Distribution of Mordell-Weil ranks of families of elliptic curves

Bartosz Naskrecki

doi:10.4064/bc108-0-16

Distribution of Mordell-Weil ranks of families of elliptic curves

Bartosz Naskrecki

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Abstract

We discuss the distribution of Mordell–Weil ranks of the family of elliptic curves y^2 = (x + αf^2)(x + βg^2)(x + γh^2) where f, g, h are coprime polynomials that parametrize the projective smooth conic a^2 + b^2 = c^2 and α, β, γ are elements from Qbar. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.

Original language	English
Pages (from-to)	201-229
Number of pages	29
Journal	Banach Center Publications
Volume	108
Early online date	14 Jun 2016
DOIs	https://doi.org/10.4064/bc108-0-16
Publication status	Published - 2016

Keywords

elliptic curve
Mordell-Weil rank
elliptic surface

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title = "Distribution of Mordell-Weil ranks of families of elliptic curves",

abstract = "We discuss the distribution of Mordell–Weil ranks of the family of elliptic curves y\textasciicircum{}2 = (x + αf\textasciicircum{}2)(x + βg\textasciicircum{}2)(x + γh\textasciicircum{}2) where f, g, h are coprime polynomials that parametrize the projective smooth conic a\textasciicircum{}2 + b\textasciicircum{}2 = c\textasciicircum{}2 and α, β, γ are elements from Qbar. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.",

keywords = "elliptic curve, Mordell-Weil rank, elliptic surface",

author = "Bartosz Naskrecki",

year = "2016",

doi = "10.4064/bc108-0-16",

language = "English",

volume = "108",

pages = "201--229",

journal = "Banach Center Publications",

issn = "0137-6934",

publisher = "Polish Academy of Sciences",

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T1 - Distribution of Mordell-Weil ranks of families of elliptic curves

AU - Naskrecki, Bartosz

PY - 2016

Y1 - 2016

N2 - We discuss the distribution of Mordell–Weil ranks of the family of elliptic curves y^2 = (x + αf^2)(x + βg^2)(x + γh^2) where f, g, h are coprime polynomials that parametrize the projective smooth conic a^2 + b^2 = c^2 and α, β, γ are elements from Qbar. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.

AB - We discuss the distribution of Mordell–Weil ranks of the family of elliptic curves y^2 = (x + αf^2)(x + βg^2)(x + γh^2) where f, g, h are coprime polynomials that parametrize the projective smooth conic a^2 + b^2 = c^2 and α, β, γ are elements from Qbar. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.

KW - elliptic curve

KW - Mordell-Weil rank

KW - elliptic surface

U2 - 10.4064/bc108-0-16

DO - 10.4064/bc108-0-16

M3 - Article (Academic Journal)

SN - 0137-6934

VL - 108

SP - 201

EP - 229

JO - Banach Center Publications

JF - Banach Center Publications

ER -

Distribution of Mordell-Weil ranks of families of elliptic curves

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