Continued fractions for rational torsion

TL;DR

The paper introduces a method using continued fractions in function fields to discover new families of hyperelliptic curves over rationals with specified torsion in their Jacobians, including a novel genus two family with torsion order eleven.

Contribution

It presents a new approach leveraging continued fractions to find hyperelliptic curves with prescribed torsion, correcting previous errors and providing a genuinely new family of curves.

Findings

New infinite family of genus two curves with torsion order eleven.

Method successfully finds hyperelliptic curves with specific torsion properties.

Correction of earlier family to ensure novelty and accuracy.

Abstract

We exhibit a method to use continued fractions in function fields to find new families of hyperelliptic curves over the rationals with given torsion order in their Jacobians. To show the utility of the method, we exhibit a new infinite family of curves over $\mathbb Q$ with genus two whose Jacobians have torsion order eleven. {\bf In this updated version, we correct an error in the initial paper:} The ``new" family claimed in the original Theorem~1 was pointed out by Professor D.~Lorenzini to have elements isomorphic with elements in Flynn's family (as defined in the paper); his guess that the families were the same up to element-wise isomorphism is correct. Here, we give a family that is new; the old proof, now free of clerical error (we had replaced our $g_u(x)$ by $1+g_u(x)$ when determining Igusa invariants), holds. Changes from the original version are flagged by {\color{red} UPDATED}; there are three of these: the new family $g_u(x)$; its partial quotients; the way to solve in the naive approach to find this new family. (We also added an acknowledgement section.) Finally, as an appendix we include a PDF export of Maple calculations verifying our correction.