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 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 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 A Study of Doubly Warped Product Immersions in a Nearly Trans-Sasakian Manifold with Slant Factor

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ali H. Alkhaldi ◽  
Aliya Naaz Siddiqui ◽  
Kamran Ahmad ◽  
Akram Ali
Keyword(s):  
Cohomology Class ◽  
Warped Product ◽  
Sasakian Manifold ◽  
Sasakian Manifolds ◽  
De Rham Cohomology ◽  
De Rham

In this article, we discuss the de Rham cohomology class for bislant submanifolds in nearly trans-Sasakian manifolds. Moreover, we give a classification of warped product bislant submanifolds in nearly trans-Sasakian manifolds with some nontrivial examples in the support. Next, it is of great interest to prove that there does not exist any doubly warped product bislant submanifolds other than warped product bislant submanifolds in nearly trans-Sasakian manifolds. Some immediate consequences are also obtained.

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.

Classification of supersingular abelian varieties

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

Torsion Elements and the Classification of Vector Bundles

1977 ◽  
Vol 29 (2) ◽  
pp. 327-332
Author(s):  
Robert D. Little
Keyword(s):  
Algebraic Topology ◽  
Cohomology Class ◽  
Cohomology Group ◽  
Vector Bundles ◽  
Geometric Construction ◽  
Integral Cohomology

There are many situations in algebraic topology when a geometric construction is possible if, and only if, a certain integral cohomology class, an obstruction is zero. When attempts are made to compute the obstruction, it often happens that it is relatively easy to show that m times the obstruction is zero, where m is an integer, and consequently the geometric construction is possible if the cohomology group in question has no elements of order m.

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.

scholarly journals On compatibility between isogenies and polarizations of abelian varieties

2017 ◽  
Vol 13 (03) ◽  
pp. 673-704 ◽  
Author(s):  
Martin Orr
Keyword(s):  
Abelian Variety ◽  
Abelian Varieties ◽  
Shimura Varieties ◽  
Division Algebras ◽  
Hermitian Forms ◽  
Endomorphism Algebras

We discuss the notion of polarized isogenies of abelian varieties, that is, isogenies which are compatible with given principal polarizations. This is motivated by problems of unlikely intersections in Shimura varieties. Our aim is to show that certain questions about polarized isogenies can be reduced to questions about unpolarized isogenies or vice versa. Our main theorem concerns abelian varieties [Formula: see text] which are isogenous to a fixed abelian variety [Formula: see text]. It establishes the existence of a polarized isogeny [Formula: see text] whose degree is polynomially bounded in [Formula: see text], if there exist both an unpolarized isogeny [Formula: see text] of degree [Formula: see text] and a polarized isogeny [Formula: see text] of unknown degree. As a further result, we prove that given any two principally polarized abelian varieties related by an unpolarized isogeny, there exists a polarized isogeny between their fourth powers. The proofs of both theorems involve calculations in the endomorphism algebras of the abelian varieties, using the Albert classification of these endomorphism algebras and the classification of Hermitian forms over division algebras.

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