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Free modules of vector-valued modular forms and applications to Jacobi forms
Abstract
The space M(ρ;ℰ) of weakly holomorphic modular forms for a dimension-r complex representation ρ of SL2(ℤ) with growth prescribed by ℰ at the cusp is a free module over ℂ[E4,E6]. We first prove that a ℂ[E4,E6]-linearly independent homogeneous family F1,…,Fr in this module gives a basis if and only if ∑i=1r wt(Fi)=12trℰ. We then extend this criterion to Jacobi forms. As applications, we determine the generator weights of free modules of O(D8d+)-invariant Jacobi forms of index D8d+(p), where D8d+ is an even unimodular lattice extending D8d, d≥2, and p=2,3,5.
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BibTeX
@misc{li2026free,
title = {{Free modules of vector-valued modular forms and applications to Jacobi forms}},
author = {Jialin Li and Haowu Wang and Bohan Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39270},
eprint = {2610.39270},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39270},
}
RIS
TY - UNPB
TI - Free modules of vector-valued modular forms and applications to Jacobi forms
AU - Jialin Li
AU - Haowu Wang
AU - Bohan Zhang
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39270
UR - https://arxiv.org/abs/2610.39270
AB - The space M(ρ;ℰ) of weakly holomorphic modular forms for a dimension-r complex representation ρ of SL_2(ℤ) with growth prescribed by ℰ at the cusp is a free module over ℂ[E_4,E_6]. We first prove that a ℂ[E_4,E_6]-linearly independent homogeneous family F_1,…,F_r in this module gives a basis if and only if ∑_i=1^r wt(F_i)=12trℰ. We then extend this criterion to Jacobi forms. As applications, we determine the generator weights of free modules of O(D_8d^+)-invariant Jacobi forms of index D_8d^+(p), where D_8d^+ is an even unimodular lattice extending D_8d, d≥2, and p=2,3,5.
ER -