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

Harris B. Daniels, Álvaro Lozano-Robledo, Jackson S. Morrow

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve, let $\overline{\mathbb{Q}}$ be a fixed algebraic closure of $\mathbb{Q}$, and let $G_{\mathbb{Q}}=\text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. The action of $G_{\mathbb{Q}}$ on the adelic Tate module of $E$ induces the adelic Galois representation $ρ_E\colon G_{\mathbb{Q}} \to \text{GL}(2,\widehat{\mathbb{Z}}).$ 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 $\mathbb{Q}$ where the entanglement occurs over an abelian extension.

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

Abstract

Let be an elliptic curve, let be a fixed algebraic closure of , and let be the absolute Galois group of . The action of on the adelic Tate module of induces the adelic Galois representation The goal of this paper is to explain how the image of 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 where the entanglement occurs over an abelian extension.

Paper Structure

This paper contains 32 sections, 23 theorems, 57 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main contributions
  3. Related results
  4. Outline of paper
  5. Conventions
  6. Comments on code
  7. Acknowledgments
  8. Group theoretic definition of entanglement
  9. Classification of abelian entanglements
  10. Abelian and Weil entanglements
  11. Discriminant entanglements
  12. The Serre (or discriminant) entanglement
  13. Other discriminant $(2,q)$-entanglements
  14. CM entanglements
  15. Fake CM entanglements
  16. ...and 17 more sections

Key Result

Theorem 1

Let $E/\mathbb Q$ be an elliptic curve, and let $p$ and $q$ be distinct primes such that $E$ has an abelian $(p,q)$-entanglement of type $S$ (Definition defn:abelian-type), for some finite abelian group $S$. Then, there is a finite set $J\subseteq \mathbb Q$, such that if $j(E)\not\in J$ and the ent where Moreover, the set $J$ contains $\{-5^2/2, -5^2\cdot 241^3/2^3,-5\cdot 29^3/2^5,5\cdot 211^3/

Figures (2)

  • Figure 2.1: A diagram describing group-theoretical entanglements. Lines denote maps, together with the subgroup that one has to quotient the group on top by to get to the group below.
  • Figure 3.1: An abelian $(a,b)$-entanglement of type $S$, and a Weil $(a,b)$-entanglement of type $T$, where $c=\mathop{\mathrm{lcm}}\nolimits(a,b)$ and $d=\gcd(a,b)$.

Theorems & Definitions (76)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Remark 1.2
  • Remark 1.3
  • Theorem 5
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • ...and 66 more