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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