Eisenstein series for the Weil representation

TL;DR

This paper computes the Fourier expansion of vector valued Eisenstein series for the Weil representation, introducing twisted series with improved multiplicative properties and explicit algebraic formulas for their coefficients.

Contribution

It defines twisted Eisenstein series with better multiplicative properties and derives explicit algebraic formulas for their Fourier coefficients in terms of L-values and representation numbers.

Findings

Fourier coefficients are rational numbers.

Twisted Eisenstein series are eigenforms of Hecke operators.

Explicit formulas involve Dirichlet L-values and lattice representation numbers.

Abstract

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 have better multiplicative properties than 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.