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.

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 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.

On étale covers of curves

1999 ◽  
Vol 127 (1) ◽  
pp. 1-5
Author(s):  
S. KAMIENNY ◽  
J. L. WETHERELL
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Abelian Varieties ◽  
Ring Of Integers ◽  
The Relationship

Let K be a number field with ring of integers R. For each integer g>1 we consider the collection of abelian, étale R-coverings f[ratio ]Y→X, where X and Y are connected proper curves over R and the genus of X is g. We ask the following question: is there a positive integer B = B(K, g) which bounds the degree of such coverings? In this note we provide partial results towards such a bound and study the relationship with bounds on torsion in abelian varieties.

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.

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.

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.

Bounding the Iwasawa invariants of Selmer groups

2020 ◽  
pp. 1-33
Author(s):  
Sören Kleine
Keyword(s):  
Number Field ◽  
Analogous Result ◽  
Abelian Varieties ◽  
Fixed Number ◽  
Selmer Groups ◽  
Iwasawa Invariants ◽  
Ordinary Reduction

Abstract We study the growth of p-primary Selmer groups of abelian varieties with good ordinary reduction at p in ${{Z}}_p$ -extensions of a fixed number field K. Proving that in many situations the knowledge of the Selmer groups in a sufficiently large number of finite layers of a ${{Z}}_p$ -extension over K suffices for bounding the over-all growth, we relate the Iwasawa invariants of Selmer groups in different ${{Z}}_p$ -extensions of K. As applications, we bound the growth of Mordell–Weil ranks and the growth of Tate-Shafarevich groups. Finally, we derive an analogous result on the growth of fine Selmer groups.

Some Results on L-Indistinguishability for SL(r)

1983 ◽  
Vol 35 (6) ◽  
pp. 1075-1109 ◽  
Author(s):  
Freydoon Shahidi
Keyword(s):  
Positive Integer ◽  
Weyl Group ◽  
Recent Work ◽  
Eisenstein Series ◽  
Number Field ◽  
Parabolic Subgroup ◽  
Cusp Form ◽  
Constant Multiple ◽  
Levi Decomposition ◽  
Image Position

Fix a positive integer r. Let AF be the ring of adeles of a number field F. For a parabolic subgroup P of SLr, we fix a Levi decomposition P = MN, and we letLet be the Weyl group of . It follows from a recent work of James Arthur [1,2] (also cf. [3]) that, among the terms appearing in the trace formula for SLr(AF), coming from the Eisenstein series, are those which are a constant multiple (depending only on M and w) of1where σ is a cusp form on M(AF) satisfying wσ ≅ σ,and in the notation of [2, 3]).

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.

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.

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