Atnaujinkite slapukų nuostatas

Generalizations of Thomae's Formula for Zn Curves 2011 ed. [Kietas viršelis]

,
  • Formatas: Hardback, 354 pages, aukštis x plotis: 235x155 mm, weight: 1550 g, XVII, 354 p., 1 Hardback
  • Serija: Developments in Mathematics 21
  • Išleidimo metai: 24-Nov-2010
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 1441978461
  • ISBN-13: 9781441978462
Kitos knygos pagal šią temą:
Generalizations of Thomae's Formula for Zn Curves 2011 ed.Didesnis paveikslėlis
  • Formatas: Hardback, 354 pages, aukštis x plotis: 235x155 mm, weight: 1550 g, XVII, 354 p., 1 Hardback
  • Serija: Developments in Mathematics 21
  • Išleidimo metai: 24-Nov-2010
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 1441978461
  • ISBN-13: 9781441978462
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9781441978462.html
Raktažodžiai:
Includes several focused proofs developed in a generalized context that is accessible to researchers in related mathematical fields such as algebraic geometry, complex analysis, and number theory. This book is suitable for mathematicians with an interest in complex analysis, algebraic geometry or number theory as well as physicists.

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.



This book provides a comprehensive overview of the theory of theta functions, and the necessary background for understanding and proving the Thomae formulae and their relationship to the Zn curves.

The book is intended for graduate students in mathematics studying complex analysis, algebraic geometry, and number theory as well as mathematical physicists and physicists studying conformal field theory.

Recenzijos

From the reviews:

This book provides a detailed exposition of Thomaes formula for cyclic covers of CP1, in the non-singular case and in the singular case for Zn curves of a particular shape. This book is written for graduate students as well as young researchers . In any case, the reader should be acquainted with complex analysis (in several variables), Riemann surfaces, and some elementary algebraic geometry. It is a very readable book. The theory is always illustrated with examples in a very generous mathematical style. (Juan M. Cervińo Mathematical Reviews, Issue 2012 f)

In the book under review, the authors present the background necessary to understand and then prove Thomaes formula for Zn curves. The point of view of the book is to work out Thomae formulae for Zn curves from first principles, i.e. just using Riemanns theory of theta functions. the elementary approach which is chosen in the book makes it a nice development of Riemanns ideas and accessible to graduate students and young researchers. (Christophe Ritzenthaler, Zentralblatt MATH, Vol. 1222, 2011)

