$\sqrt{2}$ over $K$, and we show how to adapt Bending’s result to obtain curves over $K$ with real multiplication by $\sqrt{2}$ over a quadratic extension of $K$. In Section \[S-Galois\] we give some Galois restrictions on our generalization of Bending’s construction that ensure that the curves we construct have two rational Richelot isogenies to curves with isomorphic Jacobians. In Section \[S-application\] we show that there is a positive-rank elliptic surface whose points give rise to pairs of genus-$2$ curves with isomorphic Jacobians, and we prove Theorem \[T-simple2\]. In Section \[S-genus3\] we prove Theorem \[T-nonsimple3\]. Finally, in Section \[S-examples\] we provide some explicit examples of curves over ${{\mathbb Q}}$ produced by our theorems, and we show that the Jacobian of a curve over ${{\mathbb Q}}$ does not determine whether or not the curve has rational points, or even whether or not it has real points.

We relied heavily on the computer algebra system Magma [@Magma] while working on this paper. Some of our Magma routines are available on the web: to find them, start at [http://alumni.caltech.edu/\~however/biblio.html]{}

and follow the links related to this paper.

Abelian surfaces with non-isomorphic polarizations {#S-curves}
==================================================

Weil [@Weil] showed that an abelian surface with an indecomposable 