PreprintCC BY 4.0arXiv
On the Ramanujan-Ligozat modular unit
Abstract
Let χ be the quadratic Dirichlet character of conductor N associated with a given real quadratic field k and let Xχ(N) be the double covering of X0(N) associated with χ. In this note, we study a modular unit f on the curve Xχ(N). We prove that the transform of f by the Fricke involution wN is the level N analogue of the Rogers-Ramanujan continued fraction introduced by Lee and Park. Furthermore, if N is a positive prime discriminant, then f is Ligozat's modular unit, which is known from Mazur's work on the Eisenstein ideal. We extend Ligozat's construction to any positive fundamental discriminant N.
Related
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- The strong Ramanujan bound for Drinfeld modular forms
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BibTeX
@misc{castanobernard2026ramanujanligozat,
title = {{On the Ramanujan-Ligozat modular unit}},
author = {Carlos Castano-Bernard},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24307},
eprint = {2610.24307},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24307},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Ramanujan-Ligozat modular unit
AU - Carlos Castano-Bernard
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.24307
UR - https://arxiv.org/abs/2610.24307
AB - Let χ be the quadratic Dirichlet character of conductor N associated with a given real quadratic field k and let X_χ(N) be the double covering of X_0(N) associated with χ. In this note, we study a modular unit f on the curve X_χ(N). We prove that the transform of f by the Fricke involution w_N is the level N analogue of the Rogers-Ramanujan continued fraction introduced by Lee and Park. Furthermore, if N is a positive prime discriminant, then f is Ligozat's modular unit, which is known from Mazur's work on the Eisenstein ideal. We extend Ligozat's construction to any positive fundamental discriminant N.
ER -