Generalizations of Thomae's Formula for Zn Curves (eBook)

Introduction 8
Contents 14
Chapter 1 Riemann Surfaces 20
1.1 Basic Definitions 20
1.1.1 First Properties of Compact Riemann Surfaces 20
1.1.2 Some Examples 23
1.2 The Abel Theorem, the Riemann–Roch Theoremand Weierstrass Points 25
1.2.1 The Abel Theorem and the Jacobi Inversion Theorem 25
1.2.2 The Riemann–Roch Theorem and the Riemann–Hurwitz Formula 28
1.2.3 Weierstrass Points 29
1.3 Theta Functions 32
1.3.1 First Properties of Theta Functions 32
1.3.2 Quotients of Theta Functions 35
1.3.3 Theta Functions on Riemann Surfaces 36
1.3.4 Changing the Basepoint 41
1.3.5 Matching Characteristics 44
Chapter 2 Zn Curves 50
2.1 Nonsingular Zn Curves 50
2.1.1 Functions, Differentials and Weierstrass Points 51
2.1.2 Abel–Jacobi Images of Certain Divisors 53
2.2 Non-Special Divisors of Degree g on Nonsingular Zn Curves 55
2.2.1 An Example with n = 3 and r = 2 56
2.2.2 An Example with n = 5 and r = 3 57
2.2.3 Non-Special Divisors 59
2.2.4 Characterizing All Non-Special Divisors 64
2.3 Singular Zn Curves 67
2.3.1 Functions, Differentials and Weierstrass Points 67
2.3.2 Abel–Jacobi Images of Certain Divisors 69
2.4 Non-Special Divisors of Degree g on Singular Zn Curves 71
2.4.1 An Example with n = 3 and m = 3 71
2.4.2 Non-Special Divisors 73
2.4.3 Characterizing All Non-Special Divisors 75
2.5 Some Operators 78
2.5.1 Operators for the Nonsingular Case 79
2.5.2 Operators for the Singular Case 81
2.5.3 Properties of the Operators in Both Cases 83
2.6 Theta Functions on Zn Curves 86
2.6.1 Non-Special Divisors as Characteristics for Theta Functions 86
2.6.2 Quotients of Theta Functions with Characteristics Represented by Divisors 87
2.6.3 Evaluating Quotients of Theta Functions at Branch Points 88
2.6.4 Quotients of Theta Functions as Meromorphic Functions on Zn Curves 90
Chapter 3 Examples of Thomae Formulae 94
3.1 A Nonsingular Z3 Curve with Six Branch Points 94
3.1.1 First Identities Between Theta Constants 95
3.1.2 The Thomae Formulae 97
3.1.3 Changing the Basepoint 99
3.2 A Singular Z3 Curve with Six Branch Points 102
3.2.1 First Identities between Theta Constants 103
3.2.2 The First Part of the Poor Man’s Thomae 104
3.2.3 Completing the Poor Man’s Thomae 108
3.2.4 The Thomae Formulae 111
3.2.5 Relation with the General Singular Case 113
3.2.6 Changing the Basepoint 115
3.3 A One-Parameter Family of Singular Zn Curves with Four Branch Points 117
3.3.1 Divisors and Operators 117
3.3.2 First Identities Between Theta Constants 119
3.3.3 Even n 123
3.3.4 An Example with n = 10 126
3.3.5 Thomae Formulae for Even n 128
3.3.6 Odd n 133
3.3.7 An Example with n = 9 136
3.3.8 Thomae Formulae for Odd n 137
3.3.9 Changing the Basepoint 139
3.3.10 Relation with the General Singular Case 143
3.4 Nonsingular Zn Curves with r = 1 and Small n 144
3.4.1 The Set of Divisors as a Principal Homogenous Space for Sn 1 144
3.4.2 The Case n = 4 145
3.4.3 Changing the Basepoint for n = 4 148
3.4.4 The Case n = 3 150
3.4.5 The Problem with n = 5 151
3.4.6 The Case n = 5 153
3.4.7 The Orbits for n = 5 156
3.4.8 Changing the Basepoint for n = 5 157
Chapter 4 Thomae Formulae for Nonsingular Zn Curves 161
4.0.1 A Useful Notation 161
4.1 The Poor Man’s Thomae Formulae 163
