K-groups of reciprocity functors for  and abelian varieties

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.

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ON SELMER RANK PARITY OF TWISTS

Journal of the Australian Mathematical Society ◽

10.1017/s1446788716000306 ◽

2016 ◽

Vol 102 (3) ◽

pp. 316-330 ◽

Cited By ~ 1

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.

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Isogenies Between K3 Surfaces Over

International Mathematics Research Notices ◽

10.1093/imrn/rnaa176 ◽

2020 ◽

Author(s):

Ziquan Yang

Keyword(s):

Abelian Varieties ◽

K3 Surfaces ◽

Main Step ◽

Shimura Varieties ◽

K3 Surface ◽

Perfect Field ◽

Finite Height ◽

Mild Assumption

Abstract We generalize Mukai and Shafarevich’s definitions of isogenies between K3 surfaces over ${\mathbb{C}}$ to an arbitrary perfect field and describe how to construct isogenous K3 surfaces over $\bar{{\mathbb{F}}}_p$ by prescribing linear algebraic data when $p$ is large. The main step is to show that isogenies between Kuga–Satake abelian varieties induce isogenies between K3 surfaces, in the context of integral models of Shimura varieties. As a byproduct, we show that every K3 surface of finite height admits a CM lifting under a mild assumption on $p$.

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On a Theorem of Ziv Ran concerning Abelian Varieties Which Are Product of Jacobians

Journal of Mathematics ◽

10.1155/2015/431098 ◽

2015 ◽

Vol 2015 ◽

pp. 1-5

Author(s):

Cristian Anghel ◽

Nicolae Buruiana

Keyword(s):

Abelian Variety ◽

Abelian Varieties ◽

Arbitrary Characteristic

We give a new proof for a theorem of Ziv Ran which generalizes some results of Matsusaka and Hoyt. These results provide criteria for an Abelian variety to be a Jacobian or a product of Jacobians. The advantage of our method is that it works in arbitrary characteristic.

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THE MULTILINEAR SUPPORT PROBLEM FOR PRODUCTS OF ABELIAN VARIETIES AND TORI

International Journal of Number Theory ◽

10.1142/s1793042112500145 ◽

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.

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Abelian varieties isogenous to a power of an elliptic curve

Compositio Mathematica ◽

10.1112/s0010437x17007990 ◽

2018 ◽

Vol 154 (5) ◽

pp. 934-959 ◽

Cited By ~ 2

Author(s):

Bruce W. Jordan ◽

Allan G. Keeton ◽

Bjorn Poonen ◽

Eric M. Rains ◽

Nicholas Shepherd-Barron ◽

...

Keyword(s):

Elliptic Curve ◽

Abelian Variety ◽

Opposite Direction ◽

Abelian Varieties ◽

Sufficient Conditions ◽

Higher Dimensional ◽

Free Left ◽

Necessary And Sufficient ◽

Finitely Presented ◽

Torsion Free

Let $E$ be an elliptic curve over a field $k$. Let $R:=\operatorname{End}E$. There is a functor $\mathscr{H}\!\mathit{om}_{R}(-,E)$ from the category of finitely presented torsion-free left $R$-modules to the category of abelian varieties isogenous to a power of $E$, and a functor $\operatorname{Hom}(-,E)$ in the opposite direction. We prove necessary and sufficient conditions on $E$ for these functors to be equivalences of categories. We also prove a partial generalization in which $E$ is replaced by a suitable higher-dimensional abelian variety over $\mathbb{F}_{p}$.

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Splitting properties of the reduction of semi-abelian varieties

International Journal of Number Theory ◽

10.1142/s1793042116501359 ◽

2016 ◽

Vol 12 (08) ◽

pp. 2241-2264

Author(s):

Alan Hertgen

Keyword(s):

Abelian Variety ◽

Abelian Varieties ◽

Residue Field ◽

Characteristic Exponent ◽

Jacobian Variety ◽

Ring Of Integers ◽

Identity Component ◽

Genus Formula ◽

Valuation Field

Let [Formula: see text] be a complete discrete valuation field. Let [Formula: see text] be its ring of integers. Let [Formula: see text] be its residue field which we assume to be algebraically closed of characteristic exponent [Formula: see text]. Let [Formula: see text] be a semi-abelian variety. Let [Formula: see text] be its Néron model. The special fiber [Formula: see text] is an extension of the identity component [Formula: see text] by the group of components [Formula: see text]. We say that [Formula: see text] has split reduction if this extension is split. Whereas [Formula: see text] has always split reduction if [Formula: see text] we prove that it is no longer the case if [Formula: see text] even if [Formula: see text] is tamely ramified. If [Formula: see text] is the Jacobian variety of a smooth proper and geometrically connected curve [Formula: see text] of genus [Formula: see text], we prove that for any tamely ramified extension [Formula: see text] of degree greater than a constant, depending on [Formula: see text] only, [Formula: see text] has split reduction. This answers some questions of Liu and Lorenzini.

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Index conditions and cup-product maps on Abelian varieties

International Journal of Mathematics ◽

10.1142/s0129167x14500360 ◽

2014 ◽

Vol 25 (04) ◽

pp. 1450036 ◽

Cited By ~ 2

Author(s):

Nathan Grieve

Keyword(s):

Asymptotic Behavior ◽

Abelian Variety ◽

Vector Bundles ◽

Abelian Varieties ◽

General Setting ◽

Index Theorem ◽

Line Bundles ◽

One Dimensional ◽

Cup Product ◽

Product Maps

We study questions surrounding cup-product maps which arise from pairs of non-degenerate line bundles on an abelian variety. Important to our work is Mumford's index theorem which we use to prove that non-degenerate line bundles exhibit positivity analogous to that of ample line bundles. As an application we determine the asymptotic behavior of families of cup-product maps and prove that vector bundles associated to these families are asymptotically globally generated. To illustrate our results we provide several examples. For instance, we construct families of cup-product problems which result in a zero map on a one-dimensional locus. We also prove that the hypothesis of our results can be satisfied, in all possible instances, by a particular class of simple abelian varieties. Finally, we discuss the extent to which Mumford's theta groups are applicable in our more general setting.

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Non-archimedean canonical measures on abelian varieties

Compositio Mathematica ◽

10.1112/s0010437x09004679 ◽

2010 ◽

Vol 146 (3) ◽

pp. 683-730 ◽

Cited By ~ 9

Author(s):

Walter Gubler

Keyword(s):

Line Bundle ◽

Abelian Variety ◽

Abelian Varieties ◽

Convex Geometry ◽

Explicit Description ◽

Analytic Space ◽

Ground Field ◽

Berkovich Analytic Space

AbstractFor a closedd-dimensional subvarietyXof an abelian varietyAand a canonically metrized line bundleLonA, Chambert-Loir has introduced measuresc1(L∣X)∧don the Berkovich analytic space associated toAwith respect to the discrete valuation of the ground field. In this paper, we give an explicit description of these canonical measures in terms of convex geometry. We use a generalization of the tropicalization related to the Raynaud extension ofAand Mumford’s construction. The results have applications to the equidistribution of small points.

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Complete addition laws on abelian varieties

LMS Journal of Computation and Mathematics ◽

10.1112/s1461157012001027 ◽

2012 ◽

Vol 15 ◽

pp. 308-316 ◽

Cited By ~ 1

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 .

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