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

Abstract

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 $d_K \equiv 1$ (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 $\textsf{K}_3$ of the $3$-adic field $\mathbb{Q}_3$. 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 $K=\mathbb{Q}(\sqrt{-d})$ if $d_K \equiv 1$ (mod $3$) and the class number $h(K)$ is not divisible by $3$. If $3 \mid h(K)$, congruence conditions for the trace of specific elements of $\Omega_f$ are exhibited which imply the existence of a point of infinite order in $Fer_3(K)$.