scholarly journals THE MULTILINEAR SUPPORT PROBLEM FOR PRODUCTS OF ABELIAN VARIETIES AND TORI

2012 ◽  
Vol 08 (01) ◽  
pp. 255-264
Author(s):  
ANTONELLA PERUCCA
Keyword(s):  
Abelian Variety ◽  
Number Field ◽  
Abelian Varieties ◽  
Rational Points ◽  
Dependency Relation

Let G be the product of an abelian variety and a torus defined over a number field K. The aim of this paper is detecting the dependence among some given rational points of G by studying their reductions modulo all primes of K. We show that if some simple conditions on the order of the reductions of the points are satisfied then there must be a dependency relation over the ring of K-endomorphisms of G. We generalize Larsen's result on the support problem to several points on products of abelian varieties and tori.

There are No Transcendental Brauer–Manin Obstructions on Abelian Varieties

2018 ◽  
Vol 2020 (9) ◽  
pp. 2684-2697
Author(s):  
Brendan Creutz
Keyword(s):  
Abelian Variety ◽  
Number Field ◽  
Abelian Varieties ◽  
Rational Points ◽  
Brauer Group

Abstract Suppose X is a torsor under an abelian variety A over a number field. We show that any adelic point of X that is orthogonal to the algebraic Brauer group of X is orthogonal to the whole Brauer group of X. We also show that if there is a Brauer–Manin obstruction to the existence of rational points on X, then there is already an obstruction coming from the locally constant Brauer classes. These results had previously been established under the assumption that A has finite Tate–Shafarevich group. Our results are unconditional.

scholarly journals ON SELMER RANK PARITY OF TWISTS

2016 ◽  
Vol 102 (3) ◽  
pp. 316-330 ◽  
Author(s):  
MAJID HADIAN ◽  
MATTHEW WEIDNER
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Arbitrary Number ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Hyperelliptic Curves ◽  
Torsion Subgroup ◽  
Tate Conjecture ◽  
Quadratic Twists

In this paper we study the variation of the $p$-Selmer rank parities of $p$-twists of a principally polarized Abelian variety over an arbitrary number field $K$ and show, under certain assumptions, that this parity is periodic with an explicit period. Our result applies in particular to principally polarized Abelian varieties with full $K$-rational $p$-torsion subgroup, arbitrary elliptic curves, and Jacobians of hyperelliptic curves. Assuming the Shafarevich–Tate conjecture, our result allows one to classify the rank parities of all quadratic twists of an elliptic or hyperelliptic curve after a finite calculation.

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 Determinants of subquotients of Galois representations associated with abelian varieties

2013 ◽  
Vol 13 (3) ◽  
pp. 517-559 ◽  
Author(s):  
Eric Larson ◽  
Dmitry Vaintrob
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Riemann Hypothesis ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Number Fields ◽  
Torsion Points ◽  
Prime Degree ◽  
Rho A

AbstractGiven an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\ell $, the ${\ell }^{n} $-torsion points of $A$ give rise to a representation ${\rho }_{A, {\ell }^{n} } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ( \mathbb{Z} / {\ell }^{n} \mathbb{Z} )$. In particular, we get a mod-$\ell $representation ${\rho }_{A, \ell } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{F} }_{\ell } )$ and an $\ell $-adic representation ${\rho }_{A, {\ell }^{\infty } } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{Z} }_{\ell } )$. In this paper, we describe the possible determinants of subquotients of these two representations. These two lists turn out to be remarkably similar.Applying our results in dimension $g= 1$, we recover a generalized version of a theorem of Momose on isogeny characters of elliptic curves over number fields, and obtain, conditionally on the Generalized Riemann Hypothesis, a generalization of Mazur’s bound on rational isogenies of prime degree to number fields.

Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves

2014 ◽  
Vol 66 (5) ◽  
pp. 1167-1200 ◽  
Author(s):  
Victor Rotger ◽  
Carlos de Vera-Piquero
Keyword(s):  
Abelian Variety ◽  
Moduli Space ◽  
Quadratic Field ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Main Idea ◽  
Galois Representation ◽  
Rational Points ◽  
Shimura Curves ◽  
Fields Of Moduli

AbstractThe purpose of this note is to introduce a method for proving the non-existence of rational points on a coarse moduli space X of abelian varieties over a given number field K in cases where the moduli problem is not fine and points in X(K) may not be represented by an abelian variety (with additional structure) admitting a model over the field K. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired by the work of Ellenberg and Skinner on the modularity of ℚ-curves, is that one may still attach a Galois representation of Gal(/K) with values in the quotient group GL(Tℓ(A))/ Aut(A) to a point P = [A] ∈ X(K) represented by an abelian variety A/, provided Aut(A) lies in the centre of GL(Tℓ(A)). We exemplify our method in the cases where X is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over ℚ.

