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Automorphisms of general modular curves
Abstract
We prove that for a geometrically connected modular curve XH of prime level larger than $311$, all automorphisms are modular, i.e., they arise from the moduli interpretation of XH. The proof relies on a general criterion involving the gonality of XH and the dimension of the CM part of the Jacobian of XH. For arbitrary level, we give algorithms to compute the modular automorphisms and, in many cases, to prove that there are no other automorphisms. Finally, we apply these algorithms to the curves in the database LMFDB.
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BibTeX
@misc{dose2026automorphisms,
title = {{Automorphisms of general modular curves}},
author = {Valerio Dose and Guido Maria Lido and Pietro Mercuri},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08998},
eprint = {2610.08998},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08998},
}
RIS
TY - UNPB
TI - Automorphisms of general modular curves
AU - Valerio Dose
AU - Guido Maria Lido
AU - Pietro Mercuri
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08998
UR - https://arxiv.org/abs/2610.08998
AB - We prove that for a geometrically connected modular curve X_H of prime level larger than $311$, all automorphisms are modular, i.e., they arise from the moduli interpretation of X_H. The proof relies on a general criterion involving the gonality of X_H and the dimension of the CM part of the Jacobian of X_H. For arbitrary level, we give algorithms to compute the modular automorphisms and, in many cases, to prove that there are no other automorphisms. Finally, we apply these algorithms to the curves in the database LMFDB.
ER -