Open cases for RT(Q,4) specified by inertia-decomposition configurations and congruence conditions

Establish the Rasmussen–Tamagawa conjecture RT(Q,4) in the five remaining cases characterized by the invariants (n_d) from the inertia action at ℓ, namely when the decomposition 2g = 8 equals one of: 2·φ(3)+φ(8), 2·φ(6)+φ(8), φ(16), φ(20), or φ(24), together with the respective congruence conditions on ℓ: ℓ ≡ 13 (mod 24), ℓ ≡ 13 (mod 24), ℓ ≡ 9 (mod 16), ℓ ≡ 11 (mod 20), or ℓ ≡ 13 (mod 24), by proving that A(Q, 4, ℓ) is empty for all sufficiently large primes ℓ in each case.

Background

Using the structure of the action on special fibers of Néron models, Rasmussen–Tamagawa associate to any [A] ∈ A(K, g, ℓ) integers n_d (indexed by d | e_λ) satisfying 2g = ∑{d|eλ} n_d φ(d) and e_λ = lcm{ d | n_d > 0 }. For K = Q, congruence constraints on ℓ additionally arise.

For g ≤ 3 these constraints suffice to prove RT(Q, g). For g = 4, however, Proposition 7.3 of Rasmussen–Tamagawa (as summarized in this paper) identifies five specific configurations (given by φ-sums and congruences) for which the conjecture remains unresolved. Proving emptiness in these five cases would settle RT(Q, 4).