2014/10/31 by Patrick Morton
Mathematics · #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #Algebraic number #Algebraic number field #Class (philosophy) #Cubic form #Discriminant #Fermat number #Fermat's Last Theorem #Fermat's little theorem #Field (mathematics) #Meromorphic and Entire Functions #Quadratic equation #Quadratic field #math.NT
paper · pdf · doi:10.1142/s179304211650055x
published as International Journal of Number Theory Vol. 12, No. 4 (2016) 853-902
arxiv created 2015/06/20 · openalex publication_date 2015/08/21 · arxiv updated 2016/04/15 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Explicit solutions of the cubic Fermat equation are constructed in ring class fields [Formula: see text], with conductor [Formula: see text] prime to [Formula: see text], of any imaginary quadratic field [Formula: see text] whose discriminant satisfies [Formula: see text] (mod [Formula: see text]), in terms of the Dedekind [Formula: see text]-function. As [Formula: see text] and [Formula: see text] 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 [Formula: see text] of the [Formula: see text]-adic field [Formula: see text]. 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 [Formula: see text] if [Formula: see text] (mod [Formula: see text]) and the class number [Formula: see text] is not divisible by [Formula: see text]. If [Formula: see text], congruence conditions for the trace of specific elements of [Formula: see text] are exhibited which imply the existence of a point of infinite order in [Formula: see text].