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 $\mathbb{Q}$. 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.

18 pages; 2 tables