ON TWISTS OF THE MODULAR CURVES $X(p)$

Abstract

Let $p{>}2$ be a prime, and let $k$ be a field of characteristic zero, linearly disjoint from the $p$th cyclotomic extension of $\Q$. Given a projective Galois representation $\varrho \colon \Gal(\kbar/k) {\To} \PGL_2(\F_p)$ with cyclotomic determinant, two twists $X_\varrho(p)$ and $ X'_\varrho(p)$ of a certain rational model of the modular curve $X(p)$ can be attached to it. The $k$-rational points of these twists classify the elliptic curves $E/k$ such that ${\rhobar}_{E, p}{=}\varrho$, where ${\rhobar}_{E, p}$ denotes the projective Galois representation associated with the $p$-torsion module $E[p]$. The octahedral ($p{=}3$) and icosahedral ($p{=}5$) genus-zero cases are discussed in further detail.