorie des Nombres de Bordeaux 20.1 (2008): 131-163. <http://eudml.org/doc/10825>.

@article{Jorgenson2008,
abstract = {Higher-order non-holomorphic Eisenstein series associated to a Fuchsian group $\Gamma $ are defined by twisting the series expansion for classical non-holomorphic Eisenstein series by powers of modular symbols. Their functional identities include multiplicative and additive factors, making them distinct from classical Eisenstein series. In this article we prove the meromorphic continuation of these series and establish their functional equations which relate values at $s$ and $1-s$. In addition, we construct high rank vector bundles $\mathcal\{V\}$ from certain unipotent representations $\pi $ of $\Gamma $ and show that higher-order non-holomorphic Eisenstein series can be viewed as components of certain eigensections, $\mathbb\{E\}$, of $\mathcal\{V\}$. With this viewpoint the functional identities of these higher-order series are formally identical to the classical case. Going further, we prove bounds on the Fourier coefficients of the higher-order non-holomorphic Eisenstein series.},
affiliation = {Department of Mathematics City College of New York Convent Avenue at 138th Street New York, NY 1003; Department of Mathematics Bronx Community College University Avenue and West 181st Street Bronx, NY 1045},
author = {Jorgenson, Jay, O’Sullivan, Cormac},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {1},
pages = {131-163},
publisher = {Université Bordeaux 1},
title = {Unipotent vector bundles and higher-order non-holomorphic Eisenstein series},
url = {http://eudml.org/doc/10825},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Jorgenson, Jay
AU - O’Sullivan, Cormac
TI - Unipotent vector bundles and higher-order non-holomorphic Eisenstein series
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 1
SP - 131
EP - 163
AB - Higher-order non-holomorphic Eisenstein series associated to a Fuchsian group $\Gamma $ are defined by twisting the series expansion for classical non-holomorphic Eisenstein series by powers of modular symbols. Their functional identities include multiplicative and additive factors, making them distinct from classical Eisenstein series. In this article we prove the meromorphic continuation of these series and establish their functional equations which relate values at $s$ and $1-s$. In addition, we construct high rank vector bundles $\mathcal{V}$ from certain unipotent representations $\pi $ of $\Gamma $ and show that higher-order non-holomorphic Eisenstein series can be viewed as components of certain eigensections, $\mathbb{E}$, of $\mathcal{V}$. With this viewpoint the functional identities of these higher-order series are formally identical to the classical case. Going further, we prove bounds on the Fourier coefficients of the higher-order non-holomorphic Eisenstein series.
LA - eng
UR - http://eudml.org/doc/10825
ER -

References

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