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 $$y² = x(x-a<sup>2)(x-b<sup>2)(x-c<sup>2)(x-u<sup>2)(x-v<sup>2)$$ where $a,b,c,u,v$ are positive integers that satisfy $$a² + b² + c² = u² + v<sup>2$$ and $$a⁴ + b⁴ + c⁴ = u⁴ + v<sup>4$$. 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.