scholarly journals Selmer groups of quadratic twists of elliptic curves

1999 ◽  
Vol 314 (1) ◽  
pp. 1-17 ◽  
Author(s):  
Kevin James ◽  
Ken Ono
2008 ◽  
Vol 219 (2) ◽  
pp. 523-553 ◽  
Author(s):  
Maosheng Xiong ◽  
Alexandru Zaharescu

2008 ◽  
Vol 04 (03) ◽  
pp. 403-422
Author(s):  
KAZUO MATSUNO

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.


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

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER

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.


2020 ◽  
pp. 1-18
Author(s):  
Anwesh Ray ◽  
R. Sujatha

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.


2014 ◽  
Vol 66 (4) ◽  
pp. 826-843 ◽  
Author(s):  
Byoung Du (B. D.) Kim

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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