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

Computing Galois representations of modular abelian surfaces

2014 ◽  
Vol 17 (A) ◽  
pp. 36-48 ◽  
Author(s):  
Jinxiang Zeng
Keyword(s):  
Abelian Variety ◽  
Efficient Algorithm ◽  
Galois Representations ◽  
Abelian Surfaces ◽  
Genus Two

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f\in S_2(\Gamma _0(N))$ be a normalized newform such that the abelian variety $A_f$ attached by Shimura to $f$ is the Jacobian of a genus-two curve. We give an efficient algorithm for computing Galois representations associated to such newforms.

The wall-crossing behavior for Bridgeland’s stability conditions on abelian and K3 surfaces

2018 ◽  
Vol 2018 (735) ◽  
pp. 1-107 ◽  
Author(s):  
Hiroki Minamide ◽  
Shintarou Yanagida ◽  
Kōta Yoshioka
Keyword(s):  
Stability Condition ◽  
Derived Category ◽  
The Other ◽  
Abelian Surface ◽  
Stability Conditions ◽  
Abelian Surfaces ◽  
Coherent Sheaves ◽  
Fundamental Properties ◽  
Wall Crossing ◽  
Wall Crossing Formula

AbstractThe wall-crossing behavior for Bridgeland’s stability conditions on the derived category of coherent sheaves on K3 or abelian surface is studied. We introduce two types of walls. One is called the wall for categories, where thet-structure encoded by stability condition is changed. The other is the wall for stabilities, where stable objects with prescribed Mukai vector may get destabilized. Some fundamental properties of walls and chambers are studied, including the behavior under Fourier–Mukai transforms. A wall-crossing formula of the counting of stable objects will also be derived. As an application, we will explain previous results on the birational maps induced by Fourier–Mukai transforms on abelian surfaces. These transformations turns out to coincide with crossing walls of certain property.

scholarly journals Kummer surfaces associated to (1, 2)-polarized abelian surfaces

2011 ◽  
Vol 202 ◽  
pp. 127-143
Author(s):  
Afsaneh Mehran
Keyword(s):  
Double Cover ◽  
Pezzo Surface ◽  
Abelian Surface ◽  
Kummer Surface ◽  
Abelian Surfaces ◽  
Del Pezzo Surface ◽  
Singular Fibers ◽  
Kummer Surfaces ◽  
Del Pezzo

AbstractThe aim of this paper is to describe the geometry of the generic Kummer surface associated to a (1, 2)-polarized abelian surface. We show that it is the double cover of a weak del Pezzo surface and that it inherits from the del Pezzo surface an interesting elliptic fibration with twelve singular fibers of typeI2.

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].

Elliptic curves on abelian surfaces

1994 ◽  
Vol 84 (1) ◽  
pp. 199-223 ◽  
Author(s):  
Ernst Kani
Keyword(s):  
Elliptic Curves ◽  
Abelian Surfaces

scholarly journals Corrigendum: Abelian n-division fields of elliptic curves and Brauer groups of product Kummer & abelian surfaces

2020 ◽  
Vol 8 ◽  
Author(s):  
Anthony Várilly-Alvarado ◽  
Bianca Viray
Keyword(s):  
Elliptic Curves ◽  
Galois Representations ◽  
Brauer Groups ◽  
Abelian Surfaces ◽  
High Level

There is an error in the statement and proof of [VAV17, Proposition 5.1] that affects the statements of [VAV17, Corollaries 5.2 and 5.3]. In this note, we correct the statement of [VAV17, Proposition 5.1] and explain how to rectify subsequent statements. In brief, for a statement about abelian Galois representations of a fixed level, ‘abelian’ should be replaced with ‘liftable abelian’ (Definition 1). Statements about abelian Galois representations of arbitrarily high level, however, remain unchanged because such representations give rise to liftable abelian Galois representations of smaller, but still arbitrarily high, level. Hence the main theorems of the paper remain unchanged.

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.

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