eigenform f and a prime ideal Î» of\n\nmodular if there are an\n\nGL2(k) is Of such that Ï�f,Î» is a type-\n\nâ†’\n\nD\n\nlifting of Â¯Ï�.\n\nD\n\nRemarks. (i) A representation with a type- fore if a representation is\n\nlifting must itself be type-\n\n. There-\n\nD\n\nD and modular.\n\nmodular, then it is both type-\n\nD\n\nD\n\n(ii) Conversely, if Â¯Ï� is type-\n\n, modular, and satisï¬�es (ii) of Theorem 5.3 below, -modular, by work of Ribet and others (see [28]). This plays an important\n\nD\n\nthen Â¯Ï� is D role in Wilesâ€™ work.\n\n12\n\nK. RUBIN AND A. SILVERBERG\n\n3.5. Mazur Conjecture.\n\nDeï¬�nition. A representation Â¯Ï� : GQ Â¯Ï�\n\nGL2(k) is called absolutely irreducible if\n\nâ†’\n\nÂ¯k is irreducible.\n\nâŠ—\n\nThe following variant of a conjecture of Mazur (see Conjecture 18 of [23]; see\n\nalso Conjecture 4.2 below) implies the Semistable Modular Lifting Conjecture.\n\nConjecture 3.2 (Mazur). Suppose p is an odd prime, k is a ï¬�nite ï¬�eld of charac- GL2(k) is an absolutely irreducible teristic p, lifting of Â¯Ï� to the ring of integers of\n\nis deformation data, and Â¯Ï� : GQ -modular representation. Then every type-\n\nD\n\nâ†’ D\n\nD a ï¬�nite extension of Qp is modular.\n\nRemark . Loosely speaking, Conjecture 3.2 says that if Â¯Ï� is modular, then every lifting which â€œlooks modularâ€� is modular.\n\nDeï¬�nition. An elliptic curve E over Q has good (respectively, bad ) reduction at a prime q if E is nonsingular (respectively, singular) modulo q. An elliptic curve E over Q has ordinary (respectively, supersingular) reduction at q if E has good reduction at q and E[q] has (respectively, does not have) a subgroup of order q stable under the inertia group Iq.\n\nProposition 3.3. Conjecture 3.2 implies Conjecture 2.1.\n\nProof. Suppose p is an odd prime and E is a semistable elliptic curve over Q which satisï¬�es (a) and (b) of Conjecture 2.1. We will apply Conjecture 3.2 with Â¯Ï� = Â¯Ï�E,p. Write Ï„ for complex conjugation. Then Ï„ 2 = 1, and
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: eigenform f and a prime ideal Î» of\n\nmodular if there are an\n\nGL2(k) is Of such that Ï�f,Î» is a type-\n\nâ†’\n\nD\n\nlifting of Â¯Ï�.\n\nD\n\nRemarks. (i) A representation with a type- fore if a representation is\n\nlifting must itself be type-\n\n. There-\n\nD\n\nD and modular.\n\nmodular, then it is both type-\n\nD\n\nD\n\n(ii) Conversely, if Â¯Ï� is type-\n\n, modular, and satisï¬�es (ii) of Theorem 5.3 below, -modular, by work of Ribet and others (see [28]). This plays an important\n\nD\n\nthen Â¯Ï� is D role in Wilesâ€™ work.\n\n12\n\nK. RUBIN AND A. SILVERBERG\n\n3.5. Mazur Conjecture.\n\nDeï¬�nition. A representation Â¯Ï� : GQ Â¯Ï�\n\nGL2(k) is called absolutely irreducible if\n\nâ†’\n\nÂ¯k is irreducible.\n\nâŠ—\n\nThe following variant of a conjecture of Mazur (see Conjecture 18 of [23]; see\n\nalso Conjecture 4.2 below) implies the Semistable Modular Lifting Conjecture.\n\nConjecture 3.2 (Mazur). Suppose p is an odd prime, k is a ï¬�nite ï¬�eld of charac- GL2(k) is an absolutely irreducible teristic p, lifting of Â¯Ï� to the ring of integers of\n\nis deformation data, and Â¯Ï� : GQ -modular representation. Then every type-\n\nD\n\nâ†’ D\n\nD a ï¬�nite extension of Qp is modular.\n\nRemark . Loosely speaking, Conjecture 3.2 says that if Â¯Ï� is modular, then every lifting which â€œlooks modularâ€� is modular.\n\nDeï¬�nition. An elliptic curve E over Q has good (respectively, bad ) reduction at a prime q if E is nonsingular (respectively, singular) modulo q. An elliptic curve E over Q has ordinary (respectively, supersingular) reduction at q if E has good reduction at q and E[q] has (respectively, does not have) a subgroup of order q stable under the inertia group Iq.\n\nProposition 3.3. Conjecture 3.2 implies Conjecture 2.1.\n\nProof. Suppose p is an odd prime and E is a semistable elliptic curve over Q which satisï¬�es (a) and (b) of Conjecture 2.1. We will apply Conjecture 3.2 with Â¯Ï� = Â¯Ï�E,p. Write Ï„ for complex conjugation. Then Ï„ 2 = 1, and