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Rational torsion on simple genus two Jacobians
Abstract
We exhibit new subgroups of rational torsion points in geometrically simple Jacobians of genus-two curves over ℚ. The largest group, which has order 96 and invariants [2,2,2,12], is realized by curves of the form y2 = x(x−a2)(x−b2)(x−c2)(x−u2)(x−v2) where $a,b,c,u,v$ are positive integers that satisfy a2 + b2 + c2 = u2 + v2 and a4 + b4 + c4 = u4 + v4. We also find realizations of the groups [2,2,20], [2,2,4,4], [2,2,2,8], [2,4,8], and [6,6]. Finally, we record, to the best of our knowledge, all known subgroups that arise in genus-two Jacobians over ℚ, in the geometrically simple case and in general.
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BibTeX
@misc{balakrishnan2026rational,
title = {{Rational torsion on simple genus two Jacobians}},
author = {Jennifer S. Balakrishnan and Filip Najman and Ari Shnidman and Andrew V. Sutherland},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28543},
eprint = {2608.28543},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28543},
}
RIS
TY - UNPB
TI - Rational torsion on simple genus two Jacobians
AU - Jennifer S. Balakrishnan
AU - Filip Najman
AU - Ari Shnidman
AU - Andrew V. Sutherland
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28543
UR - https://arxiv.org/abs/2608.28543
AB - We exhibit new subgroups of rational torsion points in geometrically simple Jacobians of genus-two curves over ℚ. The largest group, which has order 96 and invariants [2,2,2,12], is realized by curves of the form y^2 = x(x−a^2)(x−b^2)(x−c^2)(x−u^2)(x−v^2) where $a,b,c,u,v$ are positive integers that satisfy a^2 + b^2 + c^2 = u^2 + v^2 and a^4 + b^4 + c^4 = u^4 + v^4. We also find realizations of the groups [2,2,20], [2,2,4,4], [2,2,2,8], [2,4,8], and [6,6]. Finally, we record, to the best of our knowledge, all known subgroups that arise in genus-two Jacobians over ℚ, in the geometrically simple case and in general.
ER -