Modular forms of half-integral weight on ?0(4)

Theta lifts and local Maass forms

“…We use pr to denote the projection operator (see Section 2.3 of [23]) into Kohnen's plus space. It is useful to recall that if F is modular in Kohnen's plus space for Γ, then its Fourier expansions at the cusps 0 and 1 2 are determined by the expansion at i∞ (see [23] for a proof in the holomorphic case). Like the Hecke operators, the projection operator pr is Hermitian with respect to the regularized inner product, i.e., (2.5) G| pr, H reg = G, H| pr reg .…”

Jacobi forms over complex quadratic fields via the cubic Casimir operators

“…Our construction is of the same flavor as the method of Kohnen [14] (see also Section 4 of Pitale [17]). It is likely that our results can be extended to other complex quadratic fields, but for simplicity we restrict to the case that 3.1.…”

“…Modular forms of half-integral weight. Elstrodt, Grunewald, and Mennicke [8] give a good overview of automorphic functions over complex quadratic fields K. We extend their notion of an automorphic form and we consider modular forms of half-integral weight over K = Q(i), and in particular, we introduce an analog of Kohnen's [14] plus space. Kojima [15] also discusses modular forms of half-integral weight over complex quadratic fields, but our point of view is quite different.…”

“…We proceed as in Kohnen [14] to construct a lift from said analog to the space of Jacobi forms over K = Q(i), which gives rise to new examples of Jacobi forms over K = Q(i). Our work may be viewed as the first step toward a rigorous theory of Jacobi forms over complex quadratic fields.…”

Linear relations between Fourier coefficients of special Siegel modular forms

“…c(m)e 2πimz (z ∈ H = upper half-plane) be a cuspidal Hecke eigenform of weight k + 1 2 and level 4 contained in the "plus" space which corresponds to f under the Shimura correspondence [9,12].…”