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

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 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 The geometric average size of Selmer groups over function fields

2021 ◽  
Vol 15 (3) ◽  
pp. 673-709
Author(s):  
Aaron Landesman
Keyword(s):  
Function Fields ◽  
Selmer Groups ◽  
Geometric Average ◽  
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

scholarly journals CONTROL THEOREMS FOR ELLIPTIC CURVES OVER FUNCTION FIELDS

2009 ◽  
Vol 05 (02) ◽  
pp. 229-256 ◽  
Author(s):  
A. BANDINI ◽  
I. LONGHI
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Extension ◽  
Function Field ◽  
Function Fields ◽  
Global Field ◽  
Selmer Groups ◽  
Characteristic P ◽  
Main Conjecture ◽  
Fitting Ideal

Let F be a global field of characteristic p > 0, 𝔽/F a Galois extension with [Formula: see text] and E/F a non-isotrivial elliptic curve. We study the behavior of Selmer groups SelE(L)l (l any prime) as L varies through the subextensions of 𝔽 via appropriate versions of Mazur's Control Theorem. In the case l = p, we let 𝔽 = ∪ 𝔽d where 𝔽d/F is a [Formula: see text]-extension. We prove that Sel E(𝔽d)p is a cofinitely generated ℤp[[ Gal (ℤd/F)]]-module and we associate to its Pontrjagin dual a Fitting ideal. This allows to define an algebraic L-function associated to E in ℤp[[Gal(ℤ/F)]], providing an ingredient for a function field analogue of Iwasawa's Main Conjecture for elliptic curves.

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

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 Selmer groups for elliptic curves in \mathbb{Z}_l^d-extensions of function fields of characteristic p

2009 ◽  
Vol 59 (6) ◽  
pp. 2301-2327 ◽  
Author(s):  
Andrea Bandini ◽  
Ignazio Longhi
Keyword(s):  
Elliptic Curves ◽  
Function Fields ◽  
Selmer Groups ◽  
Characteristic P

scholarly journals Elliptic Curves with Large Rank over Function Fields

2002 ◽  
Vol 155 (1) ◽  
pp. 295 ◽  
Author(s):  
Douglas Ulmer
Keyword(s):  
Elliptic Curves ◽  
Function Fields ◽  
Large Rank

scholarly journals On Elliptic Curves with Complex Multiplication as Factors of the Jacobians of Modular Function Fields

1971 ◽  
Vol 43 ◽  
pp. 199-208 ◽  
Author(s):  
Goro Shimura
Keyword(s):  
Elliptic Curves ◽  
Cusp Form ◽  
Mellin Transform ◽  
Quadratic Field ◽  
Congruence Subgroup ◽  
Function Fields ◽  
Complex Multiplication ◽  
Modular Function ◽  
Imaginary Quadratic Field

1. As Hecke showed, every L-function of an imaginary quadratic field K with a Grössen-character γ is the Mellin transform of a cusp form f(z) belonging to a certain congruence subgroup Γ of SL2(Z). We can normalize γ so that

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