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Field name	Details
ISBN	9781441978479
ISBN	9781441978462
Author	Farkas, Hershel M.
Title	Generalizations of Thomae's Formula for Zn Curves.
Description	1 online resource (367 pages)
Contents	GENERALIZATIONS OF THOMAE'S FORMULA FOR Zn CURVES -- Introduction -- Contents -- Chapter 1 Riemann Surfaces -- 1.1 Basic Definitions -- 1.1.1 First Properties of Compact Riemann Surfaces -- 1.1.2 Some Examples -- 1.2 The Abel Theorem, the Riemann-Roch Theoremand Weierstrass Points -- 1.2.1 The Abel Theorem and the Jacobi Inversion Theorem -- 1.2.2 The Riemann-Roch Theorem and the Riemann-Hurwitz Formula -- 1.2.3 Weierstrass Points -- 1.3 Theta Functions -- 1.3.1 First Properties of Theta Functions -- 1.3.2 Quotients of Theta Functions -- 1.3.3 Theta Functions on Riemann Surfaces -- 1.3.4 Changing the Basepoint -- 1.3.5 Matching Characteristics -- Chapter 2 Zn Curves -- 2.1 Nonsingular Zn Curves -- 2.1.1 Functions, Differentials and Weierstrass Points -- 2.1.2 Abel-Jacobi Images of Certain Divisors -- 2.2 Non-Special Divisors of Degree g on Nonsingular Zn Curves -- 2.2.1 An Example with n = 3 and r = 2 -- 2.2.2 An Example with n = 5 and r = 3 -- 2.2.3 Non-Special Divisors -- 2.2.4 Characterizing All Non-Special Divisors -- 2.3 Singular Zn Curves -- 2.3.1 Functions, Differentials and Weierstrass Points -- 2.3.2 Abel-Jacobi Images of Certain Divisors -- 2.4 Non-Special Divisors of Degree g on Singular Zn Curves -- 2.4.1 An Example with n = 3 and m = 3 -- 2.4.2 Non-Special Divisors -- 2.4.3 Characterizing All Non-Special Divisors -- 2.5 Some Operators -- 2.5.1 Operators for the Nonsingular Case -- 2.5.2 Operators for the Singular Case -- 2.5.3 Properties of the Operators in Both Cases -- 2.6 Theta Functions on Zn Curves -- 2.6.1 Non-Special Divisors as Characteristics for Theta Functions -- 2.6.2 Quotients of Theta Functions with Characteristics Represented by Divisors -- 2.6.3 Evaluating Quotients of Theta Functions at Branch Points -- 2.6.4 Quotients of Theta Functions as Meromorphic Functions on Zn Curves -- Chapter 3 Examples of Thomae Formulae.
	3.1 A Nonsingular Z3 Curve with Six Branch Points -- 3.1.1 First Identities Between Theta Constants -- 3.1.2 The Thomae Formulae -- 3.1.3 Changing the Basepoint -- 3.2 A Singular Z3 Curve with Six Branch Points -- 3.2.1 First Identities between Theta Constants -- 3.2.2 The First Part of the Poor Man's Thomae -- 3.2.3 Completing the Poor Man's Thomae -- 3.2.4 The Thomae Formulae -- 3.2.5 Relation with the General Singular Case -- 3.2.6 Changing the Basepoint -- 3.3 A One-Parameter Family of Singular Zn Curves with Four Branch Points -- 3.3.1 Divisors and Operators -- 3.3.2 First Identities Between Theta Constants -- 3.3.3 Even n -- 3.3.4 An Example with n = 10 -- 3.3.5 Thomae Formulae for Even n -- 3.3.6 Odd n -- 3.3.7 An Example with n = 9 -- 3.3.8 Thomae Formulae for Odd n -- 3.3.9 Changing the Basepoint -- 3.3.10 Relation with the General Singular Case -- 3.4 Nonsingular Zn Curves with r = 1 and Small n -- 3.4.1 The Set of Divisors as a Principal Homogenous Space for Sn−1 -- 3.4.2 The Case n = 4 -- 3.4.3 Changing the Basepoint for n = 4 -- 3.4.4 The Case n = 3 -- 3.4.5 The Problem with n ≥ 5 -- 3.4.6 The Case n = 5 -- 3.4.7 The Orbits for n = 5 -- 3.4.8 Changing the Basepoint for n = 5 -- Chapter 4 Thomae Formulae for Nonsingular Zn Curves -- 4.0.1 A Useful Notation -- 4.1 The Poor Man's Thomae Formulae -- 4.1.1 First Identities Between Theta Constants -- 4.1.2 Symmetrization over R and the Poor Man's Thomae -- 4.1.3 Reduced Formulae -- 4.2 Example with n = 5 and General r -- 4.2.1 Correcting the Expressions Involving C−1 -- 4.2.2 Correcting the Expressions Not Involving C−1 -- 4.2.3 Reduction and the Thomae Formulae for n = 5 -- 4.3 Invariance also under N -- 4.3.1 The Description of hΔ for Odd n -- 4.3.2 N-Invariance for Odd n -- 4.3.3 The Description of hΔ for Even n -- 4.3.4 N-Invariance for Even n.
