Galois module structure probability for Mordell–Weil groups over quadratic extensions

Establish that for an elliptic curve E/Q with rank 1 and full rational 2-torsion, under the large-image condition on the Galois action on E[4], and within a Frobenian twist subfamily F_{b,L}, the limiting proportion of squarefree d for which the Mordell–Weil lattice E(Q(√d))/torsion is isomorphic (as a Z[C2]-module) to the regular representation Z[C2] equals 1/2, 1/8, 5/64, and 29/1024 when the systematic Selmer subspace dimension n_b equals 1, 2, 3, and 4 respectively.

Background

The authors relate the event E(Q(√d))/E(Q(√d))tors ≅ Z[C2] to a norm condition for a Mordell–Weil generator and to a Selmer lifting condition in the Frobenian family F{b,L}. Adapting their equidistribution heuristic for Selmer elements yields an identical set of limiting probabilities as in the genus-1 curve setting, depending only on the dimension n_b of the systematic subspace.

This conjecture complements their proven lower bounds (Theorem 1.7) and aligns with the random-matrix-based predictions developed in Section 6.