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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