	4.4 Thomae Formulae for Nonsingular Zn Curves -- 4.4.1 The Case r ≥ 2 -- 4.4.2 Changing the Basepoint for r ≥ 2 -- 4.4.3 The Case r = 1 -- 4.4.4 Changing the Basepoint for r = 1 -- Chapter 5 Thomae Formulae for Singular Zn Curves -- 5.1 The Poor Man's Thomae Formulae -- 5.1.1 First Identities Between Theta Constants Based on the Branch Point R -- 5.1.2 Symmetrization over R -- 5.1.3 First Identities Between Theta Constants Based on the Branch Point S -- 5.1.4 Symmetrization over S -- 5.1.5 The Poor Man's Thomae -- 5.1.6 Reduced Formulae -- 5.2 Example with n = 5 and General m -- 5.2.1 Correcting the Expressions Involving C−1 and D−1 -- 5.2.2 Correcting the Expressions Not Involving C−1 and D−1 -- 5.2.3 Reduction and the Thomae Formulae for n = 5 -- 5.3 Invariance also under N -- 5.3.1 The Description of hΔ for Odd n -- 5.3.2 N-Invariance for Odd n -- 5.3.3 The Description of hΔ for Even n -- 5.3.4 N-Invariance for Even n -- 5.4 Thomae Formulae for Singular Zn Curves -- 5.4.1 The Thomae Formulae -- 5.4.2 Changing the Basepoint -- Chapter 6 Some More Singular Zn Curves -- 6.1 A Family of Zn Curves with Four Branch Points and a Symmetric Equation -- 6.1.1 Functions, Differentials, Weierstrass Points and Abel-Jacobi Images -- 6.1.2 Non-Special Divisors in an Example of n = 7 -- 6.1.3 Non-Special Divisors in the General Case -- 6.1.4 Operators -- 6.1.5 First Identities Between Theta Constants -- 6.1.6 The Poor Man's Thomae (Unreduced and Reduced) -- 6.1.7 The Thomae Formulae in the Case n = 7 -- 6.1.8 The Thomae Formulae in the General Case -- 6.1.9 Changing the Basepoint -- 6.2 A Family of Zn Curves with Four Branch Points and an Asymmetric Equation -- 6.2.1 An Example with n = 10 -- 6.2.2 Non-Special Divisors for n = 10 -- 6.2.3 The Basic Data for General n -- 6.2.4 Non-Special Divisors for General n -- 6.2.5 Operators and Theta Quotients.
	6.2.6 Thomae Formulae for t = 1 -- 6.2.7 Thomae Formulae for t = 2 -- 6.2.8 Changing the Basepoint -- Appendix A -- Constructions and Generalizations for the Nonsingular and Singular Cases -- A.1 The Proper Order to do the Corrections in the Nonsingular Case -- A.2 Nonsingular Case, Odd n -- A.3 Nonsingular Case, Even n -- A.4 The Proper Order to do the Corrections in the Singular Case -- A.5 Singular Case, Odd n -- A.6 Singular Case, Even n -- A.7 The General Family -- A.8 Proof of Theorem A.2 -- Appendix B -- The Construction and Basepoint Change Formulae for the Symmetric Equation Case -- B.1 Description of the Process -- B.2 The Case n ≡ 1(mod 4) -- B.3 The Case n ≡ 3(mod 4) -- B.4 The Operators for the Other Basepoints -- B.5 The Expressions for hΔ for the Other Basepoints -- References -- List of Symbols -- Index.
Subject	Differential equations, Partial
	Electronic books. -- local
	Functions of complex variables
	Functions, Special
	Geometry, Algebraic
	Mathematics
	Number theory
Other Author	Electronic books.
Other name(s)	Zemel, Shaul
Ebook Link	Find Ebook Central in MyGUtech
Links to Related Works	Subject References: Differential equations, Partial . Electronic books. -- local . Functions of complex variables . Functions, Special . Geometry, Algebraic . Mathematics . Number theory . Authors: Farkas, Hershel M. . Zemel, Shaul .

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Generalizations of Thomae's Formula for Zn Curves.

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