When is an abelian surface isomorphic or isogeneous to a product of elliptic curves?

1990 ◽  
Vol 203 (1) ◽  
pp. 293-299 ◽  
Author(s):  
Wolfgang M. Ruppert
Keyword(s):  
Elliptic Curves ◽  
Abelian Surface

scholarly journals The cover associated to a (1,3)-polarized bielliptic abelian surface and its branch locus

1999 ◽  
Vol 42 (2) ◽  
pp. 375-392 ◽  
Author(s):  
Gianfranco Casnati
Keyword(s):  
Elliptic Curves ◽  
Abelian Surface ◽  
Branch Locus

Let A be an abelian surface and let |D| be a polarization of type (1,3) on A. If (A,|D|) is not a product of elliptic curves, such a polarization induces a finite morphism Q: A →p2C of degree 6. In this paper we describe the branch locus of Q when A is bielliptic in thesense of K. Hulek and S. H. Weintraub (see [13]), generalizing the results proved by Ch. Birkenhake and H. Lange in [4].

scholarly journals Hyper-K\"ahler Fourfolds Fibered by Elliptic Products

2018 ◽  
Vol Volume 2 ◽  
Author(s):  
Ljudmila Kamenova
Keyword(s):  
Elliptic Curves ◽  
Abelian Surface ◽  
Abelian Surfaces ◽  
Genus Two

Every fibration of a projective hyper-K\"ahler fourfold has fibers which are Abelian surfaces. In case the Abelian surface is a Jacobian of a genus two curve, these have been classified by Markushevich. We study those cases where the Abelian surface is a product of two elliptic curves, under some mild genericity hypotheses. Comment: 8 pages, EPIGA published version

scholarly journals ABELIAN -DIVISION FIELDS OF ELLIPTIC CURVES AND BRAUER GROUPS OF PRODUCT KUMMER & ABELIAN SURFACES

2017 ◽  
Vol 5 ◽  
Author(s):  
ANTHONY VÁRILLY-ALVARADO ◽  
BIANCA VIRAY
Keyword(s):  
Elliptic Curves ◽  
Abelian Surface ◽  
Uniform Bound ◽  
Brauer Groups ◽  
Fixed Degree ◽  
Related Family ◽  
The Family ◽  
Kummer Surfaces ◽  
Principal Homogeneous Space ◽  
Group Modulo

Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $\text{Br}\,Y/\text{Br}_{1}\,Y$ is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Néron–Severi lattices. Over a field of characteristic $0$, we prove that the existence of a strong uniform bound on the size of the odd torsion of $\text{Br}Y/\text{Br}_{1}Y$ is equivalent to the existence of a strong uniform bound on integers $n$ for which there exist non-CM elliptic curves with abelian $n$-division fields. Using the same methods we show that, for a fixed prime $\ell$, a number field $k$ of fixed degree $r$, and a fixed discriminant of the geometric Néron–Severi lattice, $\#(\text{Br}Y/\text{Br}_{1}Y)[\ell ^{\infty }]$ is bounded by a constant that depends only on $\ell$, $r$, and the discriminant.

scholarly journals Visualizing elements of order in the Tate–Shafarevich group of an elliptic curve

2016 ◽  
Vol 19 (A) ◽  
pp. 100-114
Author(s):  
Tom Fisher
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curves ◽  
Abelian Surface

We study the elliptic curves in Cremona’s tables that are predicted by the Birch–Swinnerton-Dyer conjecture to have elements of order $7$ in their Tate–Shafarevich group. We show that in many cases these elements are visible in an abelian surface or abelian 3-fold.

Elliptic Curves

1997 ◽  
Author(s):  
Henry McKean ◽  
Victor Moll
Keyword(s):  
Elliptic Curves

The Joint Universality for L-Functions of Elliptic Curves

2004 ◽  
Vol 9 (4) ◽  
pp. 331-348
Author(s):  
V. Garbaliauskienė
Keyword(s):  
Elliptic Curves ◽  
Rational Numbers ◽  
Joint Universality ◽  
Universality Theorem ◽  
Joint Universality Theorem

A joint universality theorem in the Voronin sense for L-functions of elliptic curves over the field of rational numbers is proved.

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