Joint distribution for the Selmer ranks of the congruent number curves

Ilija S. Vrećica

Czechoslovak Mathematical Journal (2020)

  • Volume: 70, Issue: 1, page 105-119
  • ISSN: 0011-4642

Abstract

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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 .

How to cite

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Vrećica, Ilija S.. "Joint distribution for the Selmer ranks of the congruent number curves." Czechoslovak Mathematical Journal 70.1 (2020): 105-119. <http://eudml.org/doc/297190>.

@article{Vrećica2020,
abstract = {We determine the distribution over square-free integers $n$ of the pair $(\dim _\{\mathbb \{F\}_2\}\{\rm Sel\}^\Phi (E_n/\mathbb \{Q\}),\dim _\{\mathbb \{F\}_2\} \{\rm Sel\}^\{\widehat\{\Phi \}\}(E_n^\{\prime \}/\mathbb \{Q\}))$, where $E_n$ is a curve in the congruent number curve family, $E_n^\{\prime \}\colon y^2=x^3+4n^2x$ is the image of isogeny $\Phi \colon E_n\rightarrow E_n^\{\prime \}$, $\Phi (x,y)=(y^2/x^2,y(n^2-x^2)/x^2)$, and $\widehat\{\Phi \}$ is the isogeny dual to $\Phi $.},
author = {Vrećica, Ilija S.},
journal = {Czechoslovak Mathematical Journal},
keywords = {elliptic curve; congruent number problem; Selmer group},
language = {eng},
number = {1},
pages = {105-119},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Joint distribution for the Selmer ranks of the congruent number curves},
url = {http://eudml.org/doc/297190},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Vrećica, Ilija S.
TI - Joint distribution for the Selmer ranks of the congruent number curves
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 1
SP - 105
EP - 119
AB - We determine the distribution over square-free integers $n$ of the pair $(\dim _{\mathbb {F}_2}{\rm Sel}^\Phi (E_n/\mathbb {Q}),\dim _{\mathbb {F}_2} {\rm Sel}^{\widehat{\Phi }}(E_n^{\prime }/\mathbb {Q}))$, where $E_n$ is a curve in the congruent number curve family, $E_n^{\prime }\colon y^2=x^3+4n^2x$ is the image of isogeny $\Phi \colon E_n\rightarrow E_n^{\prime }$, $\Phi (x,y)=(y^2/x^2,y(n^2-x^2)/x^2)$, and $\widehat{\Phi }$ is the isogeny dual to $\Phi $.
LA - eng
KW - elliptic curve; congruent number problem; Selmer group
UR - http://eudml.org/doc/297190
ER -

References

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