Q}}$; in addition, for any given number field only finitely many of the examples can be defined over that field. Furthermore, all of the explicit examples in characteristic $0$ known before now involve curves with geometrically reducible Jacobians, and the arithmetic of such curves differs qualitatively from that of curves whose Jacobians are irreducible.

In this paper we address these concerns by providing three new explicit families of pairs of non-isomorphic curves with isomorphic Jacobians. Each family is parametrized by an open subset of ${{\mathbb P}}^1$, so each family gives an infinite number of examples over ${{\mathbb Q}}$. Also, the Jacobians of the curves in one of the families are typically absolutely simple. Using examples from these families, we show that the Jacobian of a genus-$2$ curve over ${{\mathbb Q}}$ does not determine whether or not the curve has rational points, or indeed whether or not the curve has real points. Liu, Lorenzini, and Raynaud [@LLR] have used our results to show that the Jacobian of a genus-$2$ curve over ${{\mathbb Q}}$ does not determine the number of components on the reduction of a minimal model of the curve modulo a prime.

Our first family of pairs of curves can 