Wolfgang M. Schmidt

Kendrick Press, Inc. (2004) xii+333pp. Paperback $75.00. ISBN 0-09740427-1-4.

In 1948 André Weil published the proof of the Riemann hypothesis for function fields in one variable over a finite ground field, a landmark in both number theory and algebraic geometry. Applications included hitherto unattainable bounds for exponential sums, in particular Kloosterman sums. Later Grothendieck and Deligne employed profound innovations in algebraic geometry to carry Weil's work much further.

It came as a surprise to the number theory community when Sergei Stepanov, beginning in 1969, gave elementary proofs of many of Weil's results. Stepanov drew inspiration from the work of Axel Thue (1909) in Diophantine approximation.

Proofs of Weil's theorem in full generality, based on Stepanov's ideas, were given independently by Wolfgang Schmidt and Enrico Bombieri in 1973. The present book contains accounts of both methods. Schmidt's method, which is more elementary, is discussed in Chapters 1-6. This part of the book is close to the first edition (Springer, 1976). Three previously unpublished chapters cover Bombieri's proof, with material on 'Valuations and Places' (Chapter 7) and 'The Riemann-Roch Theorem' (Chapter 8). All chapters are based on the author's lectures at the University of Colorado. The text has been reset in a modern typeface.

The author develops the necessary tools in leisurely style without the need for substantial prerequisites, and includes many well chosen examples.

'… a most valuable resource for students and researchers'. (Y. Bugeaud, Math. Reviews.)