1)(135t2 + 54t + 5) (27t2 + 10t + 1)(27t2 + 18t + 1)\n\n.\n\n(cid:18)\n\n(cid:19)\n\nThis explicitly gives inï¬�nitely many modular elliptic curves over Q which are\n\nnonisomorphic over C.\n\n(For deï¬�nitions of complex multiplication, discriminant, and j-invariant, see any\n\nstandard reference on elliptic curves, such as [39].)\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n21\n\nAppendix A. Galois groups and Frobenius elements\n\nWrite GQ = Gal( Â¯Q/Q). If q is a prime number and\n\nis a prime ideal dividing\n\nQ\n\nq in the ring of integers of Â¯Q, there is a ï¬�ltration\n\nGQ\n\nDQ âŠƒ\n\nIQ\n\nâŠƒ where the decomposition group DQ and the inertia group IQ are deï¬�ned by\n\nDQ = IQ =\n\nÏƒ\n\nGQ : Ïƒ âˆˆ Q âˆˆ DQ : Ïƒx\n\n=\n\n,\n\n{\n\nQ} x (mod\n\nÏƒ\n\n) for all algebraic integers x }\n\n.\n\nâ‰¡ { There are natural identiï¬�cations\n\nQ\n\nDQ/IQ âˆ¼= Gal( Â¯Fq/Fq),\n\nDQ âˆ¼= Gal( Â¯Qq/Qq),\n\nxq of GQ\n\nand FrobQ âˆˆ Gal( Â¯Fq/Fq). If and\n\nDQ/IQ denotes the inverse image of the canonical generator x\n\n7â†’ for some Ïƒ\n\nâ€² is another prime ideal above q, then\n\nâ€² = Ïƒ\n\nQ DQâ€² = ÏƒDQÏƒâˆ’1,\n\nQ\n\nQ\n\nâˆˆ\n\nFrobQâ€² = ÏƒFrobQÏƒâˆ’1.\n\nIQâ€² = ÏƒIQÏƒâˆ’1,\n\nSince we will care about these objects only up to conjugation, we will write Dq and GQ for any representative of a FrobQ. If Ï� is a represen- Iq. We will write Frobq âˆˆ tation of GQ which is unramiï¬�ed at q, then trace(Ï�(Frobq)) and det(Ï�(Frobq)) are well deï¬�ned independent of any choices.\n\nAppendix B. Some details on the proof of Proposition 2.4\n\nB.1. The modular curve X0(15) can be viewed as a curve deï¬�ned over Q in such a way that the noncusp rational points correspond to isomorphism classes (over C) E( Â¯Q) is a subgroup of pairs (Eâ€², 42), of order 15 stable under GQ. An equation for X0(15) is y2 = x(x + 32)(x the elliptic curve discussed in 1. There are eight rational points on X0(15), four of Â§ which are cusps. There are four modular elliptic curves, corresponding
Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place.
Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: 1)(135t2 + 54t + 5) (27t2 + 10t + 1)(27t2 + 18t + 1)\n\n.\n\n(cid:18)\n\n(cid:19)\n\nThis explicitly gives inï¬�nitely many modular elliptic curves over Q which are\n\nnonisomorphic over C.\n\n(For deï¬�nitions of complex multiplication, discriminant, and j-invariant, see any\n\nstandard reference on elliptic curves, such as [39].)\n\nA REPORT ON WILESâ€™ CAMBRIDGE LECTURES\n\n21\n\nAppendix A. Galois groups and Frobenius elements\n\nWrite GQ = Gal( Â¯Q/Q). If q is a prime number and\n\nis a prime ideal dividing\n\nQ\n\nq in the ring of integers of Â¯Q, there is a ï¬�ltration\n\nGQ\n\nDQ âŠƒ\n\nIQ\n\nâŠƒ where the decomposition group DQ and the inertia group IQ are deï¬�ned by\n\nDQ = IQ =\n\nÏƒ\n\nGQ : Ïƒ âˆˆ Q âˆˆ DQ : Ïƒx\n\n=\n\n,\n\n{\n\nQ} x (mod\n\nÏƒ\n\n) for all algebraic integers x }\n\n.\n\nâ‰¡ { There are natural identiï¬�cations\n\nQ\n\nDQ/IQ âˆ¼= Gal( Â¯Fq/Fq),\n\nDQ âˆ¼= Gal( Â¯Qq/Qq),\n\nxq of GQ\n\nand FrobQ âˆˆ Gal( Â¯Fq/Fq). If and\n\nDQ/IQ denotes the inverse image of the canonical generator x\n\n7â†’ for some Ïƒ\n\nâ€² is another prime ideal above q, then\n\nâ€² = Ïƒ\n\nQ DQâ€² = ÏƒDQÏƒâˆ’1,\n\nQ\n\nQ\n\nâˆˆ\n\nFrobQâ€² = ÏƒFrobQÏƒâˆ’1.\n\nIQâ€² = ÏƒIQÏƒâˆ’1,\n\nSince we will care about these objects only up to conjugation, we will write Dq and GQ for any representative of a FrobQ. If Ï� is a represen- Iq. We will write Frobq âˆˆ tation of GQ which is unramiï¬�ed at q, then trace(Ï�(Frobq)) and det(Ï�(Frobq)) are well deï¬�ned independent of any choices.\n\nAppendix B. Some details on the proof of Proposition 2.4\n\nB.1. The modular curve X0(15) can be viewed as a curve deï¬�ned over Q in such a way that the noncusp rational points correspond to isomorphism classes (over C) E( Â¯Q) is a subgroup of pairs (Eâ€², 42), of order 15 stable under GQ. An equation for X0(15) is y2 = x(x + 32)(x the elliptic curve discussed in 1. There are eight rational points on X0(15), four of Â§ which are cusps. There are four modular elliptic curves, corresponding