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 = Q(√-d) with -d Ξ 1 (mod 8) are shown to be periodic points of a fixed algebraic function T(z) defined on the punctured disk 0 < |z|2 ≥ 1/2 of the maximal unramified, algebraic extension K2 of the 2-adic field Q2. 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.