manifold Ë†X The map that forgets the subgroup X deï¬�ned over Q and of degree [Î“(5) : Î“(5)\n\nË†S correspond to isomorphism classes of triples Eâ€²[5] is a GQ-\n\n\n\nâˆ’\n\n) where Eâ€² is an elliptic curve over Q, Ï† : E[5]\n\nC\n\nâ†’\n\nEâ€²[3] is a GQ-stable subgroup of order 3.\n\nC âŠ‚ âˆ’\n\nË†S is four copies of H/(Î“(5)\n\nÎ“0(3)).\n\nâ€¢\n\nâˆ© induces a surjective morphism Î¸ : Ë†X\n\nC\n\nâ†’\n\nÎ“0(3)] = 4.\n\nâˆ©\n\nLet Ë†X 0 be the component of Ë†X which maps to X 0. The function ï¬�eld of X 0 is Q(t), and the function ï¬�eld of Ë†X 0 is Q(t)[x]/f (t, x) where f (t, x) Q(t)[x] is irreducible and has degree 4 in x. If tâ€² Q is suï¬ƒciently close 5-adically to the value of t which corresponds to E, then the corresponding elliptic curve is semistable at Q so that f (t1, x) is 5. By the Hilbert Irreducibility Theorem we can ï¬�nd a t1 âˆˆ irreducible in Q[x]. It is possible to ï¬�x a prime â„“ = 5 such that f (t1, x) has no roots modulo â„“. If tâ€² Q is suï¬ƒciently close â„“-adically to t1, then f (tâ€², x) has no rational roots, and thus tâ€² corresponds to a rational point of X 0 which is not the image of a rational point of Ë†X 0. Therefore there are inï¬�nitely many elliptic curves Eâ€² over Q which are semistable at 5 and satisfy\n\nâˆˆ\n\nâˆˆ\n\n6\n\nâˆˆ\n\n(i) Eâ€²[5] âˆ¼= E[5] as GQ-modules, and (ii) Eâ€²[3] has no subgroup of order 3 stable under GQ.\n\nIt follows from (i) and the semistability of E that Eâ€² is semistable at all primes = 5, and thus Eâ€² is semistable. We therefore have inï¬�nitely many semistable q elliptic curves Eâ€² which satisfy the desired conditions.\n\n6\n\nAppendix C. Representation types\n\nSuppose A is a complete noetherian local Zp-algebra and Ï� : GQ\n\nGL2(A) is a |Dp for the restriction of Ï� to the decomposition group Dp.\n\nâ†’\n\nrepresentation. Write Ï� We say Ï� is\n\nordinary at p if Ï�\n\n|Dp is (after a change of basis, if necessary) of the form ï¬‚at at p if Ï� is not ordinary, and for every ideal
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: manifold Ë†X The map that forgets the subgroup X deï¬�ned over Q and of degree [Î“(5) : Î“(5)\n\nË†S correspond to isomorphism classes of triples Eâ€²[5] is a GQ-\n\n\n\nâˆ’\n\n) where Eâ€² is an elliptic curve over Q, Ï† : E[5]\n\nC\n\nâ†’\n\nEâ€²[3] is a GQ-stable subgroup of order 3.\n\nC âŠ‚ âˆ’\n\nË†S is four copies of H/(Î“(5)\n\nÎ“0(3)).\n\nâ€¢\n\nâˆ© induces a surjective morphism Î¸ : Ë†X\n\nC\n\nâ†’\n\nÎ“0(3)] = 4.\n\nâˆ©\n\nLet Ë†X 0 be the component of Ë†X which maps to X 0. The function ï¬�eld of X 0 is Q(t), and the function ï¬�eld of Ë†X 0 is Q(t)[x]/f (t, x) where f (t, x) Q(t)[x] is irreducible and has degree 4 in x. If tâ€² Q is suï¬ƒciently close 5-adically to the value of t which corresponds to E, then the corresponding elliptic curve is semistable at Q so that f (t1, x) is 5. By the Hilbert Irreducibility Theorem we can ï¬�nd a t1 âˆˆ irreducible in Q[x]. It is possible to ï¬�x a prime â„“ = 5 such that f (t1, x) has no roots modulo â„“. If tâ€² Q is suï¬ƒciently close â„“-adically to t1, then f (tâ€², x) has no rational roots, and thus tâ€² corresponds to a rational point of X 0 which is not the image of a rational point of Ë†X 0. Therefore there are inï¬�nitely many elliptic curves Eâ€² over Q which are semistable at 5 and satisfy\n\nâˆˆ\n\nâˆˆ\n\n6\n\nâˆˆ\n\n(i) Eâ€²[5] âˆ¼= E[5] as GQ-modules, and (ii) Eâ€²[3] has no subgroup of order 3 stable under GQ.\n\nIt follows from (i) and the semistability of E that Eâ€² is semistable at all primes = 5, and thus Eâ€² is semistable. We therefore have inï¬�nitely many semistable q elliptic curves Eâ€² which satisfy the desired conditions.\n\n6\n\nAppendix C. Representation types\n\nSuppose A is a complete noetherian local Zp-algebra and Ï� : GQ\n\nGL2(A) is a |Dp for the restriction of Ï� to the decomposition group Dp.\n\nâ†’\n\nrepresentation. Write Ï� We say Ï� is\n\nordinary at p if Ï�\n\n|Dp is (after a change of basis, if necessary) of the form ï¬‚at at p if Ï� is not ordinary, and for every ideal