TORSION OF ABELIAN VARIETIES OVER LARGE ALGEBRAIC EXTENSIONS OF

2017 ◽  
Vol 234 ◽  
pp. 46-86
Author(s):  
MOSHE JARDEN ◽  
SEBASTIAN PETERSEN
Keyword(s):  
Galois Group ◽  
Abelian Variety ◽  
Haar Measure ◽  
Rational Point ◽  
Algebraic Closure ◽  
Prime Numbers ◽  
Absolute Galois Group ◽  
Finitely Generated ◽  
Fixed Field ◽  
Almost All

Let$K$be a finitely generated extension of$\mathbb{Q}$, and let$A$be a nonzero abelian variety over$K$. Let$\tilde{K}$be the algebraic closure of$K$, and let$\text{Gal}(K)=\text{Gal}(\tilde{K}/K)$be the absolute Galois group of$K$equipped with its Haar measure. For each$\unicode[STIX]{x1D70E}\in \text{Gal}(K)$, let$\tilde{K}(\unicode[STIX]{x1D70E})$be the fixed field of$\unicode[STIX]{x1D70E}$in$\tilde{K}$. We prove that for almost all$\unicode[STIX]{x1D70E}\in \text{Gal}(K)$, there exist infinitely many prime numbers$l$such that$A$has a nonzero$\tilde{K}(\unicode[STIX]{x1D70E})$-rational point of order$l$. This completes the proof of a conjecture of Geyer–Jarden from 1978 in characteristic 0.

UNDECIDABILITY OF FAMILIES OF RINGS OF TOTALLY REAL INTEGERS

2008 ◽  
Vol 04 (05) ◽  
pp. 835-850 ◽  
Author(s):  
MOSHE JARDEN ◽  
CARLOS R. VIDELA
Keyword(s):  
Positive Integer ◽  
Galois Group ◽  
Haar Measure ◽  
Absolute Galois Group ◽  
First Order ◽  
Fixed Ring ◽  
The Absolute ◽  
Almost All ◽  
Totally Real

Let ℤtrbe the ring of totally real integers, Gal(ℚ) the absolute Galois group of ℚ, and e a positive integer. For each σ = (σ1,…,σe) ∈ Gal (ℚ)elet ℤtr(σ) be the fixed ring in ℤtrof σ1,…,σe. Then, the theory of all first order sentences θ that are true in ℤtr(σ) for almost all σ ∈ Gal (ℚ)e(in the sense of the Haar measure) is undecidable.

scholarly journals Independence of -adic Galois representations over function fields

2013 ◽  
Vol 149 (7) ◽  
pp. 1091-1107 ◽  
Author(s):  
Wojciech Gajda ◽  
Sebastian Petersen
Keyword(s):  
Galois Group ◽  
Finite Type ◽  
Galois Representations ◽  
Function Fields ◽  
Algebraic Closure ◽  
Finite Extension ◽  
Prime Numbers ◽  
Positive Answer ◽  
Absolute Galois Group ◽  
The Family

AbstractLet$K$be a finitely generated extension of$\mathbb {Q}$. We consider the family of$\ell $-adic representations ($\ell $varies through the set of all prime numbers) of the absolute Galois group of$K$, attached to$\ell $-adic cohomology of a separated scheme of finite type over$K$. We prove that the fields cut out from the algebraic closure of$K$by the kernels of the representations of the family are linearly disjoint over a finite extension of K. This gives a positive answer to a question of Serre.

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.

Independence of ℓ \ell -adic representations of geometric Galois groups

2018 ◽  
Vol 2018 (736) ◽  
pp. 69-93 ◽  
Author(s):  
Gebhard Böckle ◽  
Wojciech Gajda ◽  
Sebastian Petersen
Keyword(s):  
Galois Group ◽  
Finite Type ◽  
Algebraic Closure ◽  
Finite Extension ◽  
Field Extension ◽  
Galois Groups ◽  
Closed Field ◽  
Absolute Galois Group ◽  
Arbitrary Characteristic ◽  
Cohomology Modules

AbstractLetkbe an algebraically closed field of arbitrary characteristic, let{K/k}be a finitely generated field extension and letXbe a separated scheme of finite type overK. For each prime{\ell}, the absolute Galois group ofKacts on the{\ell}-adic étale cohomology modules ofX. We prove that this family of representations varying over{\ell}is almost independent in the sense of Serre, i.e., that the fixed fields inside an algebraic closure ofKof the kernels of the representations for all{\ell}become linearly disjoint over a finite extension ofK. In doing this, we also prove a number of interesting facts on the images and on the ramification of this family of representations.

scholarly journals Complete addition laws on abelian varieties

2012 ◽  
Vol 15 ◽  
pp. 308-316 ◽  
Author(s):  
Christophe Arene ◽  
David Kohel ◽  
Christophe Ritzenthaler
Keyword(s):  
Finite Number ◽  
Elliptic Curve ◽  
Galois Group ◽  
Abelian Variety ◽  
Abelian Varieties ◽  
Rational Points ◽  
Absolute Galois Group ◽  
Projective Embedding ◽  
Complete Set

AbstractWe prove that under any projective embedding of an abelian variety A of dimension g, a complete set of addition laws has cardinality at least g+1, generalizing a result of Bosma and Lenstra for the Weierstrass model of an elliptic curve in ℙ2. In contrast, we prove, moreover, that if k is any field with infinite absolute Galois group, then there exists for every abelian variety A/k a projective embedding and an addition law defined for every pair of k-rational points. For an abelian variety of dimension 1 or 2, we show that this embedding can be the classical Weierstrass model or the embedding in ℙ15, respectively, up to a finite number of counterexamples for ∣k∣≤5 .

