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

  • Harris B. Daniels

    Amherst College, USA
  • Alvaro Lozano-Robledo

    University of Connecticut, Storrs, USA
  • Jackson S. Morrow

    Université de Montréal, Canada
Towards a classification of entanglements of Galois representations attached to elliptic curves cover
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Abstract

Let E/QE/\mathbb{Q} be an elliptic curve, let Q\overline{\mathbb{Q}} be a fixed algebraic closure of Q\mathbb{Q}, and let GQ=Gal(Q/Q)G_{\mathbb{Q}}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) be the absolute Galois group of Q\mathbb{Q}. The action of GQG_{\mathbb{Q}} on the adelic Tate module of EE induces the adelic Galois representation ρE ⁣:GQGL(2,Z^).\rho_E\colon G_{\mathbb{Q}} \to \text{GL}(2,\widehat{\mathbb{Z}}).

The goal of this paper is to explain how the image of ρE\rho_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\mathbb{Q} where the entanglement occurs over an abelian extension.

Cite this article

Harris B. Daniels, Alvaro Lozano-Robledo, Jackson S. Morrow, Towards a classification of entanglements of Galois representations attached to elliptic curves. Rev. Mat. Iberoam. 39 (2023), no. 3, pp. 803–844

DOI 10.4171/RMI/1424