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