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