Computing Theta Characteristics and their Galois action on general curves of moderate genus
Abstract
In this article, we present an algorithm that, given a smooth curve X defined over ℚ, returns an explicit arithmetic representation of both odd and even Theta Characteristics split by parity. It allows us to retrieve their Galois action and étale algebra. The output includes the Theta Hyperplanes of the Theta Characteristics when they are finite, their associated proof functions as well as the dictionary of their quadratic forms. The algorithm works in a rather large generality as it handles any one-equation (potentially singular) model of smooth curves. The most binding requirement remains the necessity for an explicit degree 1 rational divisor on the curve or more likely a rational point. We implemented the algorithm and give examples up to genus $5$.
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BibTeX
@misc{misme2026computing,
title = {{Computing Theta Characteristics and their Galois action on general curves of moderate genus}},
author = {Thibaut Misme},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29449},
eprint = {2610.29449},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29449},
}
RIS
TY - UNPB
TI - Computing Theta Characteristics and their Galois action on general curves of moderate genus
AU - Thibaut Misme
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29449
UR - https://arxiv.org/abs/2610.29449
AB - In this article, we present an algorithm that, given a smooth curve X defined over ℚ, returns an explicit arithmetic representation of both odd and even Theta Characteristics split by parity. It allows us to retrieve their Galois action and étale algebra. The output includes the Theta Hyperplanes of the Theta Characteristics when they are finite, their associated proof functions as well as the dictionary of their quadratic forms. The algorithm works in a rather large generality as it handles any one-equation (potentially singular) model of smooth curves. The most binding requirement remains the necessity for an explicit degree 1 rational divisor on the curve or more likely a rational point. We implemented the algorithm and give examples up to genus $5$.
ER -