are four modular elliptic curves, corresponding to a modular form for Î“0(50) (see p. 86 of [1]), which lie in the four distinct C-isomorphism classes that correspond to the noncusp rational points on X0(15).\n\n) where Eâ€² is an elliptic curve over Q and\n\nC\n\nC âŠ‚\n\nâˆ’\n\nTherefore every elliptic curve over Q with a GQ-stable subgroup of order 15 is modular. Equivalently, if E is an elliptic curve over Q and both Â¯Ï�E,3 and Â¯Ï�E,5 are reducible, then E is modular.\n\nB.2. Fix a semistable elliptic curve E over Q. We will show that there are inï¬�nitely many semistable elliptic curves Eâ€² over Q such that\n\n(i) Â¯Ï�Eâ€²,5 is isomorphic to Â¯Ï�E,5, and (ii) Â¯Ï�Eâ€²,3 is irreducible. Let\n\n1 0 0 1\n\na b c d\n\na b c d\n\nSL2(Z) :\n\n(mod 5) }\n\n.\n\nÎ“(5) =\n\nâ‰¡\n\nâˆˆ\n\n{\n\nLet X be the twist of the classical modular curve X(5) (see [35]) by the cocycle (cid:0) induced by Â¯Ï�E,5, and let S be the set of cusps of X. Then X is a curve deï¬�ned over Q which has the following properties. The rational points on X âˆ’ (Eâ€², Ï†) where Eâ€² is an elliptic curve over Q and Ï† : E[5] module isomorphism.\n\n(cid:1)\n\n(cid:0)\n\n(cid:1)\n\n(cid:1)\n\n(cid:0)\n\nS correspond to isomorphism classes of pairs Eâ€²[5] is a GQ-\n\n\n\nâ†’\n\n22\n\nK. RUBIN AND A. SILVERBERG\n\nS is four copies of H/Î“(5), so each component of\n\nAs a complex manifold X X has genus zero.\n\n\n\nâˆ’\n\nLet X 0 be the component of X containing the rational point corresponding to (E, identity). Then X 0 is a curve of genus zero deï¬�ned over Q with a rational point, so it has inï¬�nitely many rational points. We want to show that inï¬�nitely many of these points correspond to semistable elliptic curves Eâ€² with Â¯Ï�Eâ€²,3 irreducible.\n\nThere is another modular curve Ë†X deï¬�ned over Q, with a ï¬�nite set Ë†S of cusps,\n\nwhich has the following properties. The rational points on Ë†X (Eâ€², Ï†, module isomorphism, and As a complex manifold Ë†X The map that forgets the subgroup X
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: are four modular elliptic curves, corresponding to a modular form for Î“0(50) (see p. 86 of [1]), which lie in the four distinct C-isomorphism classes that correspond to the noncusp rational points on X0(15).\n\n) where Eâ€² is an elliptic curve over Q and\n\nC\n\nC âŠ‚\n\nâˆ’\n\nTherefore every elliptic curve over Q with a GQ-stable subgroup of order 15 is modular. Equivalently, if E is an elliptic curve over Q and both Â¯Ï�E,3 and Â¯Ï�E,5 are reducible, then E is modular.\n\nB.2. Fix a semistable elliptic curve E over Q. We will show that there are inï¬�nitely many semistable elliptic curves Eâ€² over Q such that\n\n(i) Â¯Ï�Eâ€²,5 is isomorphic to Â¯Ï�E,5, and (ii) Â¯Ï�Eâ€²,3 is irreducible. Let\n\n1 0 0 1\n\na b c d\n\na b c d\n\nSL2(Z) :\n\n(mod 5) }\n\n.\n\nÎ“(5) =\n\nâ‰¡\n\nâˆˆ\n\n{\n\nLet X be the twist of the classical modular curve X(5) (see [35]) by the cocycle (cid:0) induced by Â¯Ï�E,5, and let S be the set of cusps of X. Then X is a curve deï¬�ned over Q which has the following properties. The rational points on X âˆ’ (Eâ€², Ï†) where Eâ€² is an elliptic curve over Q and Ï† : E[5] module isomorphism.\n\n(cid:1)\n\n(cid:0)\n\n(cid:1)\n\n(cid:1)\n\n(cid:0)\n\nS correspond to isomorphism classes of pairs Eâ€²[5] is a GQ-\n\n\n\nâ†’\n\n22\n\nK. RUBIN AND A. SILVERBERG\n\nS is four copies of H/Î“(5), so each component of\n\nAs a complex manifold X X has genus zero.\n\n\n\nâˆ’\n\nLet X 0 be the component of X containing the rational point corresponding to (E, identity). Then X 0 is a curve of genus zero deï¬�ned over Q with a rational point, so it has inï¬�nitely many rational points. We want to show that inï¬�nitely many of these points correspond to semistable elliptic curves Eâ€² with Â¯Ï�Eâ€²,3 irreducible.\n\nThere is another modular curve Ë†X deï¬�ned over Q, with a ï¬�nite set Ë†S of cusps,\n\nwhich has the following properties. The rational points on Ë†X (Eâ€², Ï†, module isomorphism, and As a complex manifold Ë†X The map that forgets the subgroup X