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Heavenly elliptic curves over quadratic fields

Cam McLeman, Christopher Rasmussen

Abstract

An abelian variety $A/K$ is heavenly at $\ell$ if the extension $K(A[\ell^\infty])/K(μ_{\ell^{\infty}}\!)$ is both pro-$\ell$ and unramified away from $\ell$. It is known that for a fixed quadratic field $K$, the number of $K$-isomorphism classes of heavenly elliptic curves is finite, even running over all primes $\ell$. We prove a complementary result, that for a fixed prime $\ell\geq 7$, there are only finitely many such classes, even running over all quadratic fields. This naturally raises the question of whether to expect a finiteness result when both $K$ and $\ell$ are allowed to vary. We demonstrate similarities in the behavior of heavenly elliptic curves and elliptic curves with complex multiplication, in terms of their Frobenius traces modulo $\ell$. We determine the complete list of heavenly elliptic curves defined over quadratic fields with complex multiplication and with irrational $j$-invariant (up to isomorphism). We include various extensions of our results to higher degree fields and higher-dimensional abelian varieties where possible.

Heavenly elliptic curves over quadratic fields

Abstract

An abelian variety is heavenly at if the extension is both pro- and unramified away from . It is known that for a fixed quadratic field , the number of -isomorphism classes of heavenly elliptic curves is finite, even running over all primes . We prove a complementary result, that for a fixed prime , there are only finitely many such classes, even running over all quadratic fields. This naturally raises the question of whether to expect a finiteness result when both and are allowed to vary. We demonstrate similarities in the behavior of heavenly elliptic curves and elliptic curves with complex multiplication, in terms of their Frobenius traces modulo . We determine the complete list of heavenly elliptic curves defined over quadratic fields with complex multiplication and with irrational -invariant (up to isomorphism). We include various extensions of our results to higher degree fields and higher-dimensional abelian varieties where possible.

Paper Structure

This paper contains 17 sections, 33 theorems, 56 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. New results
  3. Conjectures on heavenly elliptic curves
  4. Outline
  5. Arithmetic of abelian varieties
  6. General notions
  7. The invariant eA(l)
  8. The totient identity
  9. The fields Yama and Ten and Ihara's question
  10. Heavenly abelian varieties
  11. New finiteness results
  12. Balanced abelian varieties
  13. Bounds on balanced heavenly abelian varieties
  14. Comparison of trace behavior
  15. Heavenly elliptic curves with complex multiplication.
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\ell\geq 7$ be prime. The collection of elliptic curves which are defined over a quadratic field and which are heavenly at $\ell$ represent only finitely many $\overline{\mathbb{Q}}$-isomorphism classes.

Figures (1)

  • Figure 1: The set $\mathcal{R}$ inside $\mathbb{N} \times \mathcal{F}$

Theorems & Definitions (68)

  • Theorem 1.1: Corollary \ref{['cor:HQ21_finite']}
  • Theorem 1.2: Theorem \ref{['thm:Hkd1l_finite']}
  • Theorem 1.3: Corollary \ref{['cor:balanced_ap_mod_ell']}
  • Theorem 1.4: Theorem \ref{['thm:heavenly_cm_calculation']}
  • Conjecture 1
  • Conjecture 2
  • Conjecture 3
  • Proposition 2.1: Serre:1972
  • Proposition 2.2: Prop. 6.2, Cor. 6.4, Rasmussen-Tamagawa:2017
  • Corollary 2.3
  • ...and 58 more