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



2019 ◽  
Vol 101 (1) ◽  
pp. 299-327 ◽  
Author(s):  
Manjul Bhargava ◽  
Noam Elkies ◽  
Ari Shnidman


2021 ◽  
Vol 15 (3) ◽  
pp. 673-709
Author(s):  
Aaron Landesman




2009 ◽  
Vol 05 (02) ◽  
pp. 229-256 ◽  
Author(s):  
A. BANDINI ◽  
I. LONGHI

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.



2005 ◽  
Vol 117 (1) ◽  
pp. 1-33 ◽  
Author(s):  
Gang Yu


2021 ◽  
Vol 157 (7) ◽  
pp. 1538-1583
Author(s):  
Ananth N. Shankar ◽  
Arul Shankar ◽  
Xiaoheng Wang

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.





2002 ◽  
Vol 155 (1) ◽  
pp. 295 ◽  
Author(s):  
Douglas Ulmer


1971 ◽  
Vol 43 ◽  
pp. 199-208 ◽  
Author(s):  
Goro Shimura

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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