Index Baker’s bound, 24 Bernoulli number, 102 canonical height, 43, 44 congruence subgroup, 90 congruent number problem, 2, 45, 133 conjecture of Lang, 43 Mordell, 20 Ogg, 32 parity, 133 Taniyama-Shimura-Weil, xii, 7, 127, 137, 139 Birch and Swinnerton-Dyer, 6, 127–129, 132, 133, 135 the rank, 31 cusp, 37 cusp form for SL(2, Z), 101 for a congruence subgroup, 106 newform, 117 descent, 49 Diophantine equation, 17 Dirichlet L-function, 12 character, 12 discriminant, 36, 148 Eisenstein series, 82, 101, 116 q-expansion, 102 of level N, 108 elliptic curve, 1, 20, 147 L-function, 123 analytic rank, 133, 135 conductor, 126, 136, 139 discriminant, 36 free part, 30 group structure, 24 minimal discriminant, 36 minimal model, 36 modular, 137, 139, 141 modular parametrization, 140 Mordell-Weil group, 28 over finite fields, 35 rank, 30, 45, 59, 128, 129, 135 real period, 128 regulator, 48, 128 root number, 127, 133 semistable, 139, 141 Tamagawa numbers, 128 torsion subgroup, 30, 32, 128 Weierstrass equation, 21 elliptic function, 82 Weierstrass ℘-function, 82 elliptic height matrix, 47–49 elliptic regulator, 47 Euler product, 12 extended upper half-plane H∗, 88 193
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