scholarly journals A characterization of ordinary abelian varieties by the Frobenius push-forward of the structure sheaf

2016 ◽  
Vol 366 (3-4) ◽  
pp. 1067-1087 ◽  
Author(s):  
Akiyoshi Sannai ◽  
Hiromu Tanaka
Keyword(s):  
Abelian Varieties ◽  
Structure Sheaf ◽  
Push Forward

scholarly journals A Characterization of Ordinary Abelian Varieties by the Frobenius Push-Forward of the Structure Sheaf II

2018 ◽  
Vol 2019 (19) ◽  
pp. 5975-5988
Author(s):  
Sho Ejiri ◽  
Akiyoshi Sannai
Keyword(s):  
Abelian Variety ◽  
Projective Variety ◽  
Abelian Varieties ◽  
Smooth Projective Variety ◽  
Line Bundles ◽  
Characteristic P ◽  
Structure Sheaf ◽  
Direct Sum ◽  
Push Forward

Abstract In this paper, we prove that a smooth projective variety X of characteristic p > 0 is an ordinary abelian variety if and only if KX is pseudo-effective and $F_{*}^{e}{\mathcal {O}}_{X}$ splits into a direct sum of line bundles for an integer e with pe > 2.

scholarly journals GALOIS THEORY OF THE CANONICAL THETA STRUCTURE

2011 ◽  
Vol 07 (01) ◽  
pp. 173-202
Author(s):  
ROBERT CARLS
Keyword(s):  
Galois Theory ◽  
Theoretical Foundation ◽  
Abelian Varieties ◽  
Geometric Mean ◽  
Null Point ◽  
Point Counting ◽  
Characteristic 2 ◽  
Theoretic Characterization ◽  
Generalized Arithmetic

In this article, we give a Galois-theoretic characterization of the canonical theta structure. The Galois property of the canonical theta structure translates into certain p-adic theta relations which are satisfied by the canonical theta null point of the canonical lift. As an application, we prove some 2-adic theta identities which describe the set of canonical theta null points of the canonical lifts of ordinary abelian varieties in characteristic 2. The latter theta relations are suitable for explicit canonical lifting. Using the theory of canonical theta null points, we are able to give a theoretical foundation to Mestre's point counting algorithm which is based on the computation of the generalized arithmetic geometric mean sequence.

scholarly journals Characterization of abelian varieties

2001 ◽  
Vol 143 (2) ◽  
pp. 435-447 ◽  
Author(s):  
Jungkai A. Chen ◽  
Christopher D. Hacon
Keyword(s):  
Abelian Varieties

scholarly journals Characterization of products of theta divisors

2014 ◽  
Vol 150 (8) ◽  
pp. 1384-1412 ◽  
Author(s):  
Zhi Jiang ◽  
Martí Lahoz ◽  
Sofia Tirabassi
Keyword(s):  
Euler Characteristic ◽  
Abelian Varieties ◽  
The Other ◽  
Other Hand ◽  
Birational Equivalence ◽  
Points Of View ◽  
The One

AbstractWe study products of irreducible theta divisors from two points of view. On the one hand, we characterize them as normal subvarieties of abelian varieties such that a desingularization has holomorphic Euler characteristic $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}1$. On the other hand, we identify them up to birational equivalence among all varieties of maximal Albanese dimension. We also describe the structure of varieties $X$ of maximal Albanese dimension, with holomorphic Euler characteristic $1$ and irregularity $2\dim X-1$.

scholarly journals Modular Abelian Varieties Over Number Fields

2014 ◽  
Vol 66 (1) ◽  
pp. 170-196 ◽  
Author(s):  
Xavier Guitart ◽  
Jordi Quer
Keyword(s):  
Cohomology Class ◽  
Abelian Varieties ◽  
Galois Cohomology ◽  
Number Fields ◽  
Abelian Surfaces ◽  
Congruence Subgroups ◽  
Galois Number ◽  
Modular Varieties

AbstractThe main result of this paper is a characterization of the abelian varieties B/K defined over Galois number fields with the property that the L-function L(B/K; s) is a product of L-functions of non-CM newforms over ℚ for congruence subgroups of the form Γ1(N). The characterization involves the structure of End(B), isogenies between the Galois conjugates of B, and a Galois cohomology class attached to B/K.We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied, we prove the strong modularity of some particular abelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly modular varieties by twisting.

Algebraic and rigid geometry on the Schottky problem

2015 ◽  
Vol 2015 (705) ◽  
Author(s):  
Takashi Ichikawa
Keyword(s):  
Abelian Varieties ◽  
Theta Functions ◽  
Jacobian Varieties ◽  
Schottky Problem

AbstractWe study the Schottky problem by giving the KP characterization of Jacobian varieties among abelian varieties in terms of their algebraic or nonarchimedean theta functions.

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