Genus-2 Jacobians with torsion points of large order

TL;DR

This paper constructs explicit genus-2 curves over Q with Jacobians having large rational torsion points, including the largest known order of 70, using elliptic curve 'gluing' techniques.

Contribution

It introduces new methods for producing genus-2 Jacobians with large rational torsion points, including the largest known order of 70, via explicit elliptic curve gluing techniques.

Findings

Constructed genus-2 Jacobians with rational torsion points of order 70.

Produced a family of genus-2 curves with torsion points of order 48.

Provided examples with torsion points of orders 27, 28, and 39.

Abstract

We produce new explicit examples of genus-2 curves over the rational numbers whose Jacobian varieties have rational torsion points of large order. In particular, we produce a family of genus-2 curves over Q whose Jacobians have a rational point of order 48, parametrized by a rank-2 elliptic curve over Q, and we exhibit a single genus-2 curve over Q whose Jacobian has a rational point of order 70, the largest order known. We also give new examples of genus-2 Jacobians with rational points of order 27, 28, and 39. Most of our examples are produced by `gluing' two elliptic curves together along their n-torsion subgroups, where n is either 2 or 3. The 2-gluing examples arise from techniques developed by the author in joint work with Lepr\'evost and Poonen 15 years ago. The 3-gluing examples are made possible by an algorithm for explicit 3-gluing over non-algebraically closed fields recently developed by the author in joint work with Br\"oker, Lauter, and Stevenhagen.