PreprintCC BY 4.0arXiv
Noncongruence cusp forms, Veech groups and enumeration
Abstract
We prove the existence of weight 2 modular forms for some Veech groups, whose Fourier coefficients at a cusp determine the counts of numbers of branched Galois covers of punctured elliptic curves. We further prove there exist cusp forms, counting ramified genus $2$ covers, that are not invariant under any congruence subgroup of SL(2,Z). The main results may be viewed as analogs of the construction of an elliptic curve obtained as the quotient of the Jacobian of a classical modular curve, whose point counts modulo primes determine the coefficients of a Hecke eigenform, but where the uniformizing groups may be noncongruence.
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BibTeX
@misc{rached2026noncongruence,
title = {{Noncongruence cusp forms, Veech groups and enumeration}},
author = {John Rached},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28514},
eprint = {2609.28514},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28514},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Noncongruence cusp forms, Veech groups and enumeration
AU - John Rached
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.28514
UR - https://arxiv.org/abs/2609.28514
AB - We prove the existence of weight 2 modular forms for some Veech groups, whose Fourier coefficients at a cusp determine the counts of numbers of branched Galois covers of punctured elliptic curves. We further prove there exist cusp forms, counting ramified genus $2$ covers, that are not invariant under any congruence subgroup of SL(2,Z). The main results may be viewed as analogs of the construction of an elliptic curve obtained as the quotient of the Jacobian of a classical modular curve, whose point counts modulo primes determine the coefficients of a Hecke eigenform, but where the uniformizing groups may be noncongruence.
ER -