Abelian Functions
Abel's Theorem and the Allied Theory of Theta Functions

Baker's 1897 classic book on algebraic geometry and allied theory.

H. F. Baker (Author), Igor Krichever (Foreword by)

9780521498777, Cambridge University Press

Paperback, published 14 December 1995

724 pages
22.8 x 15.3 x 3.8 cm, 0.993 kg

"...offers welcome historical insight into a key point in the history of algebraic geometry." John B. Little, Mathematical Reviews

Classical algebraic geometry, inseparably connected with the names of Abel, Riemann, Weierstrass, Poincaré, Clebsch, Jacobi and other outstanding mathematicians of the last century, was mainly an analytical theory. In our century it has been enriched by the methods and ideas of topology, commutative algebra and Grothendieck's schemes seemed to have replaced once and forever the somewhat naive language of classical algebraic geometry. This book contains more than its modest title suggests. Written in 1897, its scope was as broad as it could possibly be, namely to cover the whole of algebraic geometry, and associated theories. The subject is discussed by Baker in terms of transcendental functions, and in particular theta functions. Many of the ideas put forward are of continuing relevance today, and some of the most exciting ideas from theoretical physics draw on work presented here.

1. The subject of investigation
2. The fundamental functions on a Riemann surface
3. The infinities of rational functions
4. Specification of a general form of Riemann's integrals
5. Certain forms of the fundamental equation of the Riemann surface
6. Geometrical investigations
7. Coordination of simple elements
8. Abel's theorem
9. Jacobi's inversion problem
10. Riemann's theta functions
11. The hyperelliptic case of Riemann's theta functions
12. A particular form of Riemann surface
13. Radical functions
14. Factorial functions
15. Relations concerning products of theta functions
16. A direct method of obtaining the equations relating theta functions
17. Theta relations with certain groups of characteristics
18. Transformation of periods
19. On systems of periods and on general Jacobian functions
20. Transformation of theta functions
21. Complex multiplication of theta functions
22. Degenerate Abelian integrals
Appendix 1. On algebraic curves in space
Appendix 2. On matrices.

Subject Areas: Differential & Riemannian geometry [PBMP]