Unipotent vector bundles and higher-order non-holomorphic Eisenstein series
Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 1, p. 131-163
Les séries d’Eisenstein non-holomorphes d’ordre supérieur associées à un groupe Fuschien Γ sont définies en tordant le développement des séries d’Eisenstein non-holomorphes classiques par des puissances des symboles modulaires. Leurs équations fonctionnelles contiennent des facteurs multiplicatifs et additifs, ce qui les différencient des séries d’Eisenstein classiques. Dans cet article, nous prouvons l’existence d’un prolongement méromorphe pour ces séries et établissons leurs équations fonctionnelles reliant les valeurs en s et 1-s. De plus, nous construisons des fibrés vectoriels de rang elevé 𝒱 pour certaines représentations unipotentes π de Γ et montrons que les séries d’Eisenstein non-holomorphes d’ordre supérieur peuvent être interprétées comme des composantes de certaines sections propres, 𝔼, de 𝒱. Avec ce point-de-vue, les équations fonctionnelles de ces séries d’ordre supérieur sont formellement identiques au cas classique. Allant plus loin, nous prouvons des bornes pour les coefficients de Fourier des séries d’Eisenstein non-holomorphes d’ordre supérieur.
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.
@article{JTNB_2008__20_1_131_0,
     author = {Jorgenson, Jay and O'Sullivan, Cormac},
     title = {Unipotent vector bundles and higher-order non-holomorphic Eisenstein series},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {1},
     year = {2008},
     pages = {131-163},
     doi = {10.5802/jtnb.619},
     mrnumber = {2434161},
     zbl = {pre05543194},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2008__20_1_131_0}
}
Jorgenson, Jay; O’Sullivan, Cormac. Unipotent vector bundles and higher-order non-holomorphic Eisenstein series. Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 1, pp. 131-163. doi : 10.5802/jtnb.619. https://www.numdam.org/item/JTNB_2008__20_1_131_0/

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