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 pth cyclotomic extension of Q. Given a projective Galois representation G a l ( k ¯ / k ) → P G L 2 ( F p ) with cyclotomic determinant, two twists Xϱ(p) and X′ϱ(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 ρ ¯ E , p = ρ , where ρ ¯ 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. 2000 Mathematics Subject Classification 11G05, 14G05, 11R32.