266 Index

Definition Index

Γ-invariant on the left, 206

k-sharblies, 233

q-expansion, 4

Q-rank, 245

abelian variety attached to f, 178

action of Hecke operators, 139

antiholomorphic, 137

arithmetic group, 208

associate proper Q-parabolic subgroups

of G, 212

automorphic form, 209

automorphy factor, 205

Bernoulli numbers, 16

Bianchi groups, 247

Borel conjecture, 212

boundary map, 40, 134

bounded domains, 211

cellular decomposition, 219

character of the modular form, 160

Cholesky decomposition, 214

codimension, 219

complex upper half plane, 1

conductor, 71

congruence subgroup, 4, 208

congruence subgroup problem, 7

Connected, 207

critical integers, 138

cross polytope, 238

cusp form, 4

cuspidal, 209

cuspidal automorphic form, 210

cuspidal cohomology, 212

cuspidal modular symbols, 40, 134

cusps for a congruence subgroup Γ, 5

Defined over Q, 207

degeneracy map, 59, 161

diamond-bracket action, 160

diamond-bracket operators, 128, 159

dimension, 219

Dirichlet character, 64

divisor, xiii

echelon form, 103

eigenforms, 59

Eilenberg–Mac Lane, 211

Eisenstein cohomology, 212

Eisenstein series, 210

Eisenstein subspace, 83

extended modular symbols, 179

extended upper half plane, 6

fan, 218

Farey tessellation, 220

formal power series, 4

Fourier expansion, 3

generalized Bernoulli numbers, 83

generalized modular symbol, 251

Grothendieck motive, 179

group cohomology, 211

Hecke algebra, 54, 83, 128

Hecke correspondence, 225

Hecke operator, 37, 128, 226

Hecke polynomials, 241

height, 107

Hermite normal form, 120, 240

Hermitian symmetric spaces, 211

holomorphic, 2

holomorphic at ∞, 4

holomorphic at the cusp α, 7

Humbert forms, 244

hypersimplices, 246

Krylov methods, 116

Krylov subspace, 116

Laplace–Beltrami–Casimir operator, 209

left action of G, 123

left action of GL2(Q), 40

left action of SL2(Z), 133

left translations, 208

level 1, 4

level of Γ, 4

linear fractional transformations, 1

Maass forms, 210

Manin symbol, 124

meromorphic, 2

meromorphic at ∞, 4

Miller basis, 20

modular complex, 244

modular elliptic curves, 187

modular form, 4, 7

modular function, 4

modular group, 2

modular symbols, 228

modular symbols algorithm, 229

modular symbols for Γ0(N), 40

modular symbols over a ring R, 124

newform, 59, 164

new modular symbols, 143

new subspace, 59, 162

nonnormalized weight k Eisenstein series,

13

normalized Eisenstein series, 17

old modular symbols, 144

old subspace, 161

opposite, 222

perfect, 216

perfection, 244

pivot column, 103

plus one quotient, 165

primitive, 71, 215

primitive character associated to, 71

principal congruence subgroup, 208

Ramanujan function, 25

rational Jordan form, 114

rational period mapping, 185

real-analytic, 210