ai are integers. The solution ( on the elliptic curve.\n\n, âˆ�\n\n) will be viewed as a point\n\nâˆ�\n\n4\n\nK. RUBIN AND A. SILVERBERG\n\nRemarks. (i) A singular point on a curve f (x, y) = 0 is a point where both partial derivatives vanish. A curve is nonsingular if it has no singular points.\n\n(ii) Two elliptic curves over Q are isomorphic if one can be obtained from the other by changing coordinates x = A2xâ€² + B, y = A3yâ€² + Cxâ€² + D, with A, B, C, D\n\nQ and dividing through by A6.\n\nâˆˆ (iii) Every elliptic curve over Q is isomorphic to one of the form\n\ny2 = x3 + a2x2 + a4x + a6\n\nwith integers ai. A curve of this form is nonsingular if and only if the cubic on the right side has no repeated roots.\n\nExample. The equation y2 = x(x + 32)(x\n\n42) deï¬�nes an elliptic curve over Q.\n\nâˆ’\n\n1.4. Modularity. Let H denote the complex upper half plane C : Im(z) > 0 } where Im(z) is the imaginary part of z. If N is a positive integer, deï¬�ne a group of matrices\n\nz\n\n{\n\nâˆˆ\n\na b c d\n\nSL2(Z) : c is divisible by N\n\n.\n\nÎ“0(N ) =\n\nâˆˆ\n\n(z) = az+b The group Î“0(N ) acts on H by linear fractional transformations cz+d . (cid:9) (cid:1) The quotient space H/Î“0(N ) is a (noncompact) Riemann surface. It can be com- pleted to a compact Riemann surface, denoted X0(N ), by adjoining a ï¬�nite set of points called cusps. The cusps are the ï¬�nitely many equivalence classes of Q âˆ�} under the action of Î“0(N ) (see Chapter 1 of [35]). The complex points of an elliptic curve can also be viewed as a compact Riemann surface.\n\na b c d\n\n(cid:8)(cid:0)\n\n(cid:1)\n\n(cid:0)\n\ni\n\nâˆª{\n\nDeï¬�nition. An elliptic curve E is modular if, for some integer N , there is a holo- morphic map from X0(N ) onto E.\n\nExample. It can be shown that there is a (holomorphic) isomorphism from X0(15) onto the elliptic curve y2 = x(x + 32)(x\n\n42).\n\nâˆ’\n\nRemark . There are many equivalent deï¬�nitions of modularity (see II.4.D of [24] and appendix of [22]). In
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: ai are integers. The solution ( on the elliptic curve.\n\n, âˆ�\n\n) will be viewed as a point\n\nâˆ�\n\n4\n\nK. RUBIN AND A. SILVERBERG\n\nRemarks. (i) A singular point on a curve f (x, y) = 0 is a point where both partial derivatives vanish. A curve is nonsingular if it has no singular points.\n\n(ii) Two elliptic curves over Q are isomorphic if one can be obtained from the other by changing coordinates x = A2xâ€² + B, y = A3yâ€² + Cxâ€² + D, with A, B, C, D\n\nQ and dividing through by A6.\n\nâˆˆ (iii) Every elliptic curve over Q is isomorphic to one of the form\n\ny2 = x3 + a2x2 + a4x + a6\n\nwith integers ai. A curve of this form is nonsingular if and only if the cubic on the right side has no repeated roots.\n\nExample. The equation y2 = x(x + 32)(x\n\n42) deï¬�nes an elliptic curve over Q.\n\nâˆ’\n\n1.4. Modularity. Let H denote the complex upper half plane C : Im(z) > 0 } where Im(z) is the imaginary part of z. If N is a positive integer, deï¬�ne a group of matrices\n\nz\n\n{\n\nâˆˆ\n\na b c d\n\nSL2(Z) : c is divisible by N\n\n.\n\nÎ“0(N ) =\n\nâˆˆ\n\n(z) = az+b The group Î“0(N ) acts on H by linear fractional transformations cz+d . (cid:9) (cid:1) The quotient space H/Î“0(N ) is a (noncompact) Riemann surface. It can be com- pleted to a compact Riemann surface, denoted X0(N ), by adjoining a ï¬�nite set of points called cusps. The cusps are the ï¬�nitely many equivalence classes of Q âˆ�} under the action of Î“0(N ) (see Chapter 1 of [35]). The complex points of an elliptic curve can also be viewed as a compact Riemann surface.\n\na b c d\n\n(cid:8)(cid:0)\n\n(cid:1)\n\n(cid:0)\n\ni\n\nâˆª{\n\nDeï¬�nition. An elliptic curve E is modular if, for some integer N , there is a holo- morphic map from X0(N ) onto E.\n\nExample. It can be shown that there is a (holomorphic) isomorphism from X0(15) onto the elliptic curve y2 = x(x + 32)(x\n\n42).\n\nâˆ’\n\nRemark . There are many equivalent deï¬�nitions of modularity (see II.4.D of [24] and appendix of [22]). In