Year: 2026
Type: preprint
Abstract: We exhibit new subgroups of rational torsion points in geometrically simple Jacobians of genus-two curves over $\mathbb Q$. 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 $\mathbb Q$, in the geometrically simple case and in general.
Source: arXiv (Cornell University)
Institutions: Boston University, Massachusetts Institute of Technology
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1717035117, Number theory and arithemtic geometry, G2450643449, G2719336981, G3408410167, G3611179903, G3678004387, G3866112975, G4938354709, Rational Points on Curves and Kim's Conjecture, G6671297155, G675437911, G7159977746, CAREER: New Directions in p-adic Heights and Rational Points on Curves, G7799207878
OpenAlex ID: W7204789477