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On Twists of the Modular Curves X ( p )
Author(s) -
Fernández Julio,
Lario JoanC.,
Rio Anna
Publication year - 2005
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609304004187
Subject(s) - mathematics , elliptic curve , prime (order theory) , torsion (gastropod) , combinatorics , galois module , zero (linguistics) , pure mathematics , discrete mathematics , medicine , linguistics , philosophy , surgery
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 G a l ( k ¯ / k ) → P G L 2 ( F p )with cyclotomic determinant, two twists X ϱ ( 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 thatρ ¯E , p = ρ , whereρ ¯E , pdenotes 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.