Rasmussen–Tamagawa conjecture (general form)

Establish that for every number field K and every positive integer g, there exists a bound L(K, g) such that for all primes ℓ > L(K, g), the set A(K, g, ℓ) of K-isomorphism classes of g-dimensional abelian varieties over K that have good reduction outside ℓ and whose ℓ-power torsion field K(A[ℓ^∞]) is a pro-ℓ extension of K(ζ_ℓ) is empty.

Background

The paper studies the finiteness and eventual emptiness of sets A(K, g, ℓ) consisting of g-dimensional abelian varieties over a number field K with good reduction outside ℓ and with constrained ℓ-power torsion extensions. This problem is encoded in the Rasmussen–Tamagawa conjecture RT(K, g).

Prior work established RT(Q, g) for g ≤ 3 unconditionally and RT(K, g) under GRH for all K and g, with several further partial results in special cases (e.g., CM/QM varieties and certain extensions). The present paper proves RT(Q, 5) and develops refinements of the Rasmussen–Tamagawa strategy.