代数函数与Abelian函数 第2版(英文影印版)
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- 作者： （美）Serge Lang
- 出版社：世界图书出版公司
- ISBN：9787510004872
- 上架时间：2009-9-8
- 出版日期：2009 年8月
- 开本：24开
- 页码：168
- 版次：2-1
- 所属分类：
数学 > 函数论 > 综合
内容简介回到顶部↑
this short book gives an introduction to algebraic and abelian functions, withemphasis on the complex analytic point of view. it could be used for a courseor seminar addressed to second year graduate students.
the goal is the same as that of the first edition, although i have made anumber of additions. i have used the weil proof of the riemann-roch theorem since it is efficient and acquaints the reader with adeles, which are a veryuseful tool pervading number theory.
the proof of the abel-jacobi theorem is that given by artin in a seminarin 1948. as far as i know, the very simple proof for the jacobi inversiontheorem is due to him. the riemann-roch theorem and the abel-jacobitheorem could form a one semester course.
the riemann relations which come at the end of the treatment of jacobi''stheorem form a bridge with the second part which deals with abelian functionsand theta functions. in may 1949, weil gave a boost to the basic theory oftheta functions in a famous bourbaki seminar talk. i have followed hisexposition of a proof of poincare that to each divisor on acomplex torus therecorresponds a theta function on the universal covering space. however, thecorrespondence between divisors and theta functions is not needed for thelinear theory of theta functions and the projective embedding of the toruswhen there exists a positive non-degenerate riemann form. therefore i havegiven the proof of existence of a theta function corresponding to a divisor onlyin the last chapter, so that it does not interfere, with the self-contained treat-ment of the linear theory.
the goal is the same as that of the first edition, although i have made anumber of additions. i have used the weil proof of the riemann-roch theorem since it is efficient and acquaints the reader with adeles, which are a veryuseful tool pervading number theory.
the proof of the abel-jacobi theorem is that given by artin in a seminarin 1948. as far as i know, the very simple proof for the jacobi inversiontheorem is due to him. the riemann-roch theorem and the abel-jacobitheorem could form a one semester course.
the riemann relations which come at the end of the treatment of jacobi''stheorem form a bridge with the second part which deals with abelian functionsand theta functions. in may 1949, weil gave a boost to the basic theory oftheta functions in a famous bourbaki seminar talk. i have followed hisexposition of a proof of poincare that to each divisor on acomplex torus therecorresponds a theta function on the universal covering space. however, thecorrespondence between divisors and theta functions is not needed for thelinear theory of theta functions and the projective embedding of the toruswhen there exists a positive non-degenerate riemann form. therefore i havegiven the proof of existence of a theta function corresponding to a divisor onlyin the last chapter, so that it does not interfere, with the self-contained treat-ment of the linear theory.
目录回到顶部↑
chapter ⅰ the riemann-roch theorem .
1. lemmas on valuations
2. the riemann-roch theorem
3. remarks on differential forms
4. residues in power series fields
5. the sum of the residues
6. the genus formula of hurwitz
7. examples
8. differentials of second kind
9. function fields and curves
10. divisor classes
chapter ⅱ the fermat curve
1. the genus
2. differentials
3. rational images of the fermat curve
4. decomposition of the divisor classes
chapter ⅲ the riemann surface
1. topology and analytic structure
2. integration on the riemann surface
chapter ⅳ the theorem of abel-jacobi
1. lemmas on valuations
2. the riemann-roch theorem
3. remarks on differential forms
4. residues in power series fields
5. the sum of the residues
6. the genus formula of hurwitz
7. examples
8. differentials of second kind
9. function fields and curves
10. divisor classes
chapter ⅱ the fermat curve
1. the genus
2. differentials
3. rational images of the fermat curve
4. decomposition of the divisor classes
chapter ⅲ the riemann surface
1. topology and analytic structure
2. integration on the riemann surface
chapter ⅳ the theorem of abel-jacobi
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