Books in the Aspects of Mathematics series

This book and the following second volume is an introduction into modern algebraic geometry. In the first volume the methods of homological algebra, theory of sheaves, and sheaf cohomology are developed. These methods are indispensable for modern algebraic geometry, but they are also fundamental for other branches of mathematics and of great interest in their own.In the last chapter of volume I these concepts are applied to the theory of compact Riemann surfaces. In this chapter the author makes clear how influential the ideas of Abel, Riemann and Jacobi were and that many of the modern methods have been anticipated by them. For this second edition the text was completely revised and corrected. The author also added a short section on moduli of elliptic curves with N-level structures. This new paragraph anticipates some of the techniques of volume II.

The aim of this book is to give an introduction to adic spaces and to develop systematically their étale cohomology. First general properties of the étale topos of an adic space are studied, in particular the points and the constructible sheaves of this topos. After this the basic results on the étale cohomology of adic spaces are proved: base change theorems, finiteness, Poincaré duality, comparison theorems with the algebraic case.

Preface The Gamma function, the zeta function, the theta function, the hyper­ geometric function, the Bessel function, the Hermite function and the Airy function, . . . are instances of what one calls special functions. These have been studied in great detail. Each of them is brought to light at the right epoch according to both mathematicians and physicists. Note that except for the first three, each of these functions is a solution of a linear ordinary differential equation with rational coefficients which has the same name as the functions. For example, the Bessel equation is the simplest non-trivial linear ordinary differential equation with an irreg­ ular singularity which leads to the theory of asymptotic expansion, and the Bessel function is used to describe the motion of planets (Kepler's equation). Many specialists believe that during the 21st century the Painleve functions will become new members of the community of special func­ tions. For any case, mathematics and physics nowadays already need these functions. The corresponding differential equations are non-linear ordinary differential equations found by P. Painleve in 1900 fqr purely mathematical reasons. It was only 70 years later that they were used in physics in order to describe the correlation function of the two dimen­ sional Ising model. During the last 15 years, more and more people have become interested in these equations, and nice algebraic, geometric and analytic properties were found.