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 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 Selmer groups of abelian varieties in extensions of function fields

2008 ◽  
Vol 261 (4) ◽  
pp. 787-804 ◽  
Author(s):  
Amílcar Pacheco
Keyword(s):  
Function Fields ◽  
Abelian Varieties ◽  
Selmer Groups

scholarly journals On the Selmer groups of abelian varieties over function fields of characteristic p > 0

2009 ◽  
Vol 146 (1) ◽  
pp. 23-43 ◽  
Author(s):  
TADASHI OCHIAI ◽  
FABIEN TRIHAN
Keyword(s):  
Number Field ◽  
Function Field ◽  
Function Fields ◽  
Abelian Varieties ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Characteristic P ◽  
Field Case ◽  
New Phenomena ◽  
Geometric Analogue

AbstractWe study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We also propose a conjecture (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field.

FUNCTIONAL EQUATION OF CHARACTERISTIC ELEMENTS OF ABELIAN VARIETIES OVER FUNCTION FIELDS (ℓ ≠ p)

2014 ◽  
Vol 10 (03) ◽  
pp. 705-735
Author(s):  
APRAMEYO PAL
Keyword(s):  
Functional Equation ◽  
Number Field ◽  
Function Field ◽  
Functional Equations ◽  
Function Fields ◽  
Growth Behavior ◽  
Iwasawa Theory ◽  
Selmer Groups ◽  
Selmer Group ◽  
Field Case

In this paper we apply methods from the number field case of Perrin-Riou [20] and Zábrádi [32] in the function field setup. In ℤℓ- and GL2-cases (ℓ ≠ p), we prove algebraic functional equations of the Pontryagin dual of Selmer group which give further evidence of the main conjectures of Iwasawa theory. We also prove some parity conjectures in commutative and non-commutative cases. As a consequence, we also get results on the growth behavior of Selmer groups in commutative and non-commutative extension of function fields.

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.

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