Everett W. Howe: Genus-2 Jacobians with torsion points of large order, Bull. London Math. Soc. 47 (2015)127–135.

There is an official version available online, as well as a preprint version: arXiv:1407.2654 [math.AG].

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 points of order 27, 28, 36, 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 Leprévost 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 Bröker, Lauter, and Stevenhagen.

The curve with a torsion point of order 70 can be written

y2 + (2x3 − 3x2 − 41x + 110) y = x3 − 51x2 + 425x + 179.