Towards a classification of entanglements of Galois representations attached to elliptic curves

2021/05/05 by Harris B. Daniels, Daniels, Harris B., Álvaro Lozano‐Robledo +3
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.2105.02060

Abstract

Let E/ℚ be an elliptic curve, let ℚ be a fixed algebraic closure of ℚ, and let G_ℚ=Gal(ℚ/ℚ) be the absolute Galois group of ℚ. The action of G_ℚ on the adelic Tate module of E induces the adelic Galois representation ρ_E\colon G_ℚ → GL(2,\widehatℤ). The goal of this paper is to explain how the image of ρ_E can be smaller than expected. To this end, we offer a group theoretic categorization of different ways in which an entanglement between division fields can be explained and prove several results on elliptic curves (and more generally, principally polarized abelian varieties) over ℚ where the entanglement occurs over an abelian extension.

Related

• Coincidences of division fields
• Entanglement of elliptic curves upon base extension
• Composite images of Galois for elliptic curves over Q & Entanglement fields
• Elliptic curves with non-abelian entanglements
• Les représentations ell-adiques associées aux courbes elliptiques sur \Q_p