2014/12/30 by Everett W. Howe · 1 citation
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Cryptography and Residue Arithmetic
paper · doi:10.1112/blms/bdu107
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, parameterized 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 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.