Geodesic
Joint distribution for the Selmer ranks of the congruent number curves
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 1, pp. 105-119 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $.
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 $.
DOI : 10.21136/CMJ.2019.0171-18
Classification : 11G05, 11N45, 14H52
Keywords: elliptic curve; congruent number problem; Selmer group 
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     author = {Vre\'cica, Ilija S.},
     title = {Joint distribution for the {Selmer} ranks of the congruent number curves},
     journal = {Czechoslovak Mathematical Journal},
     pages = {105--119},
     year = {2020},
     volume = {70},
     number = {1},
     doi = {10.21136/CMJ.2019.0171-18},
     mrnumber = {4078348},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2019.0171-18/}
}
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Vrećica, Ilija S. Joint distribution for the Selmer ranks of the congruent number curves. Czechoslovak Mathematical Journal, Tome 70 (2020) no. 1, pp. 105-119. doi: 10.21136/CMJ.2019.0171-18

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