Solutions of the cubic Fermat equation in ring class fields of imaginary quadratic fields (as periodic points of a 3-adic algebraic function)

Explicit solutions of the cubic Fermat equation are constructed in ring class fields Omega(f), with conductor f prime to 3, of any imaginary quadratic field K whose discriminant satisfies d(K) = 1 (mod 3), in terms of the Dedekind eta-function. As K and f vary, the set of coordinates of all solutions is shown to be the exact set of periodic points of a single algebraic function and its inverse defined on natural subsets of the maximal unramified, algebraic extension K-3 of the 3-adic field Q(3). This is used to give a dynamical proof of a class number relation of Deuring. These solutions are then used to give an unconditional proof of part of Aigner's conjecture: the cubic Fermat equation has a nontrivial solution in K = Q(root-d) if d(K) = 1 (mod 3) and the class number h(K) is not divisible by 3. If 3 vertical bar h(K), congruence conditions for the trace of specific elements of Omega(f) are exhibited which imply the existence of a point of infinite order in Fer(3)(K).