scholarly journals Average size of 2-Selmer groups of elliptic curves, II

2005 ◽  
Vol 117 (1) ◽  
pp. 1-33 ◽  
Author(s):  
Gang Yu
Keyword(s):  
Elliptic Curves ◽  
Selmer Groups ◽  
Average Size

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 Average size of $2$-Selmer groups of elliptic curves, I

2005 ◽  
Vol 358 (4) ◽  
pp. 1563-1584 ◽  
Author(s):  
Gang Yu
Keyword(s):  
Elliptic Curves ◽  
Selmer Groups ◽  
Average Size

Large families of elliptic curves ordered by conductor

2021 ◽  
Vol 157 (7) ◽  
pp. 1538-1583
Author(s):  
Ananth N. Shankar ◽  
Arul Shankar ◽  
Xiaoheng Wang
Keyword(s):  
Elliptic Curves ◽  
Upper Bounds ◽  
Selmer Groups ◽  
Good Reduction ◽  
Positive Proportion ◽  
The Family ◽  
Large Families ◽  
Average Size

In this paper we study the family of elliptic curves $E/{{\mathbb {Q}}}$ , having good reduction at $2$ and $3$ , and whose $j$ -invariants are small. Within this set of elliptic curves, we consider the following two subfamilies: first, the set of elliptic curves $E$ such that the quotient $\Delta (E)/C(E)$ of the discriminant divided by the conductor is squarefree; and second, the set of elliptic curves $E$ such that the Szpiro quotient $\beta _E:=\log |\Delta (E)|/\log (C(E))$ is less than $7/4$ . Both these families are conjectured to contain a positive proportion of elliptic curves, when ordered by conductor. Our main results determine asymptotics for both these families, when ordered by conductor. Moreover, we prove that the average size of the $2$ -Selmer groups of elliptic curves in the first family, again when these curves are ordered by their conductors, is $3$ . The key new ingredients necessary for the proofs are ‘uniformity estimates’, namely upper bounds on the number of elliptic curves with bounded height, whose discriminants are divisible by high powers of primes.

scholarly journals Average size of 2-Selmer groups of elliptic curves over function fields

2014 ◽  
Vol 21 (6) ◽  
pp. 1305-1339
Author(s):  
Q.P. Hô ◽  
V.B. Lê Hùng ◽  
B.C. Ngô
Keyword(s):  
Elliptic Curves ◽  
Function Fields ◽  
Selmer Groups ◽  
Average Size

ON SELMER GROUPS OF QUADRATIC TWISTS OF ELLIPTIC CURVES WITH A TWO‐TORSION OVER

Mathematika ◽  
2013 ◽  
Vol 59 (2) ◽  
pp. 303-319 ◽  
Author(s):  
Maosheng Xiong
Keyword(s):  
Elliptic Curves ◽  
Selmer Groups ◽  
Quadratic Twists

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 Selmer groups and Mordell–Weil groups of elliptic curves over towers of function fields

2006 ◽  
Vol 142 (05) ◽  
pp. 1215-1230 ◽  
Author(s):  
Jordan S. Ellenberg
Keyword(s):  
Elliptic Curves ◽  
Function Fields ◽  
Selmer Groups ◽  
Towers Of Function Fields

scholarly journals 2-Selmer groups, 2-class groups and rational points on elliptic curves

2018 ◽  
Vol 371 (7) ◽  
pp. 4631-4653
Author(s):  
Chao Li
Keyword(s):  
Elliptic Curves ◽  
Rational Points ◽  
Selmer Groups ◽  
Class Groups

Fine Selmer groups for elliptic curves with complex multiplication

2005 ◽  
pp. 327-337
Author(s):  
J. Coates ◽  
R. Sujatha
Keyword(s):  
Elliptic Curves ◽  
Complex Multiplication ◽  
Selmer Groups
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