Solutions of diophantine equations as periodic points of $p$-adic algebraic functions, I

Abstract

Solutions of the quartic Fermat equation in ring class fields of odd conductor over quadratic fields $K=\mathbb{Q}(\sqrt{-d})$ with $-d \equiv 1$ (mod $8$) are shown to be periodic points of a fixed algebraic function $T(z)$ defined on the punctured disk $0< |z|_2 \le \frac{1}{2}$ of the maximal unramified, algebraic extension $\textsf{K}_2$ of the $2$-adic field $\mathbb{Q}_2$. All ring class fields of odd conductor over imaginary quadratic fields in which the prime $p=2$ splits are shown to be generated by complex periodic points of the algebraic function $T$, and conversely, all but two of the periodic points of $T$ generate ring class fields over suitable imaginary quadratic fields. This gives a dynamical proof of a class number relation originally proved by Deuring. It is conjectured that a similar situation holds for an arbitrary prime $p$ in place of $p=2$, where the case $p=3$ has been previously proved by the author, and the case $p=5$ will be handled in Part II.