 

Patrick Morton
Solutions of diophantine equations as periodic points of padic algebraic functions. I view print


Published: 
July 26, 2016

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


Abstract
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 K_{2} of the 2adic field 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.


Author information
Dept. of Mathematical Sciences, Indiana University  Purdue University at Indianapolis (IUPUI), 402 N. Blackford St., LD 270, Indianapolis, Indiana, 46202
pmorton@iupui.edu

