Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 70, No. 1, pp. 105-119, 2020

Joint distribution for the Selmer ranks of the congruent number curves

Ilija S. Vrećica

Received April 4, 2018. Published online September 23, 2019.

Abstract: 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 $\Phi$.

Keywords: elliptic curve; congruent number problem; Selmer group

Classification MSC: 11G05, 14H52, 11N45

DOI: 10.21136/CMJ.2019.0171-18

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

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Affiliations: Ilija Vrećica, Department of Mathematics, University of Belgrade, Studentski trg 16, Belgrade, Serbia, e-mail: ilijav@matf.bg.ac.rs

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