4.1.1 First Identities Between Theta Constants 163
4.1.2 Symmetrization over R and the Poor Man’s Thomae 166
4.1.3 Reduced Formulae 168
4.2 Example with n = 5 and General r 171
4.2.1 Correcting the Expressions Involving C 1 172
4.2.2 Correcting the Expressions Not Involving C 1 173
4.2.3 Reduction and the Thomae Formulae for n = 5 175
4.3 Invariance also under N 177
4.3.1 The Description of h. for Odd n 177
4.3.2 N-Invariance for Odd n 180
4.3.3 The Description of h. for Even n 183
4.3.4 N-Invariance for Even n 185
4.4 Thomae Formulae for Nonsingular Zn Curves 187
4.4.1 The Case r = 2 188
4.4.2 Changing the Basepoint for r = 2 193
4.4.3 The Case r = 1 194
4.4.4 Changing the Basepoint for r = 1 196
Chapter 5 Thomae Formulae for Singular Zn Curves 201
5.1 The Poor Man’s Thomae Formulae 201
5.1.1 First Identities Between Theta Constants Based on the Branch Point R 202
5.1.2 Symmetrization over R 205
5.1.3 First Identities Between Theta Constants Based on the Branch Point S 208
5.1.4 Symmetrization over S 210
5.1.5 The Poor Man’s Thomae 213
5.1.6 Reduced Formulae 217
5.2 Example with n = 5 and General m 220
5.2.1 Correcting the Expressions Involving C 1 and D 1 222
5.2.2 Correcting the Expressions Not Involving C 1 and D 1 224
5.2.3 Reduction and the Thomae Formulae for n = 5 227
5.3 Invariance also under N 229
5.3.1 The Description of h. for Odd n 230
5.3.2 N-Invariance for Odd n 233
5.3.3 The Description of h. for Even n 237
5.3.4 N-Invariance for Even n 240
5.4 Thomae Formulae for Singular Zn Curves 243
5.4.1 The Thomae Formulae 244
5.4.2 Changing the Basepoint 247
Chapter 6 Some More Singular Zn Curves 251
6.1 A Family of Zn Curves with Four Branch Points and a Symmetric Equation 252
6.1.1 Functions, Differentials, Weierstrass Points and Abel–Jacobi Images 253
6.1.2 Non-Special Divisors in an Example of n = 7 255
6.1.3 Non-Special Divisors in the General Case 259
6.1.4 Operators 264
6.1.5 First Identities Between Theta Constants 266
6.1.6 The Poor Man’s Thomae (Unreduced and Reduced) 270
6.1.7 The Thomae Formulae in the Case n = 7 275
6.1.8 The Thomae Formulae in the General Case 278
6.1.9 Changing the Basepoint 283
6.2 A Family of Zn Curves with Four Branch Points and an Asymmetric Equation 286
6.2.1 An Example with n = 10 286
6.2.2 Non-Special Divisors for n = 10 289
6.2.3 The Basic Data for General n 290
6.2.4 Non-Special Divisors for General n 294
6.2.5 Operators and Theta Quotients 297
6.2.6 Thomae Formulae for t = 1 299
6.2.7 Thomae Formulae for t = 2 301
6.2.8 Changing the Basepoint 303
Appendix A 307
Constructions and Generalizations for the Nonsingular and Singular Cases 307
A.1 The Proper Order to do the Corrections in the Nonsingular Case 308
A.2 Nonsingular Case, Odd n 309
A.3 Nonsingular Case, Even n 314
A.4 The Proper Order to do the Corrections in the Singular Case 318
A.5 Singular Case, Odd n 319
A.6 Singular Case, Even n 326
A.7 The General Family 330
A.8 Proof of Theorem A.2 336
Appendix B 342
The Construction and Basepoint Change Formulae for the Symmetric Equation Case 342
B.1 Description of the Process 342
B.2 The Case n = 1(mod 4) 345
B.3 The Case n = 3(mod 4) 351
B.4 The Operators for the Other Basepoints 356
B.5 The Expressions for h. for the Other Basepoints 359
References 363
List of Symbols 364
Index 367