[Paper Review] Arithmetic of abelian varieties with constrained torsion

Christopher Rasmussen, Akio Tamagawa|arXiv (Cornell University)|Feb 6, 2013

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper establishes finiteness results for isomorphism classes of abelian varieties over a number field K with constrained ℓ-power torsion fields, proving under the Generalized Riemann Hypothesis that only finitely many such classes exist for any fixed K and dimension g. Unconditionally, finiteness is shown for semistable abelian varieties and in low-degree extensions, with uniform bounds on ℓ derived via Galois representations and analytic number theory.

ABSTRACT

Let $K$ be a number field. We present several new finiteness results for isomorphism classes of abelian varieties over $K$ whose $\ell$-power torsion fields are arithmetically constrained for some rational prime $\ell$. Such arithmetic constraints are related to an unresolved question of Ihara regarding the kernel of the canonical outer Galois representation on the pro-$\ell$ fundamental group of $P^1 - \{0,1,\infty\}$. Under GRH, we demonstrate the set of classes is finite for any fixed $K$ and any fixed dimension. Without GRH, we prove a semistable version of the result. In addition, several unconditional results are obtained when the degree of $K/\Q$ and the dimension of abelian varieties are not too large, through a careful analysis of the special fiber of such abelian varieties. In some cases, the results (viewed as a bound on the possible values of $\ell$) are uniform in the degree of the extension $K/\Q$.

Motivation & Objective

• To investigate the finiteness of isomorphism classes of abelian varieties over a number field K whose ℓ-power torsion fields are contained in a specific pro-ℓ extension related to Ihara’s conjecture.
• To determine whether the set of such abelian varieties is finite for any fixed K and dimension g, particularly focusing on the role of ℓ and the arithmetic of torsion fields.
• To establish uniform bounds on ℓ independent of K, when possible, by analyzing the Galois representations and Frobenius elements on torsion points.
• To resolve cases of Ihara’s open question about whether the field generated by ℓ-power torsion of an abelian variety lies within the minimal field of definition of the pro-ℓ tower over P¹−{0,1,∞}.

Proposed method

• Use of the canonical outer Galois representation on the pro-ℓ fundamental group of P¹−{0,1,∞} to define the field 山(K,ℓ), which captures the minimal field of definition of the pro-ℓ tower.
• Analysis of the Galois representation ρ on A[ℓ] for abelian varieties A over K with good reduction away from ℓ, leading to constraints on the indices of semistable reduction.
• Construction of a character χ(m_Q) from the Galois representation ρ, which is shown to never vanish on Frobenius elements, enabling contradiction arguments.
• Application of analytic number theory tools, including the Lambert W-function and bounds on primes in arithmetic progressions, to derive uniform bounds on ℓ.
• Leveraging the Generalized Riemann Hypothesis to ensure the existence of small primes with specific splitting behavior, used to derive contradictions for large ℓ.
• Use of class field theory and decomposition types in Galois extensions to constrain the possible degrees and ramification of K(A[ℓ^∞]) over K.

Experimental results

Research questions

• RQ1 For a fixed number field K and dimension g, is the set of K-isomorphism classes of abelian varieties A with K(A[ℓ^∞]) ⊆ 天(K,ℓ) finite for all ℓ?
• RQ2 Can uniform bounds on ℓ be established such that K(A[ℓ^∞]) ⊆ 天(K,ℓ) implies ℓ ≤ C for some constant C depending only on g and [K:Q]?
• RQ3 Does the containment K(A[ℓ^∞]) ⊆ 山(K,ℓ) hold for all abelian varieties A over K with good reduction away from ℓ, and how does this relate to Ihara’s conjecture?
• RQ4 What is the maximal ℓ for which there exist non-isotrivial abelian varieties A over K with K(A[ℓ^∞]) ⊆ 天(K,ℓ), and can this be bounded uniformly?
• RQ5 How do the properties of the Galois representation on A[ℓ] constrain the Frobenius elements and lead to contradictions for large ℓ?

Key findings

• Under the Generalized Riemann Hypothesis, the set 𝒜(K,g) of pairs ([A],ℓ) with [A] in 𝒜(K,g,ℓ) is finite for any fixed number field K and dimension g.
• Without GRH, the finiteness of 𝒜(K,g) is proven unconditionally for abelian varieties with semistable reduction over K.
• For elliptic curves over K with [K:Q] ≤ 3 and g=1, the set 𝒜(K,1,ℓ) is empty for all sufficiently large ℓ, with a uniform bound derived via the Lambert W-function.
• A uniform bound on ℓ is established: if ℓ > 16g²C₃²ⁿC₅⁴ⁿ(2n)⁴ⁿexp(C₄/C₅), then 𝒜(K,g,ℓ) is empty, with constants derived from analytic number theory.
• For K=ℚ and g=1, the set 𝒜(ℚ,1) is finite and explicitly determined to contain 50 ℚ-isomorphism classes spanning 21 isogeny classes, with only 4 classes remaining open for ℓ=11.
• The character χ(m_Q) constructed from the Galois representation on A[ℓ] is shown to never vanish on Frobenius elements, a key tool in deriving contradictions for large ℓ.