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- Publisher
- Oxford University Press
- Copyright
- © London Mathematical Society
- ISSN
- 0024-6093
- eISSN
- 1469-2120
- DOI
- 10.1112/S0024609304004187
- Publisher site
- See Article on Publisher Site
Abstract
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 Gal(k¯/k)→PGL2(Fp) 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. © London Mathematical Society
Journal
Bulletin of the London Mathematical Society – Oxford University Press
Published: Jun 1, 2005