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

scholarly journals Distribution of Selmer groups of quadratic twists of a family of elliptic curves

2008 ◽  
Vol 219 (2) ◽  
pp. 523-553 ◽  
Author(s):  
Maosheng Xiong ◽  
Alexandru Zaharescu
Keyword(s):  
Elliptic Curves ◽  
Selmer Groups ◽  
Quadratic Twists

ON THE 2-ADIC IWASAWA INVARIANTS OF ORDINARY ELLIPTIC CURVES

2008 ◽  
Vol 04 (03) ◽  
pp. 403-422
Author(s):  
KAZUO MATSUNO
Keyword(s):  
Elliptic Curves ◽  
Explicit Formula ◽  
Selmer Groups ◽  
Quadratic Twists ◽  
Iwasawa Invariants

In this paper, we give an explicit formula describing the variation of the 2-adic Iwasawa λ-invariants attached to the Selmer groups of elliptic curves under quadratic twists. To prove this formula, we extend some results known for odd primes p, an analogue of Kida's formula proved by Hachimori and the author and a formula given by Greenberg and Vatsal, to the case where p = 2.

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 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 Euler characteristics and their congruences in the positive rank setting

2020 ◽  
pp. 1-18
Author(s):  
Anwesh Ray ◽  
R. Sujatha
Keyword(s):  
Elliptic Curves ◽  
Euler Characteristic ◽  
Selmer Groups ◽  
Euler Characteristics ◽  
Rank Zero ◽  
Homology Groups ◽  
Main Conjecture ◽  
Congruence Relations ◽  
Higher Rank ◽  
Positive Rank

Abstract The notion of the truncated Euler characteristic for Iwasawa modules is an extension of the notion of the usual Euler characteristic to the case when the homology groups are not finite. This article explores congruence relations between the truncated Euler characteristics for dual Selmer groups of elliptic curves with isomorphic residual representations, over admissible p-adic Lie extensions. Our results extend earlier congruence results from the case of elliptic curves with rank zero to the case of higher rank elliptic curves. The results provide evidence for the p-adic Birch and Swinnerton-Dyer formula without assuming the main conjecture.

Signed-Selmer Groups over the ℤ2p-extension of an Imaginary Quadratic Field

2014 ◽  
Vol 66 (4) ◽  
pp. 826-843 ◽  
Author(s):  
Byoung Du (B. D.) Kim
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Selmer Groups

AbstractLet E be an elliptic curve over ℚ that has good supersingular reduction at p > 3. We construct what we call the ±/±-Selmer groups of E over the ℤ2p-extension of an imaginary quadratic field K when the prime p splits completely over K/ℚ, and prove that they enjoy a property analogous to Mazur's control theorem.Furthermore, we propose a conjectural connection between the±/±-Selmer groups and Loeffler's two-variable ±/±-p-adic L-functions of elliptic curves.

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