194 Index Fermat, 4, 6, 45 Fibonacci, 3 finite field, 35 fundamental domain of a lattice, 79 of a modular curve, 85 fundamental theorem of algebra, 160 Hasse, 39 Hasse’s bound, 39 Hecke operator Tn, 114 Um and Vm, 114 wN , 112 diamond, 112 height, 43 Hensel’s lemma, 65, 181 Hilbert, 18 10th problem, 18 homogeneous space, 59, 61, 62, 64 isomorphism of curves, 22 j-invariant, 148 Jacobi symbol, 8 Kronecker symbol, 14 L-function, 11 Euler product, 138 local factor, 124 of a modular form, 135, 137, 139 of an elliptic curve, 123, 124, 127, 135, 137, 139 of Dirichlet, 12 of Hasse-Weil, 124 root number, 127, 133 functional equation, 126, 138 lattice, 77 Legendre symbol, 14 linear independence, 46 minimal discriminant, 36 minimal model, 36 modular j-invariant, 103 modular curve, 87 algebraic model, 93 cusp, 89, 91 for SL(2, Z), 89 for a congruence subgroup, 91 modular discriminant, 103 modular form, 7 L-function, 135 cusp form, 101, 135, 137, 139 eigenform, 115, 138 for SL(2, Z), 101 for a congruence subgroup, 106 level, 105 new form, 111, 139 newform, 117, 139 normalized, 102 of an elliptic curve, 137 old form, 108 normalized eigenform, 116 modular function, 99, 100 weakly, 100 Néron-Tate pairing, 47, 48 node, 37 PARI/GP, 147 Parity conjecture, 133 Petersson inner product, 110 point at infinity, 21 rank, 30, 128, 129 analytic, 133 reduction of an elliptic curve, 37, 124, 126 additive, 37 good, 37 non-split multiplicative, 37, 125 split multiplicative, 37, 38, 124 regular prime, 141 regulator of an elliptic curve, 48, 128 Riemann zeta function, 12 Sage, 147 Selmer group, 66–68 semistable, 139, 141 Shafarevich-Tate group, 66–68, 128 singular curve, 35–37, 72, 176 cusp, 37 node, 37 smooth curve, 20, 35, 176 theorem of

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