K-varieties and Galois representations

2024/12/04 by Enric Florit, Florit, Enric, Ariel Pacetti +1
Computer Science · Engineering · Mathematics · #Polynomial and algebraic computation #Advanced Numerical Analysis Techniques #Commutative Algebra and Its Applications

paper · pdf · doi:10.48550/arxiv.2412.03184

Abstract

In a remarkable article Ribet showed how to attach rational 2-dimensional representations to elliptic \mathbb Q-curves. An abelian variety A is a (weak) K-variety if it is isogenous to all of its Gal_K-conjugates. In this article we study the problem of attaching an absolutely irreducible ℓ-adic representation of Gal_K to an abelian K-variety, which sometimes has smaller dimension than expected. When possible, we also construct a Galois-equivariant pairing, which restricts the image of this representation. As an application of our construction, we prove modularity of abelian surfaces over \mathbb Q with potential quaternionic multiplication.

Related

• Galois action on the ℚ-isogeny classes of abelian surfaces with quaternionic multiplication
• The arithmetic of QM-abelian surfaces through their Galois representations
• Abelian varieties over Q and modular forms
• Explicit surjectivity of Galois representations attached to abelian surfaces and GL₂-varieties
• Equivalence of conjectures on heavenly elliptic curves