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In this expository text we sketch some interrelations between several famous conjectures in number theory and algebraic geometry that have intrigued math- ematicians for a long period of time. Starting from Fermat's Last Theorem one is naturally led to introduce L- functions, the main, motivation being the calculation of class numbers. In partic- ular, Kummer showed that the class numbers of cyclotomic fields play a decisive role in the corroboration of Fermat's Last Theorem for a large class of exponents. Before Kummer, Dirichlet had already successfully applied his L-functions to the proof of the theorem on arithmetic progressions. Another prominent appearance of an L-function is Riemann's paper where the now famous Riemann Hypothesis was stated. In short, nineteenth century number theory showed that much, if not all, of number theory is reflected by properties of L-functions. Twentieth century number theory, class field theory and algebraic geome- try only strengthen the nineteenth century number theorists's view. We just mention the work of E. H~cke, E. Artin, A. Weil and A. Grothendieck with his collaborators. Heeke generalized Dirichlet's L-functions to obtain results on the distribution of primes in number fields. Artin introduced his L-functions as a non-abelian generalization of Dirichlet's L-functions with a generalization of class field theory to non-abelian Galois extensions of number fields in mind.

The Riemann-Hilbert problem (Hilbert's 21st problem) belongs to the theory of linear systems of ordinary differential equations in the complex domain. The problem concerns the existence of a Fuchsian system with prescribed singularities and monodromy. Hilbert was convinced that such a system always exists. However, this turned out to be a rare case of a wrong forecast made by him. In 1989 the second author (A. B.) discovered a counterexample, thus obtaining a negative solution to Hilbert's 21st problem in its original form.

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 book ...is a storehouse of useful information for the mathematicians interested in foliation theory." (John Cantwell, Mathematical Reviews 1992)

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.

This volume is dedicated to Pierre Dolbeault, with contributions to complex analysis and analytic geometry. Contributors include Pierre Lelong, V. Ancona, B. Gaveau, B. Berndtsson, E.M. Chirka, E.L. Stout, J.P. Demailly, K. Diederich, G. Herbort, P. Dolbeault and G. Henkin.

Bringing together experts in the field of harmonic maps and integrable systems to give a coherent account of this subject, this book starts with introductory articles. Harmonic maps are maps between Riemannian or pseudo-Riemannian manifolds which extremize a natural energy integral.

"A publication of the Max-Planck-Institut f'ur Mathematik, Bonn."

"A publication of the Max-Planck-Institut f'ur Mathematik, Bonn."