[PDF] On the Selmer groups and Mordell-Weil groups of elliptic curves $ y^{2} = x (x \pm p) (x \pm q) $ over imaginary quadratic number fields of class number one | Semantic Scholar

Search 206,587,598 papers from all fields of science

Search

Sign InCreate Free Account

Corpus ID: 119628029

On the Selmer groups and Mordell-Weil groups of elliptic curves $ y^{2} = x (x \pm p) (x \pm q) $ over imaginary quadratic number fields of class number one

@article{Li2012OnTS,
title={On the Selmer groups and Mordell-Weil groups of elliptic curves \$ y^\{2\} = x (x \pm p) (x \pm q) \$ over imaginary quadratic number fields of class number one},
author={Xiumei Li},
journal={arXiv: Number Theory},
year={2012}
}

Let $ p $ and $ q $ be odd prime numbers with $ q - p = 2, $ the $\varphi -$Selmer groups, Shafarevich-Tate groups ($ \varphi - $ and $ 2-$part) and their dual ones as well the Mordell-Weil groups of elliptic curves $ y^{2} = x (x \pm p) (x \pm q) $ over imaginary quadratic number fields of class number one are determined explicitly in many cases.

It is proved that the Mordell-Weil group E(E/Q) having rank 0 and dimension 0 is proved, and the Kodaira symbol, the torsion subgroup $E(K)_{tors}$ for any number field $K$ are obtained.Expand

textabstractIn this paper the family of elliptic curves over Q given by the equation y2 = (x + p)(x2 + p2) is studied. It is shown that for p a prime number = ±3 mod 8, the only rational solution to… Expand

where f(X,Z) is a binary quartic form (or quartic for short) with integer coefficients. One wishes to know whether equation (1) has a Q-rational point and if so to exhibit one. One can often show… Expand

1 p-adic Numbers.- 2 Finite Extensions of the Field of p-adic Numbers.- 3 Construction of Universal p-adic Fields.- 4 Continuous Functions on Zp.- 5 Differentiation.- 6 Analytic Functions and… Expand