principal polarization is a Jacobian, so one of our goals in this paper is to write down abelian surfaces with two non-isomorphic principal polarizations. In this section we will make a few observations about such surfaces.

Suppose $B$ is an abelian surface with two principal polarizations $\mu$ and $\mu'$, which we view as isogenies from $B$ to its dual variety $\hat{B}$. The polarized varieties $(B,\mu)$ and $(B,\mu')$ are isomorphic to one another if and only if there is an automorphism $\beta$ of $B$ such that $\mu' = \hat{\beta} \mu \beta$, where $\hat{\beta}$ is the dual of $\beta$. We would like to write down an abelian surface $B$ with two non-isomorphic principal polarizations $\mu$ and $\mu'$, so we would like to avoid the existence of such an automorphism $\beta$. We will accomplish this by obtaining $\mu'$ from $\mu$ through the use of an automorphism of a surface [*isogenous*]{} to $B$. Our main tool is the following well-known construction:

Suppose $(A,\lambda)$ is a principally-polarized abelian surface over a field $K$, suppose $n$ is a positive integer, and suppose $G$ is a rank-$n^2$ subgroupscheme of the $n$-torsion $A[n]$ of $A$ that is isotropic with respect to the $\lambda$-Weil pairing on $A[n]$. Let $B$ 