Farkas / Zemel Generalizations of Thomae's Formula for Zn Curves

1. Auflage 2010
ISBN: 978-1-4419-7847-9
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band 21, 354 Seiten

Reihe: Developments in Mathematics

Previous publications on the generalization of the Thomae formulae to Zn curves have emphasized the theory's implications in mathematical physics and depended heavily on applied mathematical techniques. This book redevelops these previous results demonstrating how they can be derived directly from the basic properties of theta functions as functions on compact Riemann surfaces. 'Generalizations of Thomae's Formula for Zn Curves' includes several refocused proofs developed in a generalized context that is more accessible to researchers in related mathematical fields such as algebraic geometry, complex analysis, and number theory. This book is intended for mathematicians with an interest in complex analysis, algebraic geometry or number theory as well as physicists studying conformal field theory.

Autoren/Hrsg.

Farkas, Hershel M.

Zemel, Shaul

Inhaltsverzeichnis

1;Introduction;8
2;Contents;14
3;Chapter 1 Riemann Surfaces;20
3.1;1.1 Basic Definitions;20
3.1.1;1.1.1 First Properties of Compact Riemann Surfaces;20
3.1.2;1.1.2 Some Examples;23
3.2;1.2 The Abel Theorem, the Riemann–Roch Theoremand Weierstrass Points;25
3.2.1;1.2.1 The Abel Theorem and the Jacobi Inversion Theorem;25
3.2.2;1.2.2 The Riemann–Roch Theorem and the Riemann–Hurwitz Formula;28
3.2.3;1.2.3 Weierstrass Points;29
3.3;1.3 Theta Functions;32
3.3.1;1.3.1 First Properties of Theta Functions;32
3.3.2;1.3.2 Quotients of Theta Functions;35
3.3.3;1.3.3 Theta Functions on Riemann Surfaces;36
3.3.4;1.3.4 Changing the Basepoint;41
3.3.5;1.3.5 Matching Characteristics;44
4;Chapter 2 Zn Curves;50
4.1;2.1 Nonsingular Zn Curves;50
4.1.1;2.1.1 Functions, Differentials and Weierstrass Points;51
4.1.2;2.1.2 Abel–Jacobi Images of Certain Divisors;53
4.2;2.2 Non-Special Divisors of Degree g on Nonsingular Zn Curves;55
4.2.1;2.2.1 An Example with n = 3 and r = 2;56
4.2.2;2.2.2 An Example with n = 5 and r = 3;57
4.2.3;2.2.3 Non-Special Divisors;59
4.2.4;2.2.4 Characterizing All Non-Special Divisors;64
4.3;2.3 Singular Zn Curves;67
4.3.1;2.3.1 Functions, Differentials and Weierstrass Points;67
4.3.2;2.3.2 Abel–Jacobi Images of Certain Divisors;69
4.4;2.4 Non-Special Divisors of Degree g on Singular Zn Curves;71
4.4.1;2.4.1 An Example with n = 3 and m = 3;71
4.4.2;2.4.2 Non-Special Divisors;73
4.4.3;2.4.3 Characterizing All Non-Special Divisors;75
4.5;2.5 Some Operators;78
4.5.1;2.5.1 Operators for the Nonsingular Case;79
4.5.2;2.5.2 Operators for the Singular Case;81
4.5.3;2.5.3 Properties of the Operators in Both Cases;83
4.6;2.6 Theta Functions on Zn Curves;86
4.6.1;2.6.1 Non-Special Divisors as Characteristics for Theta Functions;86
4.6.2;2.6.2 Quotients of Theta Functions with Characteristics Represented by Divisors;87
4.6.3;2.6.3 Evaluating Quotients of Theta Functions at Branch Points;88
4.6.4;2.6.4 Quotients of Theta Functions as Meromorphic Functions on Zn Curves;90
5;Chapter 3 Examples of Thomae Formulae;94
5.1;3.1 A Nonsingular Z3 Curve with Six Branch Points;94
5.1.1;3.1.1 First Identities Between Theta Constants;95
5.1.2;3.1.2 The Thomae Formulae;97
5.1.3;3.1.3 Changing the Basepoint;99
5.2;3.2 A Singular Z3 Curve with Six Branch Points;102
5.2.1;3.2.1 First Identities between Theta Constants;103
5.2.2;3.2.2 The First Part of the Poor Man’s Thomae;104
5.2.3;3.2.3 Completing the Poor Man’s Thomae;108
5.2.4;3.2.4 The Thomae Formulae;111
5.2.5;3.2.5 Relation with the General Singular Case;113
5.2.6;3.2.6 Changing the Basepoint;115
5.3;3.3 A One-Parameter Family of Singular Zn Curves with Four Branch Points;117
5.3.1;3.3.1 Divisors and Operators;117
5.3.2;3.3.2 First Identities Between Theta Constants;119
5.3.3;3.3.3 Even n;123
5.3.4;3.3.4 An Example with n = 10;126
5.3.5;3.3.5 Thomae Formulae for Even n;128
5.3.6;3.3.6 Odd n;133
5.3.7;3.3.7 An Example with n = 9;136
5.3.8;3.3.8 Thomae Formulae for Odd n;137
5.3.9;3.3.9 Changing the Basepoint;139
5.3.10;3.3.10 Relation with the General Singular Case;143
5.4;3.4 Nonsingular Zn Curves with r = 1 and Small n;144
