Harmonic Eisenstein Series of Weight One

Abstract

In this short note, we will construct a harmonic Eisenstein series of weight one, whose image under the ξ-operator is a weight one Eisenstein series studied by Hecke [6]. 1 2 YINGKUN LĨ ϑ(τ ) and related to the holomorphic Eisenstein series ϑ(τ ) constructed by Hecke via ξθ(τ ) = ϑ(τ ),where ξ = ξ 1 is the differentiable operator introduced by Bruinier and Funke [4]. In the notion loc. cit.,θ(τ ) is a harmonic Maass form of weight one. For any k ∈ 1 2 Z, a harmonic Maass form of weight k is a real analytic functions on the upper half-plane H := {τ = u+iv : v > 0} that transforms with weight k with respect to a discrete subgroup of SL 2 (R), and is annihilated by the weight k hyperbolic LaplacianHarmonic Maass forms can be written as the sum of a holomorphic part and a non-holomorphic part. The Fourier coefficients of their holomorphic parts are expected to contain interesting arithmetic information concerning the ξ k -images of the non-holomorphic parts (see e.g.[2, 5]).In [11], Kudla, Rapoport and Yang considered an Eisenstein series, which is harmonic. The Fourier coefficients of its holomorphic part are logarithms of rational numbers, and can be interpreted as the arithmetic degree of special divisors on an arithmetic curve. In view of their work and the Kudla program [8], we expect the Fourier coefficients of the harmonic Eisenstein series we construct to have a similar interpretation as well.The idea to constructθ(τ ) is rather straightforward. If we can construct a functionΘ(τ, t) such that it is modular in τ and satisfies ξΘ(τ, t) = Θ(τ, t) for each t, then simply integrating it in t will produce the desirableθ(τ ). This idea has already been used in [3], where ξ 1/2 connected the theta kernels constructed from the Gaussian and the Kudla-Millson Schwartz form. In our setting, we will introduce an L ∞ functionφ τ , which is a ξ-preimage of the Schwartz function used in constructing Θ(τ, t) under ξ (see Prop. 3.4). We will then use this function to form a theta kernelΘ(τ, t) and integrate it to obtain the harmonic Eisenstein seriesθ(τ ) in Theorem 4.3 in the last section.

Harmonic Maass forms associated to real quadratic fields

“…Remark 5.4. When L is isotropic, a similar statement holds, and the Fourier coefficients are logarithms of rational numbers (see [25]).…”