scholarly journals Explicit surjectivity of Galois representations for abelian surfaces and GL2-varieties

2016 ◽  
Vol 460 ◽  
pp. 26-59 ◽  
Author(s):  
Davide Lombardo
Keyword(s):  
Galois Representations ◽  
Abelian Surfaces

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 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 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.

Local torsion on abelian surfaces with real multiplication by ${\bf Q}(\sqrt{5})$

2014 ◽  
Vol 10 (07) ◽  
pp. 1807-1827
Author(s):  
Adam Gamzon
Keyword(s):  
Elliptic Curve ◽  
Deformation Theory ◽  
Galois Representations ◽  
Finite Extension ◽  
Text Structure ◽  
Torsion Point ◽  
Abelian Surfaces ◽  
Real Multiplication

Fix an integer d ≥ 1. In 2008, David and Weston showed that, on average, an elliptic curve over Q picks up a nontrivial p-torsion point defined over a finite extension K of the p-adics of degree at most d for only finitely many primes p. This paper proves an analogous averaging result for principally polarized abelian surfaces A over Q with real multiplication by [Formula: see text] and a level-[Formula: see text] structure. Furthermore, we indicate how the result on abelian surfaces with real multiplication relates to the deformation theory of modular Galois representations.

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