RT(K,3) for quadratic fields K

Establish the Rasmussen–Tamagawa conjecture RT(K,3) unconditionally for every quadratic number field K; that is, prove that there exists a bound L(K) such that for all primes ℓ > L(K), the set A(K, 3, ℓ) of K-isomorphism classes of abelian threefolds over K with good reduction outside ℓ and with K(A[ℓ^∞]) a pro-ℓ extension of K(ζ_ℓ) is empty.

Background

The authors’ refinements yield partial progress toward RT(K,3) over quadratic fields but are insufficient for a full resolution. This complements known unconditional results over Q for g ≤ 3 and GRH-conditional results in general.

Completing these cases would extend the unconditional scope of the Rasmussen–Tamagawa conjecture beyond the rational field to all quadratic extensions for g = 3.