- 2 t)/(4PQ)$, let $D=1$, and let ${{\mathcal C}}'$ be the curve over $K(Q)$ defined by using this $A$, $P$, $Q$, and $D$ in Bending’s construction. Then the $\alpha_i$ are related to the $\beta_i$ by $$Q\alpha_i = \beta_i/(s-1) + 2t,$$ and the curve $Y^2 = G_1 G_2 G_3$ is isomorphic $y^2 = g_1 g_2 g_3$ via the relation $x = 2(s-1)(QX + t)$. This shows that ${{\mathcal C}}(r,s,t)$ is isomorphic to its Richelot dual over $K(Q)$. The rest of the theorem follows from Bending’s theorem.

Bending’s family of curves has three “geometric” parameters $A$, $P$, and $Q$ and one “arithmetic” parameter $D$ (which parametrizes quadratic twists of the curve determined by $A$, $P$, and $Q$). Since the moduli space ${{\mathcal M}}$ of genus-$2$ curves with real multiplication by $\sqrt{2}$ is a two-dimensional rational variety, one might hope to replace Bending’s three-geometric-parameter family with a two-parameter family. But there is an obstruction, which stems from the fact that ${{\mathcal M}}$ is a coarse moduli space and not a fine one: A $K$-rational point on ${{\mathcal M}}$ does not necessarily give rise to a curve over $K$. Indeed, Mestre [@Mestre] has shown that to every $K$-rational point $P$ on the moduli space 