On elliptic Galois representations and genus-zero modular units

Abstract

Given an odd prime p and a representation ϱ of the absolute Galois group of a number field k onto PGL 2 (𝔽 p ) with cyclotomic determinant, the moduli space of elliptic curves defined over k with p-torsion giving rise to ϱ consists of two twists of the modular curve X(p). We make here explicit the only genus-zero cases p=3 and p=5, which are also the only symmetric cases: PGL 2 (𝔽 p )≃𝒮 n for n=4 or n=5, respectively. This is done by studying the corresponding twisted Galois actions on the function field of the curve, for which a description in terms of modular units is given. As a consequence of this twisting process, we recover an equivalence between the ellipticity of ϱ and its principality, that is, the existence in its fixed field of an element α of degree n over k such that α and α 2 have both trace zero over k.