Heavenly elliptic curves over quadratic fields

2024/10/24 by Cam McLeman, McLeman, Cam, Christopher Rasmussen +1 · 2 citations
Mathematics · #11G05 #11G10 #11G15 #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2410.18389

openalex publication_date 2024/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

An abelian variety A/K is heavenly at ℓ if the extension K(A[ℓ^∞])/K(μ_ℓ^∞ ) is both pro-ℓ and unramified away from ℓ. 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 ℓ. We prove a complementary result, that for a fixed prime ℓ≥ 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 ℓ 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 ℓ. 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.

Cited by

• On the Rasmussen-Tamagawa conjecture for abelian fivefolds
• Equivalence of conjectures on heavenly elliptic curves

Related

• Equivalence of conjectures on heavenly elliptic curves
• Infinitely many primes of basic reduction for some abelian fourfolds
• Spanning the isogeny class of a power of an ordinary elliptic curve over a finite field. Application to the number of rational points of curves of genus ≤ 4
• A Uniform Version of a Finiteness Conjecture for CM Elliptic Curves
• Elliptic Curves with Everywhere Good Reduction