scholarly journals Quadratic twists of elliptic curves associated to the simplest cubic fields

1997 ◽  
Vol 73 (10) ◽  
pp. 185-186 ◽  
Author(s):  
Dongho Byeon
Keyword(s):  
Elliptic Curves ◽  
Quadratic Twists ◽  
Cubic Fields

scholarly journals Disparity in Selmer ranks of quadratic twists of elliptic curves

2013 ◽  
Vol 178 (1) ◽  
pp. 287-320 ◽  
Author(s):  
Zev Klagsbrun ◽  
Barry Mazur ◽  
Karl Rubin
Keyword(s):  
Elliptic Curves ◽  
Quadratic Twists

scholarly journals Torsion of rational elliptic curves over cubic fields

2016 ◽  
Vol 46 (6) ◽  
pp. 1899-1917 ◽  
Author(s):  
Enrique González-Jiménez ◽  
Filip Najman ◽  
José M. Tornero
Keyword(s):  
Elliptic Curves ◽  
Cubic Fields

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

scholarly journals Ranks of Elliptic Curves in Families of Quadratic Twists

2000 ◽  
Vol 9 (4) ◽  
pp. 583-590 ◽  
Author(s):  
Karl Rubin ◽  
Alice Silverberg
Keyword(s):  
Elliptic Curves ◽  
Quadratic Twists

scholarly journals Moments and distribution of central $$L$$ L -values of quadratic twists of elliptic curves

2015 ◽  
Vol 202 (3) ◽  
pp. 1029-1068 ◽  
Author(s):  
Maksym Radziwiłł ◽  
K. Soundararajan
Keyword(s):  
Elliptic Curves ◽  
Quadratic Twists

Elliptic curves with good reduction everywhere over cubic fields

2015 ◽  
Vol 11 (04) ◽  
pp. 1149-1164 ◽  
Author(s):  
Nao Takeshi
Keyword(s):  
Elliptic Curves ◽  
Infinite Family ◽  
Good Reduction ◽  
Cubic Fields

We give a criterion for cubic fields over which there exist no elliptic curves with good reduction everywhere, and we construct a certain infinite family of cubic fields over which there exist elliptic curves with good reduction everywhere.

scholarly journals Quadratic twists of elliptic curves

2014 ◽  
Vol 110 (2) ◽  
pp. 357-394 ◽  
Author(s):  
John Coates ◽  
Yongxiong Li ◽  
Ye Tian ◽  
Shuai Zhai
Keyword(s):  
Elliptic Curves ◽  
Quadratic Twists

scholarly journals The Selmer groups and the ambiguous ideal class groups of cubic fields

1996 ◽  
Vol 54 (2) ◽  
pp. 267-274
Author(s):  
Yen-Mei J. Chen
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Upper Bound ◽  
Selmer Groups ◽  
Ideal Class ◽  
Selmer Group ◽  
Class Groups ◽  
Ideal Class Groups ◽  
Cubic Fields ◽  
Rational Isogeny

In this paper, we study a family of elliptic curves with CM by which also admits a ℚ-rational isogeny of degree 3. We find a relation between the Selmer groups of the elliptic curves and the ambiguous ideal class groups of certain cubic fields. We also find some bounds for the dimension of the 3-Selmer group over ℚ, whose upper bound is also an upper bound of the rank of the elliptic curve.

scholarly journals Selmer groups of quadratic twists of elliptic curves

1999 ◽  
Vol 314 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Kevin James ◽  
Ken Ono
Keyword(s):  
Elliptic Curves ◽  
Selmer Groups ◽  
Quadratic Twists
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