RT(K,2) for quadratic fields K

Establish the Rasmussen–Tamagawa conjecture RT(K,2) 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, 2, ℓ) of K-isomorphism classes of abelian surfaces over K with good reduction outside ℓ and with K(A[ℓ^∞]) a pro-ℓ extension of K(ζ_ℓ) is empty.

Background

The paper introduces two new ingredients—small quadratic nonresidues and explicit quadratic residues—to extend known techniques. These yield nontrivial partial results for RT(K,2) when K is quadratic, but do not yet lead to a complete proof.

While RT(Q, g) for g ≤ 3 is known, and RT(K, g) under GRH holds in general, the unconditional case for quadratic fields and g = 2 remains unresolved here.