Index 267

reduced, 234

reducing point, 230

regular, 219

relative to the cusps, 39

restriction of scalars, 207

right action of SL2(Z), 44, 125

right translation, 209

satisfies condition Cn, 131

self-adjoint homogeneous cone, 248

Semisimple, 207

set of cusps, 5

Set of real points, 207

sharbly complex, 233

sigma function, 15

slowly increasing, 209

split form of SLn, 207

split symplectic group, 208

standard fundamental domain, 17

star involution, 141

strong deformation retract, 219

symplectic sharbly complex, 250

tilings, 245

topological cell, 218

transportable, 182

unimodular, 229

virtual cohomological dimension, 215

Voronoˇ ı decomposition, 219

Voronoˇ ı polyhedron, 215

Voronoˇ ı reduction algorithm, 218

weakly modular function, 3, 5

Weierstrass ℘-function, 14

weight, 3, 4, 7

weight k modular symbols for G, 123

weight k right action, 5

well-rounded retract, 219

width of the cusp, 6, 8

SAGE Index

SAGE , ix, xii, 2, 15, 16, 20, 22, 26, 30,

41, 43, 45, 51, 52, 56, 58, 63, 65–67,

74, 77, 78, 85, 89, 95, 106, 144, 161,

163, 198

M36, 28

q-expansion of Δ, 15

SL2(Z), 2

Z/NZ, 65

basis for M24, 20

basis for S2(Γ0(N)), 56

Bernoulli numbers, 16

Bernoulli numbers modulo p, 30

boundary map, 52

continued fraction convergents, 43

cuspidal submodule, 52

dimension formulas, 93

dimension Sk(Γ0(N)), 95

dimension Sk(Γ1(N)), 97

dimension with character, 101, 161

Dirichlet character tutorial, 78

Dirichlet group, 67

echelon form, 112

Eisenstein arithmetic, 26

Eisenstein series, 89

evaluation of character, 67

generalized Bernoulli numbers, 85

Hecke operators M2(Γ0(39)), 50

Hecke operators M2(Γ0(6)), 49

Hecke operator T2, 49

Heilbronn matrices, 49

Manin symbols, 45

Miller basis, 22

modular symbols, 44

modular symbols of level 11, 41

modular symbols printing, 46

rational reconstruction, 106

General Index

Basmaji’s trick, 133

Bernoulli numbers

generalized, 83

Birch and Swinnerton-Dyer conjecture,

10

boundary map, 134

computing, 51

boundary modular symbols

and Manin symbols, 134

congruent number problem, 10

conjecture

Maeda, 28

Shimura-Taniyama, 37

cusp forms

Δ, 14

for Γ, 134

higher level dimension, 92, 96

cuspidal modular symbols

and Manin symbols, 134

cusps

action of SL2(Z) on, 5

and boundary map, 134

criterion for vanishing, 136

dimension

cusp forms of higher level, 92, 96

Diophantine equations, 10

Dirichlet character, 142

and cusps, 136

Eisenstein series, 13

algorithm to enumerate, 88

and Bernoulli numbers, 83

are eigenforms, 88

basis of, 88

compute, 63

compute using SAGE, 89

Fourier expansion, 15