of genus-$2$ curves there is naturally associated a genus-$0$ curve over $K$, and $P$ corresponds to a curve over $K$ if and only if the genus-$0$ curve has a $K$-rational point.

Galois restrictions {#S-Galois}
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In order to prove Theorem \[T-simple2\] we will apply the construction outlined in Section \[S-curves\] to a Jacobian with real multiplication by $\sqrt{2}$ that we will obtain from Theorem \[T-RM\]; we will take the automorphism $\alpha$ to be $1+\sqrt{2}$. The construction requires that we find a Galois-stable maximal isotropic subgroup $G$ of the $2$-torsion of the Jacobian such that $G' = (1+\sqrt{2})(G)$ is a maximal isotropic subgroup different from $G$. This requirement imposes some restrictions on the values of $r$, $s$, and $t$ that we will be able to use in Theorem \[T-RM\]. In this section we will make these restrictions explicit, and in Section \[S-application\] we will find an elliptic surface that parametrizes a subset of the allowable values of $r$, $s$, and $t$.

Recall the basic outline of Theorem \[T-RM\]: Given three elements $r$, $s$, $t$ of our base field $K$, we define a polynomial $h = T^3 - c_2 T^2 + c_1 T - c_0$ in the polynomial ring $K[T]$, and we 