Modular forms. In this paper we will work with a deï¬�nition of modularity which uses modular forms.\n\nDeï¬�nition. If N is a positive integer, a modular form f of weight k for Î“0(N ) is C which satisï¬�es a holomorphic function f : H\n\nâ†’\n\nf (Î³(z)) = (cz + d)kf (z)\n\na b c d\n\nH,\n\n(3)\n\nÎ“0(N ) and z\n\nfor every Î³ =\n\nâˆˆ\n\nâˆˆ\n\n(cid:1)\n\n(cid:0)\n\nand is holomorphic at the cusps (see Chapter 2 of [35]).\n\n1 1 0 1\n\nÎ“0(N )), so âˆ� n=0 ane2Ï€inz, with complex numbers an and it has a Fourier expansion f (z) = (cid:1) . We say f is a cusp form if it with n vanishes at all the cusps; in particular for a cusp form the coeï¬ƒcient a0 (the value at i\n\nA modular form f satisï¬�es f (z) = f (z + 1) (apply (3) to\n\nâˆˆ\n\n(cid:0)\n\n0 because f is holomorphic at the cusp i\n\nâ‰¥\n\nâˆ�\n\nP\n\n) is zero. Call a cusp form normalized if a1 = 1.\n\nâˆ� For ï¬�xed N there are commuting linear operators (called Hecke operators) Tm, 1, on the (ï¬�nite-dimensional) vector space of cusp forms of weight\n\nfor integers m two for Î“0(N ) (see Chapter 3 of [35]). If f (z) =\n\nâ‰¥\n\nâˆ� n=1 ane2Ï€inz, then\n\nP danm/d2\n\nâˆ�\n\ne2Ï€inz\n\n(4)\n\nTmf (z) =\n\nn=1 X\n\n(d,N )=1 d|(n,m)\n\n(cid:0) X\n\n(cid:1)\n\nwhere (a, b) denotes the greatest common divisor of a and b and a b means that a divides b. The Hecke algebra T (N ) is the ring generated over Z by these operators.\n\n|\n\nDeï¬�nition. In this paper an eigenform will mean a normalized cusp form of weight two for some Î“0(N ) which is an eigenfunction for all the Hecke operators.\n\nâˆ� n=1 ane2Ï€inz is an eigenform, then Tmf = amf for all m.\n\nBy (4), if f (z) =\n\nP\n\n6\n\nK. RUBIN AND A. SILVERBERG\n\n1.7. Modularity, revisited. Suppose E is an elliptic curve over Q. If p is a prime, write Fp for the ï¬�nite ï¬�eld with p elements, and let E(Fp) denote the Fp- solutions of the equation for E (including the point at inï¬�nity). We now give a second deï¬�nition of modularity for an elliptic
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Dropbox is a file hosting service that brings everything-traditional files, cloud content, and web shortcuts together in one place. ->: Modular forms. In this paper we will work with a deï¬�nition of modularity which uses modular forms.\n\nDeï¬�nition. If N is a positive integer, a modular form f of weight k for Î“0(N ) is C which satisï¬�es a holomorphic function f : H\n\nâ†’\n\nf (Î³(z)) = (cz + d)kf (z)\n\na b c d\n\nH,\n\n(3)\n\nÎ“0(N ) and z\n\nfor every Î³ =\n\nâˆˆ\n\nâˆˆ\n\n(cid:1)\n\n(cid:0)\n\nand is holomorphic at the cusps (see Chapter 2 of [35]).\n\n1 1 0 1\n\nÎ“0(N )), so âˆ� n=0 ane2Ï€inz, with complex numbers an and it has a Fourier expansion f (z) = (cid:1) . We say f is a cusp form if it with n vanishes at all the cusps; in particular for a cusp form the coeï¬ƒcient a0 (the value at i\n\nA modular form f satisï¬�es f (z) = f (z + 1) (apply (3) to\n\nâˆˆ\n\n(cid:0)\n\n0 because f is holomorphic at the cusp i\n\nâ‰¥\n\nâˆ�\n\nP\n\n) is zero. Call a cusp form normalized if a1 = 1.\n\nâˆ� For ï¬�xed N there are commuting linear operators (called Hecke operators) Tm, 1, on the (ï¬�nite-dimensional) vector space of cusp forms of weight\n\nfor integers m two for Î“0(N ) (see Chapter 3 of [35]). If f (z) =\n\nâ‰¥\n\nâˆ� n=1 ane2Ï€inz, then\n\nP danm/d2\n\nâˆ�\n\ne2Ï€inz\n\n(4)\n\nTmf (z) =\n\nn=1 X\n\n(d,N )=1 d|(n,m)\n\n(cid:0) X\n\n(cid:1)\n\nwhere (a, b) denotes the greatest common divisor of a and b and a b means that a divides b. The Hecke algebra T (N ) is the ring generated over Z by these operators.\n\n|\n\nDeï¬�nition. In this paper an eigenform will mean a normalized cusp form of weight two for some Î“0(N ) which is an eigenfunction for all the Hecke operators.\n\nâˆ� n=1 ane2Ï€inz is an eigenform, then Tmf = amf for all m.\n\nBy (4), if f (z) =\n\nP\n\n6\n\nK. RUBIN AND A. SILVERBERG\n\n1.7. Modularity, revisited. Suppose E is an elliptic curve over Q. If p is a prime, write Fp for the ï¬�nite ï¬�eld with p elements, and let E(Fp) denote the Fp- solutions of the equation for E (including the point at inï¬�nity). We now give a second deï¬�nition of modularity for an elliptic