Heavenly elliptic curves over quadratic fields

1. Problem Statement and Context

The paper investigates heavenly abelian varieties, a concept arising from the study of pro-$\ell$ outer Galois representations attached to the projective line minus three points ($\mathbb{P}^1 \setminus \{0, 1, \infty\}$).

• Definition: An abelian variety $A$ defined over a number field $K$ is called heavenly at a prime $\ell$ if the extension $K(A[\ell^\infty])/K(\mu_{\ell^\infty})$ is both pro-$\ell$ and unramified away from $\ell$.
• Context: This definition relates to a question by Yasutaka Ihara regarding the kernel of the Galois representation on the fundamental group of $\mathbb{P}^1 \setminus \{0, 1, \infty\}$. Specifically, it asks whether the field fixed by the kernel (denoted $\mathfrak{yama}$) coincides with the maximal pro-$\ell$ extension unramified away from $\ell$ (denoted $\mathfrak{ten}$).
• Finiteness Conjectures: It is known that for a fixed $K$, there are finitely many heavenly isomorphism classes. The authors address the complementary finiteness question: If the prime $\ell$ is fixed, are there only finitely many heavenly elliptic curves when the base field $K$ varies over all quadratic fields?

2. Methodology

The authors employ a combination of arithmetic geometry, Galois representation theory, and computational number theory.

• Galois Representations and Torsion Structure: They analyze the structure of the $\ell$-torsion points $A[\ell]$ and the associated Galois representations $\rho_{A, \ell}$. They utilize the Tate-Oort classification of finite flat group schemes to relate the characters of the Galois action to fundamental characters.
• The "Balanced" Condition: A central technical tool is the introduction of balanced abelian varieties. An elliptic curve is balanced at $\ell$ if its Tate-Oort numbers (exponents of the fundamental character) satisfy a specific symmetry condition ($j_r = e/2$). The authors prove that for sufficiently large $\ell$, heavenly curves must be balanced.
• Siegel's Theorem and Integral Points: To prove finiteness results, they map the set of heavenly curves to $S$-integral points on modular curves ($Y_1(\ell)$). They utilize Levin's generalization of Siegel's theorem, which states that if a curve has sufficiently many points at infinity, the set of $S$-integral points of bounded degree is finite.
• Complex Multiplication (CM) Analysis: For curves with CM, they leverage the theory of Hecke characters and genus theory to characterize exactly when a CM curve is heavenly. This involves analyzing the splitting behavior of primes in the compositum of the CM field and the base field.
• Computational Verification: The authors use SageMath and MAGMA to:
  • Search for solutions to congruences derived from the Weil bounds.
  • Enumerate all possible isogeny classes of CM curves over quadratic fields.
  • Verify the "heavenly" condition for specific curves listed in their tables.

3. Key Contributions and Results

A. Finiteness Results (Theorems 1.1, 1.2, 3.1)

The paper establishes that for a fixed prime $\ell \ge 7$, there are only finitely many $\mathbb{Q}$-isomorphism classes of elliptic curves defined over any quadratic field that are heavenly at $\ell$.

• Generalization: This extends to fields of degree $d$ over a base field $K_0$ provided $\ell > 2d + 1$.
• Mechanism: The proof relies on showing that heavenly curves correspond to $S$-integral points on $Y_1(\ell)$ of degree at most $d$. Since $\ell - 1 > 2d$, the number of cusps on $X_1(\ell)$ exceeds twice the degree, ensuring finiteness via Levin's theorem.

B. Trace Behavior and CM Similarities (Theorem 1.3, Corollary 6.9)

The authors demonstrate a striking similarity between heavenly curves and curves with Complex Multiplication (CM) regarding their Frobenius traces ($a_p$).

• Result: For a heavenly curve $E/K$ over a quadratic field and a prime $p \nmid \ell$:
  • If $p$ is inert in $K(\sqrt{-\ell})$, then $a_p \equiv 0 \pmod \ell$.
  • If $p$ splits in $K(\sqrt{-\ell})$, then $a_p^2 \equiv 4N(p) \pmod \ell$.
• Significance: This behavior mirrors the trace properties of CM curves. This leads to Conjecture A: For $\ell > 19$, every heavenly elliptic curve over a quadratic field has Complex Multiplication.

C. Classification of Heavenly CM Curves (Theorem 1.4, Theorem 7.3)

The authors provide a complete classification of heavenly elliptic curves over quadratic fields that possess CM and have an irrational $j$-invariant.

• Constraints: Such curves can only exist for primes $\ell \in \{2, 3, 7, 11, 23, 31, 47\}$.
• Explicit List: They determine the exact isogeny classes and provide explicit Weierstrass equations for all such curves (listed in Table A.2).
• Method: They use genus theory to restrict the possible CM fields and conductors, then perform a computational search for quadratic twists that satisfy the "good reduction away from $\ell$" and "pro-$\ell$" conditions.

D. Bounds and Non-Existence (Section 8)

The paper establishes strong non-existence results for large $\ell$.

• Lemma 8.1 & 8.2: They show that for large $\ell$, the set of primes where a heavenly curve could exist is extremely restrictive.
• Proposition 8.3: Conditional on the Extended Riemann Hypothesis (ERH), for any quadratic field $K$ with discriminant $|\Delta_K| < 5 \cdot 10^5$, there are no heavenly elliptic curves for any prime $\ell > 163$.

4. Significance and Future Directions

• Resolution of a Specific Finiteness Case: The paper resolves the finiteness question for fixed $\ell$ and varying quadratic fields, a significant step toward the broader conjecture that the set of all heavenly pairs $(\ell, [A])$ is finite.
• Bridge to CM Theory: By proving that heavenly curves mimic the trace behavior of CM curves, the authors provide strong evidence that heavenly curves are essentially "CM-like" objects, potentially belonging to a very small, structured set.
• Computational Arithmetic Geometry: The work provides a robust framework for enumerating special arithmetic objects (heavenly curves) using a combination of theoretical bounds (Siegel/Levin) and explicit computation (MAGMA/Sage).
• Ihara's Question: While not solving Ihara's full question ($\mathfrak{yama} = \mathfrak{ten}$), the characterization of heavenly varieties deepens the understanding of the arithmetic of the kernel of the Galois representation on $\mathbb{P}^1 \setminus \{0, 1, \infty\}$.

In summary, the paper successfully narrows the search space for heavenly elliptic curves, proving their finiteness over varying quadratic fields for fixed large primes, and completely classifying the subset of these curves that possess Complex Multiplication.