We propose a solution to the hyperelliptic Schottky problem, based on the use of Jacobian Nullwerte and symmetric models for hyperelliptic curves. Both ingredients are interesting on its own, since the first provide period matrices which can be geometrically described, and the second have remarkable arithmetic properties.

We determine the distribution over square-free integers $n$ of the pair $({dim}_{{𝔽}_2}{\mathrm{Sel}}^{\Phi }(E_n/\mathbb{Q}),{dim}_{{𝔽}_2}{\mathrm{Sel}}^{\widehat{\Phi }}(E_n^{\text{'}}/\mathbb{Q}))$, where $E_n$ is a curve in the congruent number curve family, $E_n^{\text{'}}:y^2=x^3+4n^{2}x$ is the image of isogeny $\Phi :E_n\to E_n^{\text{'}}$, $\Phi (x,y)=(y^2/x^2,y(n^2-x^2)/x^2)$, and $\widehat{\Phi }$ is the isogeny dual to $\Phi$.