Digression on real abelian varieties and classification of real abelian surfaces

1989 ◽  
pp. 75-94
Author(s):  
Robert Silhol
Keyword(s):  
Abelian Varieties ◽  
Abelian Surfaces

scholarly journals Classification of nonrigid families of abelian varieties

1993 ◽  
Vol 45 (2) ◽  
pp. 159-189
Author(s):  
Masa-Hiko Saitō
Keyword(s):  
Abelian Varieties

scholarly journals Deformations of minimal cohomology classes on abelian varieties

2016 ◽  
Vol 18 (04) ◽  
pp. 1550066
Author(s):  
Luigi Lombardi ◽  
Sofia Tirabassi
Keyword(s):  
Cohomology Class ◽  
Hyperelliptic Curve ◽  
Abelian Varieties ◽  
Infinitesimal Deformations

We show that the infinitesimal deformations of Brill–Noether loci [Formula: see text] attached to a smooth non-hyperelliptic curve [Formula: see text] are in one-to-ne correspondence with the deformations of [Formula: see text]. As an application, we prove that if a Jacobian [Formula: see text] deforms together with a minimal cohomology class out the Jacobian locus, then [Formula: see text] is hyperelliptic. In particular, this provides an evidence toward a conjecture of Debarre on the classification of ppavs carrying a minimal cohomology class.

scholarly journals Principal polarizations of abelian surfaces over finite fields

1980 ◽  
Vol 77 ◽  
pp. 1-4
Author(s):  
Stuart Turner
Keyword(s):  
Zeta Function ◽  
Finite Fields ◽  
Abelian Varieties ◽  
Infinite Family ◽  
Abelian Surfaces ◽  
Jacobian Varieties

In § 1 of this note we construct abelian varieties of dimension two defined over Fpn, n odd, which admit infinitely many distinct principal polarizations. These polarizations determine an infinite family of geometrically non-isomorphic complete singular curves defined and irreducible over Fpn which have isomorphic Jacobian varieties. In § 2 we calculate the zeta function of these curves.

Classification of supersingular abelian varieties

1989 ◽  
Vol 283 (2) ◽  
pp. 333-351 ◽  
Author(s):  
Ke-Zheng Li
Keyword(s):  
Abelian Varieties

scholarly journals Abelian surfaces good away from 2

2017 ◽  
Vol 13 (04) ◽  
pp. 991-1001
Author(s):  
Christopher Rasmussen ◽  
Akio Tamagawa
Keyword(s):  
Elliptic Curves ◽  
Number Field ◽  
High Dimension ◽  
Abelian Varieties ◽  
Number Fields ◽  
Sufficient Condition ◽  
Good Reduction ◽  
Abelian Surfaces ◽  
Special Case

Fix a number field [Formula: see text] and a rational prime [Formula: see text]. We consider abelian varieties whose [Formula: see text]-power torsion generates a pro-[Formula: see text] extension of [Formula: see text] which is unramified away from [Formula: see text]. It is a necessary, but not generally sufficient, condition that such varieties have good reduction away from [Formula: see text]. In the special case of [Formula: see text], we demonstrate that for abelian surfaces [Formula: see text], good reduction away from [Formula: see text] does suffice. The result is extended to elliptic curves and abelian surfaces over certain number fields unramified away from [Formula: see text]. An explicit example is constructed to demonstrate that good reduction away from [Formula: see text] is not sufficient, at [Formula: see text], for abelian varieties of sufficiently high dimension.

scholarly journals Abelian varieties over finite fields as basic abelian varieties

2017 ◽  
Vol 29 (2) ◽  
pp. 489-500 ◽  
Author(s):  
Chia-Fu Yu
Keyword(s):  
Finite Field ◽  
Abelian Variety ◽  
Finite Fields ◽  
Mass Formula ◽  
Abelian Varieties ◽  
Closed Field ◽  
Abelian Surfaces ◽  
Algebraically Closed Field

AbstractIn this note we show that any basic abelian variety with additional structures over an arbitrary algebraically closed field of characteristic ${p>0}$ is isogenous to another one defined over a finite field. We also show that the category of abelian varieties over finite fields up to isogeny can be embedded into the category of basic abelian varieties with suitable endomorphism structures. Using this connection, we derive a new mass formula for a finite orbit of polarized abelian surfaces over a finite field.

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.

Sign in / Sign up

Export Citation Format

Share Document