scholarly journals Non-archimedean canonical measures on abelian varieties

2010 ◽  
Vol 146 (3) ◽  
pp. 683-730 ◽  
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.

scholarly journals Wonderful compactifications of Bruhat-Tits buildings

2017 ◽  
Vol Volume 1 ◽  
Author(s):  
Bertrand Remy ◽  
Amaury Thuillier ◽  
Annette Werner
Keyword(s):  
Local Field ◽  
Semisimple Group ◽  
The Other ◽  
Analytic Space ◽  
Ground Field ◽  
Berkovich Analytic Space ◽  
Euclidean Building ◽  
The Relationship

Given a split semisimple group over a local field, we consider the maximal Satake-Berkovich compactification of the corresponding Euclidean building. We prove that it can be equivariantly identified with the compactification which we get by embedding the building in the Berkovich analytic space associated to the wonderful compactification of the group. The construction of this embedding map is achieved over a general non-archimedean complete ground field. The relationship between the structures at infinity, one coming from strata of the wonderful compactification and the other from Bruhat-Tits buildings, is also investigated.

scholarly journals Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: An algebro-geometric approach

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Enrico Arbarello ◽  
Giulio Codogni ◽  
Giuseppe Pareschi
Keyword(s):  
Linear System ◽  
Line Bundle ◽  
Abelian Variety ◽  
Algebraic Curves ◽  
Abelian Varieties ◽  
Geometric Approach ◽  
Kp Equation ◽  
Basic Tool ◽  
Base Locus

Abstract We give completely algebro-geometric proofs of a theorem by T. Shiota, and of a theorem by I. Krichever, characterizing Jacobians of algebraic curves among all irreducible principally polarized abelian varieties. Shiota’s characterization is given in terms of the KP equation. Krichever’s characterization is given in terms of trisecant lines to the Kummer variety. Here we treat the case of flexes and degenerate trisecants. The basic tool we use is a theorem we prove asserting that the base locus of the linear system associated to an effective line bundle on an abelian variety is reduced. This result allows us to remove all the extra assumptions that were introduced in the theorems by the first author, C. De Concini, G.Marini, and O. Debarre, in order to achieve algebro-geometric proofs of the results above.

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 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 On the projective varieties associated with some subrings of the ring of Thetanullwerte

1994 ◽  
Vol 133 ◽  
pp. 71-83 ◽  
Author(s):  
Riccardo Salvati Manni
Keyword(s):  
Line Bundle ◽  
Abelian Variety ◽  
Torsion Points ◽  
Projective Varieties ◽  
Symmetric Line

Let (X, L) be a principally polarized abelian variety (ppav) of dimension g such that L is a symmetric line bundle, i.e. i*L ≃ L where i is the inversion map i(x) = — x. We shall denote by X[2] the two torsion points of X which are fixed by i. For any x in X[2] we have an isomorphism(1) .

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

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.

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.

scholarly journals RANK ONE CONNECTIONS ON ABELIAN VARIETIES, II

2012 ◽  
Vol 23 (12) ◽  
pp. 1250125
Author(s):  
INDRANIL BISWAS ◽  
JACQUES HURTUBISE ◽  
A. K. RAINA
Keyword(s):  
Line Bundle ◽  
Exact Sequence ◽  
Abelian Varieties ◽  
Explicit Construction ◽  
The Other ◽  
Holomorphic Line Bundle ◽  
Canonical Isomorphism ◽  
Complex Torus ◽  
Rank One

Given a holomorphic line bundle L on a compact complex torus A, there are two naturally associated holomorphic ΩA-torsors over A: one is constructed from the Atiyah exact sequence for L, and the other is constructed using the line bundle [Formula: see text], where α is the addition map on A × A, and p1 is the projection of A × A to the first factor. In [I. Biswas, J. Hurtvbise and A. K. Raina, Rank one connections on abelian varieties, Internat. J. Math.22 (2011) 1529–1543], it was shown that these two torsors are isomorphic. The aim here is to produce a canonical isomorphism between them through an explicit construction.

scholarly journals Abelian varieties isogenous to a power of an elliptic curve

2018 ◽  
Vol 154 (5) ◽  
pp. 934-959 ◽  
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 $\mathbb{F}_{p}$.

scholarly journals Splitting properties of the reduction of semi-abelian varieties

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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