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.

scholarly journals SEMI-ABELIAN SURFACES AND INTEGRABLE SYSTEMS

2001 ◽  
Vol 44 (2) ◽  
pp. 249-265
Author(s):  
Ibrahima Faye
Keyword(s):  
Abelian Variety ◽  
Integrable Systems ◽  
Subject Classification ◽  
Level Surface ◽  
Abelian Surfaces ◽  
Completely Integrable ◽  
Homogeneous Systems ◽  
Set Of Points ◽  
Primary 37J35

AbstractWe study some weight-homogeneous systems which are not algebraically completely integrable (ACI) in the sense of Adler and van Moerebeke, but whose invariant level surface completes into a semi-abelian variety by adding a set of points (thus ACI in the sense of Mumford).AMS 2000 Mathematics subject classification: Primary 37J35. Secondary 14H70; 37N05; 70E40

scholarly journals The arithmetic of QM-abelian surfaces through their Galois representations

2004 ◽  
Vol 281 (1) ◽  
pp. 124-143 ◽  
Author(s):  
Luis V. Dieulefait ◽  
Victor Rotger
Keyword(s):  
Galois Representations ◽  
Abelian Surfaces

scholarly journals Multiparty Non-Interactive Key Exchange and More From Isogenies on Elliptic Curves

2020 ◽  
Vol 14 (1) ◽  
pp. 5-14
Author(s):  
Dan Boneh ◽  
Darren Glass ◽  
Daniel Krashen ◽  
Kristin Lauter ◽  
Shahed Sharif ◽  
...  
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Efficient Algorithm ◽  
Group Structure ◽  
Mathematical Problem ◽  
Key Exchange ◽  
Cryptographic Primitives ◽  
Indistinguishability Obfuscation ◽  
Multilinear Maps ◽  
Multilinear Map

AbstractWe describe a framework for constructing an efficient non-interactive key exchange (NIKE) protocol for n parties for any n ≥ 2. Our approach is based on the problem of computing isogenies between isogenous elliptic curves, which is believed to be difficult. We do not obtain a working protocol because of a missing step that is currently an open mathematical problem. What we need to complete our protocol is an efficient algorithm that takes as input an abelian variety presented as a product of isogenous elliptic curves, and outputs an isomorphism invariant of the abelian variety.Our framework builds a cryptographic invariant map, which is a new primitive closely related to a cryptographic multilinear map, but whose range does not necessarily have a group structure. Nevertheless, we show that a cryptographic invariant map can be used to build several cryptographic primitives, including NIKE, that were previously constructed from multilinear maps and indistinguishability obfuscation.

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 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 Determinants of subquotients of Galois representations associated with abelian varieties

2013 ◽  
Vol 13 (3) ◽  
pp. 517-559 ◽  
Author(s):  
Eric Larson ◽  
Dmitry Vaintrob
Keyword(s):  
Elliptic Curves ◽  
Abelian Variety ◽  
Number Field ◽  
Riemann Hypothesis ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Number Fields ◽  
Torsion Points ◽  
Prime Degree ◽  
Rho A

AbstractGiven an abelian variety $A$ of dimension $g$ over a number field $K$, and a prime $\ell $, the ${\ell }^{n} $-torsion points of $A$ give rise to a representation ${\rho }_{A, {\ell }^{n} } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ( \mathbb{Z} / {\ell }^{n} \mathbb{Z} )$. In particular, we get a mod-$\ell $representation ${\rho }_{A, \ell } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{F} }_{\ell } )$ and an $\ell $-adic representation ${\rho }_{A, {\ell }^{\infty } } : \mathrm{Gal} ( \overline{K} / K)\rightarrow {\mathrm{GL} }_{2g} ({ \mathbb{Z} }_{\ell } )$. In this paper, we describe the possible determinants of subquotients of these two representations. These two lists turn out to be remarkably similar.Applying our results in dimension $g= 1$, we recover a generalized version of a theorem of Momose on isogeny characters of elliptic curves over number fields, and obtain, conditionally on the Generalized Riemann Hypothesis, a generalization of Mazur’s bound on rational isogenies of prime degree to number fields.

Galois Representations Over Fields of Moduli and Rational Points on Shimura Curves

2014 ◽  
Vol 66 (5) ◽  
pp. 1167-1200 ◽  
Author(s):  
Victor Rotger ◽  
Carlos de Vera-Piquero
Keyword(s):  
Abelian Variety ◽  
Moduli Space ◽  
Quadratic Field ◽  
Galois Representations ◽  
Abelian Varieties ◽  
Main Idea ◽  
Galois Representation ◽  
Rational Points ◽  
Shimura Curves ◽  
Fields Of Moduli

AbstractThe purpose of this note is to introduce a method for proving the non-existence of rational points on a coarse moduli space X of abelian varieties over a given number field K in cases where the moduli problem is not fine and points in X(K) may not be represented by an abelian variety (with additional structure) admitting a model over the field K. This is typically the case when the abelian varieties that are being classified have even dimension. The main idea, inspired by the work of Ellenberg and Skinner on the modularity of ℚ-curves, is that one may still attach a Galois representation of Gal(/K) with values in the quotient group GL(Tℓ(A))/ Aut(A) to a point P = [A] ∈ X(K) represented by an abelian variety A/, provided Aut(A) lies in the centre of GL(Tℓ(A)). We exemplify our method in the cases where X is a Shimura curve over an imaginary quadratic field or an Atkin–Lehner quotient over ℚ.

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