New York Journal of Mathematics
Volume 22 (2016) 715-740


Patrick Morton

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

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Published: July 26, 2016
Keywords: Periodic points, algebraic function, 2-adic field, ring class fields, quartic Fermat equation
Subject: 11D41,11G07,11G15,14H05

Solutions of the quartic Fermat equation in ring class fields of odd conductor over quadratic fields K=Q(\sqrt{-d}) with -d ≡ 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.

Author information

Dept. of Mathematical Sciences, Indiana University - Purdue University at Indianapolis (IUPUI), 402 N. Blackford St., LD 270, Indianapolis, Indiana, 46202