DETERMINATION OF ALGEBRAIC POINTS OF LOW DEGREE ON A FAMILY CURVES

Abstract

Let n be a positive integer. The purpose of this paper is to determine explicitly algebraic points of low degree over Q on the family curves of affine equation
y³ⁿ=x⁴ⁿ-1. An algebraic point of degree at most 2 is called an algebraic point of low degree. The essential tools for this determination are the theorem of
Chevalley-Weil, cyclotomic polynomials, rational morphisms between these curves and the special Picard curve of affine equation y³=x⁴-1.