MR0681120 (84m:14032)
Lang, Serge
Introduction to algebraic and abelian functions.
Second edition. Graduate Texts in Mathematics, 89.
Springer-Verlag, New York-Berlin, 1982. ix+169 pp. \$32.00. ISBN 0-387-90710-6
14Hxx (14-01 14K20)

Since this second edition is an extended version of the first one, only the differences are described here [for the first edition see MR 48 #6122]. Chapter I extends the former first chapter on the Riemann-Roch theorem (the case of hyperelliptic fields is included, differentials of the second kind and the notion of the divisor class group (of a function field in one variable) are introduced). Chapter II is entirely new and deals with the Fermat curve as a significant example for the notions and theorems proved in the first chapter. The genus and an explicit description of the differentials of first and second kind are obtained. After a consideration of rational images of the Fermat curve one is led to a decomposition of the group of divisor classes of the Fermat curve (following Rohrlich, based on work of Faddeev). Chapters III and IV are the previous chapters (with some additional explanations) on Riemann surfaces and the theorem of Abel-Jacobi. Chapter V is new and again deals with the Fermat curve. The period lattice (also for the related curves corresponding to the divisor class group decomposition of Chapter II) is explicitly computed in terms of a basis for the differentials of the first kind (following Rohrlich). Chapter VI coincides (with slight modifications) with the former chapter on the linear theory of theta functions. Chapter VII contains the chapter on duality theory of the first edition. Furthermore, Poincaré's complete reducibility theorem is included. Finally, the periods and the first homology group of a complex torus are considered. Chapter VIII extends the former Appendix 1 on Riemann forms and matrices. The Siegel upper half space is introduced for parametrizing the isomorphism classes of principally polarized abelian varieties. For vector spaces of normalized theta functions satisfying certain conditions an explicit basis is constructed (following Shimura). Chapter IX again is new and is devoted to the construction of abelian varieties containing a quaternion algebra in their algebra of endomorphisms. The final Chapter X coincides with the last chapter on theta functions and divisors of the first edition up to some remarks (e.g., the existence of a Riemann form on an abelian variety).

This revised edition is an excellent and very readable introduction to some basic notions in algebraic geometry.

Reviewed by Reinhard Bölling

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This is a review of first edition. It was in German, but I ran it through the Alta Vista translator, with hillarious results (e.g., "the Hurwitz' sex formula").

It begins with a short introduction to valuation theory and gives Weil's proof of the sentence of Riemann smelling. The chapter closes with the Hurwitz' sex formula. In the second chapter Riemann surfaces are examined. First the local Uniformisierungssatz is proven and dealt then with the topology and the analytic structure of the Riemann surfaces. Thus it is shown among other things that the Riemann surfaces are being connected, triangulierbar and orientable. Finally with the integration on Riemann surfaces is dealt and the Cauchy sentence is proven. On this basis in chapter III the sentence is treated from Abel Jacobi (proof after Artin). Remarks over the Riemann relations and the Pontrjagin duality complete the chapter. In the next chapter the linear theory of the theta functions is represented (after the Weil Bourbakivortrag from the year 1949). Thus among other things that there is a normalized theta function in each equivalence class of theta functions, abelsche functions will become shown introduced and it the sentence by Riemann smelling for the torus are proven. Further one finds the sentence of Lefschetz over the projektive imbedding by a not-degenerated theta function. In an appendix to the illustration of the 1-dimensionale case is implemented. To this chapter remarks over duality theory (binary abelsche variousnesses, Tate group in connection with rational and $p$ adischen representations, grief mating) close. In the end the connection is examined by theta functions and divisors. It is e.g. shown that the positive divisors on the torus can be represented by theta functions on #C^n$. The available book gives a very beautiful introduction to the theory of the abelschen functions. It will become little special knowledge presupposed and the proofs very in detail and easily understandably represented. This connected representation of the results is well suitable for students or as basis for lectures or seminars. Reviewed by Gerhard Pfister