Elliptic curves on abelian surfaces

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.

Download Full-text

Hyper-K\"ahler Fourfolds Fibered by Elliptic Products

Épijournal de Géométrie Algébrique ◽

10.46298/epiga.2018.volume2.3983 ◽

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

Download Full-text

Abelian surfaces good away from 2

International Journal of Number Theory ◽

10.1142/s179304211750052x ◽

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.

Download Full-text

Super-Isolated Elliptic Curves and Abelian Surfaces in Cryptography

Experimental Mathematics ◽

10.1080/10586458.2017.1412371 ◽

2018 ◽

Vol 28 (4) ◽

pp. 385-397 ◽

Cited By ~ 1

Author(s):

Travis Scholl

Keyword(s):

Elliptic Curves ◽

Abelian Surfaces

Download Full-text

Surface abéliennes à multiplication quaternionique munie d'une structure de niveau Γ0(N)

International Journal of Number Theory ◽

10.1142/s1793042117500725 ◽

2017 ◽

Vol 13 (05) ◽

pp. 1317-1333

Author(s):

Florence Gillibert

Keyword(s):

Elliptic Curves ◽

Prime Number ◽

Quaternion Algebra ◽

Level Structure ◽

Maximal Order ◽

Abelian Surfaces ◽

Text Type ◽

Number Formula ◽

Imaginary Field

A theorem of Mazur gives the set of possible prime degrees for rational isogenies between elliptic curves. In this paper, we are working on a similar problem in the case of abelian surfaces of [Formula: see text]-type over [Formula: see text] with quaternionic multiplication (over [Formula: see text]) endowed with a [Formula: see text] level structure. We prove the following result: for a fixed indefinite quaternion algebra [Formula: see text] of discriminant [Formula: see text] and a fixed quadratic imaginary field [Formula: see text], there exists an effective bound [Formula: see text] such that for a prime number [Formula: see text], not dividing the conductor of the order [Formula: see text], there do not exist abelian surfaces [Formula: see text] such that [Formula: see text] is a maximal order of [Formula: see text] and [Formula: see text] is endowed with a [Formula: see text] level structure.

Download Full-text