Riemann surfaces (Q5917554)

The theory of Riemann surfaces, taken in the more general sense as including algebraic, geometric, analytic and topological aspects of the study of algebraic functions is one of the great achievements of the 19th century. It is both deep and rich in its mathematical content and continues to offer numerous possibilities for contemporary research. This book, a text-book in a series aimed at students from about the third year on, does full justice to the richness of the theory. In its rather limited space of 312 pages it covers the definition of a Riemann surface and their topology, the theory of elliptic functions, the theory of algebraic function fields and their localizations and the associated Riemann surface, the theory of differentials and integration on Riemann surfaces, algebraic geometry (intersection numbers, linear systems), potential theory (with subharmonic functions), the Riemann mapping theorem and the uniformisation of Riemann surfaces, the Riemann--Roch theorem and the theory of the Jacobian variety up to the theorem of Torelli and the Schottky problem. This is a very substantial programme for the size of the book. It is achieved with intellectual honesty; almost all the results are proved within the book itself, not by quoting from other sources. The author avoids those tempting but vague geometrical arguments which are characteristic of the theory of Riemann surfaces. The treatment is consequently very concise and very satisfying. The student needs a sound grounding in complex analysis, algebra and topology before starting, at the level of the standard courses. The historical dimension of the theory of Riemann surfaces and of algebraic curves is emphasized through a series of illuminating historical remarks. These are kept short and in some cases the reviewer felt that the examination exhortation ``Discuss!'' would have been appropriate; there is little in the discussion of history, even in the history of mathematics, that is ``plain and simple''. Nevertheless there remarks should act as a stimulus to read at least some of the classics. They also help to explain the ``why'' of the theory. There are also a large number of exercises and these are very welcome. This is a book which is demanding; it will probably be tough for the average student doing a second course in complex analysis. The gifted student however will value this book and keep it long after the examinations are over and forgotten.