Introduction vii
1 Riemann Surfaces 1(30)
1.1 Basic Definitions
1(5)
1.1.1 First Properties of Compact Riemann Surfaces
1(3)
1.1.2 Some Examples
4(2)
1.2 The Abel Theorem, the Riemann—Roch Theorem and Weierstrass Points
6(7)
1.2.1 The Abel Theorem and the Jacobi Inversion Theorem
6(3)
1.2.2 The Riemann—Roch Theorem and the Riemann—Hurwitz Formula
9(1)
1.2.3 Weierstrass Points
10(3)
1.3 Theta Functions
13(18)
1.3.1 First Properties of Theta Functions
13(3)
1.3.2 Quotients of Theta Functions
16(1)
1.3.3 Theta Functions on Riemann Surfaces
17(5)
1.3.4 Changing the Basepoint
22(3)
1.3.5 Matching Characteristics
25(6)
2 Zn Curves 31(44)
2.1 Nonsingular Zn Curves
31(5)
2.1.1 Functions, Differentials and Weierstrass Points
32(2)
2.1.2 Abel—Jacobi Images of Certain Divisors
34(2)
2.2 Non-Special Divisors of Degree g on Nonsingular Zn Curves
36(12)
2.2.1 An Example with n = 3 and r = 2
37(1)
2.2.2 An Example with n = 5 and r = 3
38(2)
2.2.3 Non-Special Divisors
40(5)
2.2.4 Characterizing All Non-Special Divisors
45(3)
2.3 Singular Zn Curves
48(4)
2.3.1 Functions, Differentials and Weierstrass Points
48(2)
2.3.2 Abel—Jacobi Images of Certain Divisors
50(2)
2.4 Non-Special Divisors of Degree g on Singular Zn Curves
52(7)
2.4.1 An Example with n = 3 and m — 3
52(2)
2.4.2 Non-Special Divisors
54(2)
2.4.3 Characterizing All Non-Special Divisors
56(3)
2.5 Some Operators
59(8)
2.5.1 Operators for the Nonsingular Case
60(2)
2.5.2 Operators for the Singular Case
62(2)
2.5.3 Properties of the Operators in Both Cases
64(3)
2.6 Theta Functions on Zn Curves
67(8)
2.6.1 Non-Special Divisors as Characteristics for Theta Functions
67(1)
2.6.2 Quotients of Theta Functions with Characteristics Represented by Divisors
68(1)
2.6.3 Evaluating Quotients of Theta Functions at Branch Points
69(2)
2.6.4 Quotients of Theta Functions as Meromorphic Functions on Zn Curves
71(4)
3 Examples of Thomae Formulae 75(68)
3.1 A Nonsingular Z3 Curve with Six Branch Points
75(8)
3.1.1 First Identities Between Theta Constants
76(2)
3.1.2 The Thomae Formulae
78(2)
3.1.3 Changing the Basepoint
80(3)
3.2 A Singular Z3 Curve with Six Branch Points
83(15)
3.2.1 First Identities between Theta Constants
84(1)
3.2.2 The First Part of the Poor Man's Thomae
85(4)
3.2.3 Completing the Poor Man's Thomae
89(3)
3.2.4 The Thomae Formulae
92(2)
3.2.5 Relation with the General Singular Case
94(2)
3.2.6 Changing the Basepoint
96(2)
3.3 A One-Parameter Family of Singular Zn Curves with Four Branch Points
98(27)
3.3.1 Divisors and Operators
98(2)
3.3.2 First Identities Between Theta Constants
100(4)
3.3.3 Even n
104(3)
3.3.4 An Example with n = 10
107(2)
3.3.5 Thomae Formulae for Even n
109(5)
3.3.6 Odd a
114(3)
3.3.7 An Example with a = 9
117(1)
3.3.8 Thomae Formulae for Odd n
118(2)
3.3.9 Changing the Basepoint
120(4)
3.3.10 Relation with the General Singular Case
124(1)
3.4 Nonsingular 4 Curves with r I and Small a
125(18)
3.4.1 The Set of Divisors as a Principal Homogenous Space for Sn-1
125(1)
3.4.2 The Case a = 4
126(3)
3.4.3 Changing the Basepoint for a 4
129(2)
3.4.4 The Case n = 3
131
3.4.5 The Problem with n > 5
112(2)
3.4.6 The Case n = 5
114(3)
3.4.7 The Orbits for n = 5
117(21)
3.4.8 Changing the Basepoint for n = 5
138(5)
4 Thomae Formulae for Nonsingular 4 Curves 143(40)
4.0.1 A Useful Notation
143(2)
4.1 The Poor Man's Thomae Formulae
145(8)
4.1.1 First Identities Between Theta Constants
145(3)
