216 INDEX

Modular form, 3, 3, 4

complex multiplication, 9

half-integral weight, 11

holomorphic, 3, 11

integral weight, 3

meromorphic, 3, 11

newform, 29

p-adic, 74

weakly holomorphic, 3, 11

Modular function, 3

Nebentypus character, 4, 11

Newform, 29, 55

Newman’s Conjecture, 105

p-adic Γ-function, 199

Parkin-Shanks Conjecture, 101

Partition, 85

function p(n), 85

Petersson inner product, 29

Projection operator, 51

Quadratic twist, 22, 52

of an elliptic curve, 152

of a modular form, 22, 52

Ramanujan’s 1Ψ1 summation, 166

Ramanujan’s congruences for p(n), 87

Ramanujan-Petersson Conjecture, 31

Ramanujan’s tau-function, 41

Ramanujan’s Theta-operator, 23, 36

Rank of a partition, 88

Rogers-Fine identity, 166

Selmer group, 157

Shafarevich-Tate group, 157

Shimura correspondence, 53

Shimura-Taniyama Conjecture, 151

Siegel-Walfisz condition, 144

Sign of the functional equation, 150

Singular K3 surface, 193

Singular modulus, 121

Slash operator, 3, 51

Sturm’s Theorem, 40

Subbarao’s Conjecture, 102

Supersingular j-invariant, 36

Teichm¨ uller character, 135, 201

Theta function, 12

Theta operator, 23, 36

Traces of Hecke operators, 196

Traces of singular moduli, 124

Trinomial coeﬃcient, 196

Twist of a modular form, 22, 52

U -operator, 28, 50

V -operator, 28, 50

Valence formula, 8

Weierstrass point, 109

Weierstrass weight, 109