May 2003
title: 'Infinite families of pairs of curves over ${{\mathbb Q}}$ with isomorphic Jacobians'
---

Introduction {#S-intro}
============

Torelli’s theorem shows that a curve is completely determined by its polarized Jacobian variety, but it has been known since the late 1800s that distinct curves can have isomorphic unpolarized Jacobians. In particular, the unpolarized Jacobian of a curve may not reflect all of the curve’s geometric properties. Proving that a particular property of curves cannot always be determined from the Jacobian is equivalent to showing that there exist two curves, one with the given property and one without, whose Jacobians are isomorphic to one another. Thus, for example, the pairs of curves written down in [@Howe:PAMS] show that one cannot tell whether or not a curve of genus $3$ over the complex numbers is hyperelliptic simply by looking at its Jacobian.

One would also like to find [*arithmetic*]{} properties of curves that are not determined by the Jacobian, but from an arithmetic perspective the heretofore-known explicit examples of distinct curves with isomorphic Jacobians (catalogued in the introduction to [@Howe:PAMS]) are not entirely satisfying. The primary complaint is that none of the examples involves curves that can be defined over ${{\mathbb 