4.1.2 Symmetrization over R and the Poor Man's Thomae
148(2)
4.1.3 Reduced Formulae
150(3)
4.2 Example with n = 5 and General r
153(6)
4.2.1 Correcting the Expressions Involving C-1
154(1)
4.2.2 Correcting the Expressions Not Involving C-1
155(2)
4.2.3 Reduction and the Thomae Formulae for n = 5
157(2)
4.3 Invariance also under N
159(10)
4.3.1 The Description of hδ for Odd n
159(3)
4.3.2 N-Invariance for Odd n
162(3)
4.3.3 The Description of hδ for Even n
165(2)
4.3.4 N-Invariance for Even a
167(2)
4.4 Thomae Formulae for Nonsingular Zn Curves
169(14)
4.4.1 The Case r > or equal to 2
170(5)
4.4.2 Changing the Basepoint for r > or equal to 2
175(1)
4.4.3 The Case r = 1
176(2)
4.4.4 Changing the Basepoint for r = 1
178(5)
5 Thomae Formulae for Singular Zn Curves 183(50)
5.1 The Poor Man's Thomae Formulae
183(19)
5.1.1 First Identities Between Theta Constants Based on the Branch Point R
184(3)
5.1.2 Symmetrization over R
187(3)
5.1.3 First Identities Between Theta Constants Based on the Branch Point S
190(2)
5.1.4 Symmetrization over S
192(3)
5.1.5 The Poor Man's Thomae
195(4)
5.1.6 Reduced Formulae
199(3)
5.2 Example with n = 5 and General in
202(9)
5.2.1 Correcting the Expressions Involving C-1 and
204(2)
5.2.2 Correcting the Expressions Not Involving C-1 and D-1
206(3)
5.2.3 Reduction and the Thomae Formulae for n = 5
209(2)
5.3 Invariance also under N
211(14)
5.3.1 The Description of hδ for Odd a
212(3)
5.3.2 N-Invariance for Odd n
215(4)
5.3.3 The Description of ha for Even a
219(3)
5.3.4 N-Invariance for Even n
222(3)
5.4 Thomae Formulae for Singular Zn Curves
225(8)
5.4.1 The Thomae Formulae
226(3)
5.4.2 Changing the Basepoint
229(4)
6 Some More Singular 4 Curves 233(56)
6.1 A Family of Zn Curves with Four Branch Points and a Symmetric Equation
234(34)
6.1.1 Functions, Differentials, Weierstrass Points and Abel—Jacobi Images
235(2)
6.1.2 Non-Special Divisors in an Example of a = 7
237(4)
6.1.3 Non-Special Divisors in the General Case
241(5)
6.1.4 Operators
246(2)
6.1.5 First Identities Between Theta Constants
248(4)
6.1.6 The Poor Man's Thomae (Unreduced and Reduced)
252(5)
6.1.7 The Thomae Formulae in the Case a = 7
257(3)
6.1.8 The Thomae Formulae in the General Case
260(5)
6.1.9 Changing the Basepoint
265(3)
6.2 A Family of Zn Curves with Four Branch Points and an Asymmetric Equation
268(21)
6.2.1 An Example with n = 10
268(3)
6.2.2 Non-Special Divisors for n = 110
271(1)
6.2.3 The Basic Data for General n
272(4)
6.2.4 Non-Special Divisors for General n
276(3)
6.2.5 Operators and Theta Quotients
279(2)
6.2.6 Thomae Formulae for t = 1
281(2)
6.2.7 Thomae Formulae for t = 2
283(2)
6.2.8 Changing the Basepoint
285(4)
Appendices
A Constructions and Generalizations for the Nonsingular and Singular Cases
289(36)
A.1 The Proper Order to do the Corrections in the Nonsingular Case
290(1)
A.2 Nonsingular Case, Odd n
291(5)
A.3 Nonsingular Case, Even n
296(4)
A.4 The Proper Order to do the Corrections in the Singular Case
300(1)
A.5 Singular Case, Odd n
301(7)
A.6 Singular Case, Even n
308(4)
A.7 The General Family
312(6)
A.8 Proof of Theorem A.2
318(7)
B The Construction and Basepoint Change Formulae for the Symmetric Equation Case
325(22)
B.1 Description of the Process
325(3)
B.2 The Case n=1(mod 4)
328(6)
B.3 The Case n=3(mod 4)
334(5)
B.4 The Operators for the Other Basepoints
339(3)
B.5 The Expressions for hδ for the Other Basepoints
342(5)
References 347(2)
List of Symbols 349(4)
Index 353