scholarly journals Tate Cycles on Abelian Varieties with Complex Multiplication

2015 ◽  
Vol 67 (1) ◽  
pp. 198-213 ◽  
Author(s):  
V. Kumar Murty ◽  
Vijay M. Patankar
Keyword(s):  
Abelian Variety ◽  
Number Field ◽  
Cohomology Class ◽  
Abelian Varieties ◽  
Complex Multiplication ◽  
Special Fibre

AbstractWe consider Tate cycles on an Abelian variety A defined over a sufficiently large number field K and having complexmultiplication. We show that there is an effective bound C = C(A, K) so that to check whether a given cohomology class is a Tate class on A, it suffices to check the action of Frobenius elements at primes v of norm ≤ C. We also show that for a set of primes v of K of density 1, the space of Tate cycles on the special fibre Av of the Néron model of A is isomorphic to the space of Tate cycles on A itself.

scholarly journals Divisibility of torsion subgroups of abelian surfaces over number fields

2020 ◽  
pp. 1-33
Author(s):  
John Cullinan ◽  
Jeffrey Yelton
Keyword(s):  
Prime Number ◽  
Abelian Variety ◽  
Number Field ◽  
Abelian Varieties ◽  
Number Fields ◽  
Two Dimensional ◽  
Abelian Surfaces ◽  
Torsion Subgroups

Abstract Let A be a two-dimensional abelian variety defined over a number field K. Fix a prime number $\ell $ and suppose $\#A({\mathbf {F}_{\mathfrak {p}}}) \equiv 0 \pmod {\ell ^2}$ for a set of primes ${\mathfrak {p}} \subset {\mathcal {O}_{K}}$ of density 1. When $\ell =2$ Serre has shown that there does not necessarily exist a K-isogenous $A'$ such that $\#A'(K)_{{tor}} \equiv 0 \pmod {4}$ . We extend those results to all odd $\ell $ and classify the abelian varieties that fail this divisibility principle for torsion in terms of the image of the mod- $\ell ^2$ representation.

scholarly journals Level structures on abelian varieties and Vojta’s conjecture

2017 ◽  
Vol 153 (2) ◽  
pp. 373-394 ◽  
Author(s):  
Dan Abramovich ◽  
Anthony Várilly-Alvarado
Keyword(s):  
Positive Integer ◽  
Recent Work ◽  
Abelian Variety ◽  
Number Field ◽  
Abelian Varieties ◽  
Fixed Number ◽  
Vojta's Conjecture

Assuming Vojta’s conjecture, and building on recent work of the authors, we prove that, for a fixed number field $K$ and a positive integer $g$, there is an integer $m_{0}$ such that for any $m>m_{0}$ there is no principally polarized abelian variety $A/K$ of dimension $g$ with full level-$m$ structure. To this end, we develop a version of Vojta’s conjecture for Deligne–Mumford stacks, which we deduce from Vojta’s conjecture for schemes.

Diamonds in Torsion of Abelian Varieties

2010 ◽  
Vol 9 (3) ◽  
pp. 477-480 ◽  
Author(s):  
Moshe Jarden
Keyword(s):  
Abelian Variety ◽  
Number Field ◽  
Abelian Varieties ◽  
Abelian Extension

AbstractA theorem of Kuyk says that every Abelian extension of a Hilbertian field is Hilbertian. We conjecture that for an Abelian variety A defined over a Hilbertian field K every extension L of K in K(Ator) is Hilbertian. We prove our conjecture when K is a number field. The proof applies a result of Serre about l-torsion of Abelian varieties, information about l-adic analytic groups, and Haran's diamond theorem.

scholarly journals Ultrametric theta functions and abelian varieties

1978 ◽  
Vol 69 ◽  
pp. 65-96 ◽  
Author(s):  
Horacio Tapia-Recillas
Keyword(s):  
Positive Integer ◽  
Abelian Variety ◽  
Abelian Varieties ◽  
Complete Solution ◽  
Theta Functions ◽  
Partial Solution ◽  
Rational Points ◽  
Graded Ring ◽  
Multiplicative Subgroup ◽  
Natural Way

Let k be a field complete with respect to a non-trivial, non-archimedean valuation and let g be a positive integer. Consider the following question : if Γ is a multiplicative subgroup of Gg = (k*)g satisfying certain “Riemann conditions”, can one construct in a natural way an abelian variety defined over k having Gg/Γ as its set of k-rational points? This problem was first considered by Morikawa [3]. J. Tate provided a complete solution for g = 1 (cf. for example [6]). J. McCabe [2] gave a partial solution for g > 1. He showed how to attach to Γ a graded ring R of theta functions such that A = Proj. R is g-dimensional abelian variety over k.

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