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Joint distribution for the Selmer ranks of the congruent number curves

Ilija S. Vrećica — 2020

Czechoslovak Mathematical Journal

We determine the distribution over square-free integers n of the pair ( dim 𝔽 2 Sel Φ ( E n / ℚ ) , dim 𝔽 2 Sel Φ ^ ( E n ' / ℚ ) ) , where E n is a curve in the congruent number curve family, E n ' : y 2 = x 3 + 4 n 2 x is the image of isogeny Φ : E n → E n ' , Φ ( x , y ) = ( y 2 / x 2 , y ( n 2 - x 2 ) / x 2 ) , and Φ ^ is the isogeny dual to Φ .

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