Eisenstein series for the Weil representation

JOURNAL OF NUMBER THEORY

卷:193

Article

Schwagenscheidt, Markus¹ 

[1] Univ Cologne, Math Inst, Gyrhofstr 8b, D-50931 Cologne, Germany

关键词: Holomorphic modular forms of integral and half-integral weight;    Eisenstein series;    Fourier coefficients;    Vector valued modular forms;    Weil representation;   

DOI  :  10.1016/j.jnt.2018.05.014

来源: Elsevier

【 摘 要 】

We compute the Fourier expansion of vector valued Eisenstein series for the Weil representation associated to an even lattice. To this end, we define certain twists by Dirichlet characters of the usual Eisenstein series associated to isotropic elements in the discriminant form of the underlying lattice. These twisted functions still form a generating system for the space of Eisenstein series but their Fourier coefficients have better multiplicative properties than the coefficients of the individual Eisenstein series. We adapt a method of Bruinier and Kuss to obtain algebraic formulas for the Fourier coefficients of the twisted Eisenstein series in terms of special values of Dirichlet L-functions and representation numbers modulo prime powers of the underlying lattice. In particular, we obtain that the Fourier coefficients of the individual Eisenstein series are rational numbers. Additionally, we show that the twisted Eisenstein series are eigenforms of the Hecke operators on vector valued modular forms introduced by Bruinier and Stein. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free