scholarly journals Abelian varieties and finitely generated Galois groups

2021 ◽  
pp. 1-12
Author(s):  
Bo-Hae Im ◽  
Michael Larsen
Keyword(s):  
Abelian Varieties ◽  
Galois Groups ◽  
Finitely Generated

ON THE RANK OF ABELIAN VARIETIES OVER AMPLE FIELDS

2010 ◽  
Vol 06 (03) ◽  
pp. 579-586 ◽  
Author(s):  
ARNO FEHM ◽  
SEBASTIAN PETERSEN
Keyword(s):  
Finite Field ◽  
Abelian Variety ◽  
Galois Extension ◽  
Characteristic Zero ◽  
Rational Point ◽  
Abelian Varieties ◽  
Finitely Generated ◽  
Infinite Rank ◽  
Finite Set

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.

scholarly journals Finitely Generated Pro-pAbsolute Galois Groups over Global Fields

1999 ◽  
Vol 77 (1) ◽  
pp. 83-96 ◽  
Author(s):  
Ido Efrat
Keyword(s):  
Galois Groups ◽  
Finitely Generated ◽  
Global Fields

scholarly journals Finitely generated pro-p Galois groups of p-Henselian fields

1999 ◽  
Vol 138 (3) ◽  
pp. 215-228 ◽  
Author(s):  
Ido Efrat
Keyword(s):  
Galois Groups ◽  
Finitely Generated

On torsion of abelian varieties over large algebraic extensions of finitely generated fields

Mathematika ◽  
1984 ◽  
Vol 31 (1) ◽  
pp. 110-116 ◽  
Author(s):  
Marcel Jacobson ◽  
Moshe Jarden
Keyword(s):  
Abelian Varieties ◽  
Finitely Generated

Finitely generated pro-p-groups as Galois groups of maximalp-extensions of function fields over ℚ q

1996 ◽  
Vol 90 (1) ◽  
pp. 225-238 ◽  
Author(s):  
Christian U. Jensen ◽  
Alexander Prestel
Keyword(s):  
Function Fields ◽  
Galois Groups ◽  
Finitely Generated

scholarly journals Big monodromy theorem for abelian varieties over finitely generated fields

2013 ◽  
Vol 217 (2) ◽  
pp. 218-229 ◽  
Author(s):  
Sara Arias-de-Reyna ◽  
Wojciech Gajda ◽  
Sebastian Petersen
Keyword(s):  
Abelian Varieties ◽  
Finitely Generated

On torsion of abelian varieties over large algebraic extensions of finitely generated fields: Erratum

Mathematika ◽  
1985 ◽  
Vol 32 (2) ◽  
pp. 316-316 ◽  
Author(s):  
Marcel Jacobson ◽  
Moshe Jarden
Keyword(s):  
Abelian Varieties ◽  
Finitely Generated

scholarly journals Specializations of finitely generated subgroups of abelian varieties

1989 ◽  
Vol 311 (1) ◽  
pp. 413-413 ◽  
Author(s):  
D. W. Masser
Keyword(s):  
Abelian Varieties ◽  
Finitely Generated

scholarly journals Integral division points on curves

2013 ◽  
Vol 149 (12) ◽  
pp. 2011-2035 ◽  
Author(s):  
David Grant ◽  
Su-Ion Ih
Keyword(s):  
Elliptic Curve ◽  
Number Field ◽  
Abelian Varieties ◽  
Algebraic Closure ◽  
Effective Divisor ◽  
Dimensional Case ◽  
Finitely Generated ◽  
General Conjecture ◽  
Finite Set ◽  
Set Of Points

AbstractLet $k$ be a number field with algebraic closure $ \overline{k} $, and let $S$ be a finite set of primes of $k$ containing all the infinite ones. Let $E/ k$ be an elliptic curve, ${\mit{\Gamma} }_{0} $ be a finitely generated subgroup of $E( \overline{k} )$, and $\mit{\Gamma} \subseteq E( \overline{k} )$ the division group attached to ${\mit{\Gamma} }_{0} $. Fix an effective divisor $D$ of $E$ with support containing either: (i) at least two points whose difference is not torsion; or (ii) at least one point not in $\mit{\Gamma} $. We prove that the set of ‘integral division points on $E( \overline{k} )$’, i.e., the set of points of $\mit{\Gamma} $ which are $S$-integral on $E$ relative to $D, $ is finite. We also prove the ${ \mathbb{G} }_{\mathrm{m} } $-analogue of this theorem, thereby establishing the 1-dimensional case of a general conjecture we pose on integral division points on semi-abelian varieties.

scholarly journals Abelian varieties over fields of finite characteristic

2014 ◽  
Vol 12 (5) ◽  
Author(s):  
Yuri Zarhin
Keyword(s):  
Abelian Varieties ◽  
Finitely Generated ◽  
Torsion Points ◽  
Finite Characteristic ◽  
Galois Action ◽  
Characteristic 2

AbstractThe aim of this paper is to extend our previous results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.

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