Shimura’s vector-valued modular forms, weight changing operators, and Laplacians

Abstract

We investigate the various types of weight raising and weight lowering operators on quasi-modular forms, or equivalently on Shimura's vector-valued modular forms involving symmetric power representations. We also present all the eigenfunctions of the two possible Laplacian operators.

Shintani lifts of nearly holomorphic modular forms

“…These differential operators preserve modularity, in the sense that whereas for , which involves complex conjugation, we have It is known that , , and preserve near holomorphicity, with decreasing the depth by 1, and and increasing it by at most 1. For more on these modular forms, including their relations with quasi-modular forms and Shimura’s vector-valued modular forms, see [MR, Ze3, Ze7].…”

“…Using [Ze3, Ze7], the sum is a (vector-valued) quasi-modular form of weight and depth , and one checks that the contribution of is just for every . Moreover, Lemmas 3.16 and 3.21 and equation (3.32) show that applying this combination to , , and amounts to replacing from equation (4.13), in equation (4.9), and appearing in equation (4.10) by , , and , respectively.…”

“…We will show in Corollary 4.4 that when , the Shintani lift of a nearly holomorphic modular form is also nearly holomorphic, and establish results analogous to the harmonic case (with or without constant terms) (see Theorem 1.2 in the Introduction, as well as Theorem 4.3, Proposition 4.5, and Remark 4.6 for the general statement). One could perhaps try to give another proof of the nearly holomorphic lift result by using the isomorphism described in, e.g., [MR, Ze3, Ze7], and analyzing the effects of raising operators on theta kernels. Here, is the iterated raising operator defined in (2.2).…”

Shimura lifts of weakly holomorphic modular forms

“…It is known that L z , R κ , and ∆ κ preserve near holomorphicity, with L z decreasing the depth by 1, and R κ and ∆ κ usually increasing it by 1. For more on these modular forms, including their relations with quasi-modular forms and Shimura's vector valued modular forms, see [MR], [Ze3], and [Ze7].…”

“…Remark 4.5. Using [Ze3] and [Ze7], the sum ⌊p/2⌋ a=0 1 v a a! L a τ I k,L (τ, f ) is a (vector-valued) quasi-modular form of weight k + 1 2 and depth p 2 , and one checks that the contribution of I nh k,L,h (τ, f ) is just 0≤m∈Z+Q(h) Tr m,h (f )q m τ for every h ∈ D L .…”