scholarly journals Extensions of number fields defined by cohomology groups

1983 ◽  
Vol 92 ◽  
pp. 179-186 ◽  
Author(s):  
Hans Opolka
Keyword(s):  
Natural Number ◽  
Galois Group ◽  
Cohomology Group ◽  
Algebraic Closure ◽  
Number Fields ◽  
Cohomology Groups ◽  
Absolute Galois Group ◽  
The Absolute

Letkbe a field of characteristic 0, letbe an algebraic closure ofkand denote byGk= G(/k) the absolute Galois group ofk. Suppose that for some natural numbern≥ 3 the cohomology groupHn(Gk) Z) is trivial.

scholarly journals The Absolute Galois Group of the Field of Totally S-Adic Numbers

2009 ◽  
Vol 194 ◽  
pp. 91-147 ◽  
Author(s):  
Dan Haran ◽  
Moshe Jarden ◽  
Florian Pop
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Free Product ◽  
Absolute Galois Group ◽  
Fixed Field ◽  
Local Factors ◽  
Finite Set ◽  
The Absolute

AbstractFor a finite set S of primes of a number field K and for σ1,…, σe ∈ Gal(K) we denote the field of totally S-adic numbers by Ktot,S and the fixed field of σ1,…,σe in Ktot,S by Ktot,S(σ). We prove that foralmost all σ ∈ Gal(K)e the absolute Galois group of Ktot,S(σ) is the free product of and a free product of local factors over S.

scholarly journals The semisimplicity conjecture for A-motives

2010 ◽  
Vol 146 (3) ◽  
pp. 561-598 ◽  
Author(s):  
Nicolas Stalder
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Zariski Closure ◽  
Drinfeld Modules ◽  
Absolute Galois Group ◽  
Connected Component ◽  
Finitely Generated ◽  
Endomorphism Algebra ◽  
Monodromy Groups ◽  
The Absolute

AbstractWe prove the semisimplicity conjecture for A-motives over finitely generated fields K. This conjecture states that the rational Tate modules V𝔭(M) of a semisimple A-motive M are semisimple as representations of the absolute Galois group of K. This theorem is in analogy with known results for abelian varieties and Drinfeld modules, and has been sketched previously by Tamagawa. We deduce two consequences of the theorem for the algebraic monodromy groups G𝔭(M) associated to an A-motive M by Tannakian duality. The first requires no semisimplicity condition on M and states that G𝔭(M) may be identified naturally with the Zariski closure of the image of the absolute Galois group of K in the automorphism group of V𝔭(M). The second states that the connected component of G𝔭(M) is reductive if M is semisimple and has a separable endomorphism algebra.

Relaxation of strict parity for reducible Galois representations attached to the homology of GL(3, ℤ)

2016 ◽  
Vol 12 (02) ◽  
pp. 361-381 ◽  
Author(s):  
Avner Ash ◽  
Darrin Doud
Keyword(s):  
Irreducible Representation ◽  
Finite Field ◽  
Galois Group ◽  
Galois Representations ◽  
Algebraic Closure ◽  
Continuous Homomorphism ◽  
Two Dimensional ◽  
Absolute Galois Group ◽  
Direct Sum ◽  
Parity Condition

In this paper, we prove the following theorem: Let [Formula: see text] be an algebraic closure of a finite field of characteristic [Formula: see text]. Let [Formula: see text] be a continuous homomorphism from the absolute Galois group of [Formula: see text] to [Formula: see text]) which is isomorphic to a direct sum of a character and a two-dimensional odd irreducible representation. We assume that the image of [Formula: see text] is contained in the intersection of the stabilizers of the line spanned by [Formula: see text] and the plane spanned by [Formula: see text], where [Formula: see text] denotes the standard basis. Such [Formula: see text] will not satisfy a certain strict parity condition. Under the conditions that the Serre conductor of [Formula: see text] is squarefree, that the predicted weight [Formula: see text] lies in the lowest alcove, and that [Formula: see text], we prove that [Formula: see text] is attached to a Hecke eigenclass in [Formula: see text], where [Formula: see text] is a subgroup of finite index in [Formula: see text] and [Formula: see text] is an [Formula: see text]-module. The particular [Formula: see text] and [Formula: see text] are as predicted by the main conjecture of the 2002 paper of the authors and David Pollack, minus the requirement for strict parity.

scholarly journals Cohomology and torsion cycles over the maximal cyclotomic extension

2019 ◽  
Vol 2019 (752) ◽  
pp. 211-227
Author(s):  
Damian Rössler ◽  
Tamás Szamuely
Keyword(s):  
Fixed Points ◽  
Galois Group ◽  
Abelian Variety ◽  
Number Field ◽  
Positive Characteristic ◽  
Supporting Evidence ◽  
Classical Theorem ◽  
Absolute Galois Group ◽  
Torsion Points ◽  
Cyclotomic Extension

Abstract A classical theorem by K. Ribet asserts that an abelian variety defined over the maximal cyclotomic extension K of a number field has only finitely many torsion points. We show that this statement can be viewed as a particular case of a much more general one, namely that the absolute Galois group of K acts with finitely many fixed points on the étale cohomology with {\mathbf{Q}/\mathbf{Z}} -coefficients of a smooth proper {\overline{K}} -variety defined over K. We also present a conjectural generalization of Ribet’s theorem to torsion cycles of higher codimension. We offer supporting evidence for the conjecture in codimension 2, as well as an analogue in positive characteristic.

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