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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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

[1] Brent, R. P., McKay, B. D.: Determinants and ranks of random matrices over $\mathbb{Z}_m$. Discrete Math. 66 (1987), 35-49. | DOI | MR | JFM

[2] Faulkner, B., James, K.: A graphical approach to computing Selmer groups of congruent number curves. Ramanujan J. 14 (2007), 107-129. | DOI | MR | JFM

[3] Feng, K.: Non-congruent numbers, odd graphs and the Birch-Swinnerton-Dyer conjecture. Acta Arith. 75 (1996), 71-83. | DOI | MR | JFM

[4] Feng, K., Xiong, M.: On elliptic curves $y^2=x^3-n^2x$ with rank zero. J. Number Theory 109 (2004), 1-26. | DOI | MR | JFM

[5] Heath-Brown, D. R.: The size of Selmer groups for the congruent number problem. Invent. Math. 111 (1993), 171-195. | DOI | MR | JFM

[6] Heath-Brown, D. R.: The size of Selmer groups for the congruent number problem II. Invent. Math. 118 (1994), 331-370. | DOI | MR | JFM

[7] Kane, D., Klagsbrun, Z.: On the joint distribution of $ Sel_\Phi (E/\mathbb{Q})$ and $ Sel_{\widehat{\Phi}}(E'/\mathbb{Q})$ in quadratic twist families. Available at , 25 pages. | arXiv

[8] Koblitz, N.: Introduction to Elliptic Curves and Modular Forms. Graduate Texts in Mathematics 97, Springer, New York (1984). | DOI | MR | JFM

[9] Rhoades, R. C.: $2$-Selmer groups and the Birch-Swinnerton-Dyer conjecture for the congruent number curves. J. Number Theory 129 (2009), 1379-1391. | DOI | MR | JFM

[10] Xiong, M., Zaharescu, A.: Selmer groups and Tate-Shafarevich groups for the congruent number problem. Comment. Math. Helv. 84 (2009), 21-56. | DOI | MR | JFM

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