Harmonic Maass forms associated to real quadratic fields

Abstract

In this paper, we explicitly construct harmonic Maass forms that map to the holomorphic weight one theta series associated by Hecke to odd ray class group characters of real quadratic fields. From this construction, we give precise arithmetic information contained in the Fourier coefficients of the holomorphic part of the harmonic Maass form, establishing the main part of a conjecture of the second author.

Petersson norm of cusp forms associated to real quadratic fields

“…as a function of t ∈ D + F,N , and transforms in τ ∈ H with respect to the Weil representation ω of weight 1 on Γ. Integrating Θ L (τ, t)dt then defines a weight one modular form ϑ L (τ ) with the following Fourier expansion (see [3,6])…”

“…It is anisotropic over Q and has signature (1, 1) over R. In [6], Hecke associated to such lattice L a holomorphic, vector-valued cusp form ϑ L of weight one, whose Fourier expansion is given explicitly in terms of the elements in the dual lattice L ∨ (see §2. 3). In this note, we will calculate the Petersson norm of ϑ L , which is defined by with Γ := SL 2 (Z) and dµ(τ ) := dudv v 2 the invariant measure.…”

“…Remark 1.2. In [3], we constructed a harmonic Maass formθ L that maps to ϑ L under the differential operator ξ := 2iv∂ τ . One can express ϑ L 2 Pet as a rational linear combination of the principal part Fourier coefficients ofθ L , which are rational multiples of the fundamental unit of F .…”

“…Let ε κ ∈ O × F be the totally positive unit defined in §2. 3. It turns out that the Petersson norm of ϑ L is a rational multiple of log ε κ , and the rational multiple can be expressed in terms of Rademacher symbols Ψ(γ) of hyperbolic elements γ ∈ Γ (see §2.7).…”

Mock Modular Forms with Integral Fourier Coefficients

“…In some cases, Charollois and the first author [13] constructed mock modular forms whose shadows are weight 3 2 unary theta functions, and whose coefficients are simple finite sums of rational numbers. In particular, they obtained explicit bounds on the denominators of the coefficients.…”