5.4.1;3.4.1 The Set of Divisors as a Principal Homogenous Space for Sn 1;144
5.4.2;3.4.2 The Case n = 4;145
5.4.3;3.4.3 Changing the Basepoint for n = 4;148
5.4.4;3.4.4 The Case n = 3;150
5.4.5;3.4.5 The Problem with n = 5;151
5.4.6;3.4.6 The Case n = 5;153
5.4.7;3.4.7 The Orbits for n = 5;156
5.4.8;3.4.8 Changing the Basepoint for n = 5;157
6;Chapter 4 Thomae Formulae for Nonsingular Zn Curves;161
6.1;4.0.1 A Useful Notation;161
6.2;4.1 The Poor Man’s Thomae Formulae;163
6.2.1;4.1.1 First Identities Between Theta Constants;163
6.2.2;4.1.2 Symmetrization over R and the Poor Man’s Thomae;166
6.2.3;4.1.3 Reduced Formulae;168
6.3;4.2 Example with n = 5 and General r;171
6.3.1;4.2.1 Correcting the Expressions Involving C 1;172
6.3.2;4.2.2 Correcting the Expressions Not Involving C 1;173
6.3.3;4.2.3 Reduction and the Thomae Formulae for n = 5;175
6.4;4.3 Invariance also under N;177
6.4.1;4.3.1 The Description of h. for Odd n;177
6.4.2;4.3.2 N-Invariance for Odd n;180
6.4.3;4.3.3 The Description of h. for Even n;183
6.4.4;4.3.4 N-Invariance for Even n;185
6.5;4.4 Thomae Formulae for Nonsingular Zn Curves;187
6.5.1;4.4.1 The Case r = 2;188
6.5.2;4.4.2 Changing the Basepoint for r = 2;193
6.5.3;4.4.3 The Case r = 1;194
6.5.4;4.4.4 Changing the Basepoint for r = 1;196
7;Chapter 5 Thomae Formulae for Singular Zn Curves;201
7.1;5.1 The Poor Man’s Thomae Formulae;201
7.1.1;5.1.1 First Identities Between Theta Constants Based on the Branch Point R;202
7.1.2;5.1.2 Symmetrization over R;205
7.1.3;5.1.3 First Identities Between Theta Constants Based on the Branch Point S;208
7.1.4;5.1.4 Symmetrization over S;210
7.1.5;5.1.5 The Poor Man’s Thomae;213
7.1.6;5.1.6 Reduced Formulae;217
7.2;5.2 Example with n = 5 and General m;220
7.2.1;5.2.1 Correcting the Expressions Involving C 1 and D 1;222
7.2.2;5.2.2 Correcting the Expressions Not Involving C 1 and D 1;224
7.2.3;5.2.3 Reduction and the Thomae Formulae for n = 5;227
7.3;5.3 Invariance also under N;229
7.3.1;5.3.1 The Description of h. for Odd n;230
7.3.2;5.3.2 N-Invariance for Odd n;233
7.3.3;5.3.3 The Description of h. for Even n;237
7.3.4;5.3.4 N-Invariance for Even n;240
7.4;5.4 Thomae Formulae for Singular Zn Curves;243
7.4.1;5.4.1 The Thomae Formulae;244
7.4.2;5.4.2 Changing the Basepoint;247
8;Chapter 6 Some More Singular Zn Curves;251
8.1;6.1 A Family of Zn Curves with Four Branch Points and a Symmetric Equation;252
8.1.1;6.1.1 Functions, Differentials, Weierstrass Points and Abel–Jacobi Images;253
8.1.2;6.1.2 Non-Special Divisors in an Example of n = 7;255
8.1.3;6.1.3 Non-Special Divisors in the General Case;259
8.1.4;6.1.4 Operators;264
8.1.5;6.1.5 First Identities Between Theta Constants;266
8.1.6;6.1.6 The Poor Man’s Thomae (Unreduced and Reduced);270
8.1.7;6.1.7 The Thomae Formulae in the Case n = 7;275
8.1.8;6.1.8 The Thomae Formulae in the General Case;278
8.1.9;6.1.9 Changing the Basepoint;283
8.2;6.2 A Family of Zn Curves with Four Branch Points and an Asymmetric Equation;286
8.2.1;6.2.1 An Example with n = 10;286
8.2.2;6.2.2 Non-Special Divisors for n = 10;289
8.2.3;6.2.3 The Basic Data for General n;290
8.2.4;6.2.4 Non-Special Divisors for General n;294
8.2.5;6.2.5 Operators and Theta Quotients;297
8.2.6;6.2.6 Thomae Formulae for t = 1;299
8.2.7;6.2.7 Thomae Formulae for t = 2;301
8.2.8;6.2.8 Changing the Basepoint;303
9;Appendix A;307
9.1;Constructions and Generalizations for the Nonsingular and Singular Cases;307
9.1.1;A.1 The Proper Order to do the Corrections in the Nonsingular Case;308
9.1.2;A.2 Nonsingular Case, Odd n;309
9.1.3;A.3 Nonsingular Case, Even n;314
9.1.4;A.4 The Proper Order to do the Corrections in the Singular Case;318
9.1.5;A.5 Singular Case, Odd n;319
9.1.6;A.6 Singular Case, Even n;326
9.1.7;A.7 The General Family;330
9.1.8;A.8 Proof of Theorem A.2;336
10;Appendix B;342
10.1;The Construction and Basepoint Change Formulae for the Symmetric Equation Case;342
10.1.1;B.1 Description of the Process;342
10.1.2;B.2 The Case n = 1(mod 4);345
10.1.3;B.3 The Case n = 3(mod 4);351
10.1.4;B.4 The Operators for the Other Basepoints;356
10.1.5;B.5 The Expressions for h. for the Other Basepoints;359
11;References;363
12;List of Symbols;364
13;Index;367