scholarly journals Ranks, 2-Selmer groups, and Tamagawa numbers of elliptic curves with ℤ∕2ℤ × ℤ∕8ℤ-torsion

2019 ◽  
Vol 2 (1) ◽  
pp. 173-189
Author(s):  
Stephanie Chan ◽  
Jeroen Hanselman ◽  
Wanlin Li
Keyword(s):  
Elliptic Curves ◽  
Selmer Groups ◽  
Tamagawa Numbers

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.

scholarly journals The average size of the 3‐isogeny Selmer groups of elliptic curves y2=x3+k

2019 ◽  
Vol 101 (1) ◽  
pp. 299-327 ◽  
Author(s):  
Manjul Bhargava ◽  
Noam Elkies ◽  
Ari Shnidman
Keyword(s):  
Elliptic Curves ◽  
Selmer Groups ◽  
Average Size

scholarly journals Selmer groups of elliptic curves that can be arbitrarily large

2003 ◽  
Vol 99 (1) ◽  
pp. 148-163 ◽  
Author(s):  
Remke Kloosterman ◽  
Edward F. Schaefer
Keyword(s):  
Elliptic Curves ◽  
Selmer Groups

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.

Fine Selmer groups of elliptic curves over p-adic Lie extensions

2005 ◽  
Vol 331 (4) ◽  
pp. 809-839 ◽  
Author(s):  
J. Coates ◽  
R. Sujatha
Keyword(s):  
Elliptic Curves ◽  
Selmer Groups
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