The Riemann Hypothesis for Function Fields
Frobenius Flow and Shift Operators

An exposition of the theory of curves over a finite field, and connections to the Riemann Hypothesis for function fields.

Machiel van Frankenhuijsen (Author)

9781107685314, Cambridge University Press

Paperback / softback, published 9 January 2014

162 pages, 12 b/w illus. 1 table
22.8 x 15.2 x 1 cm, 0.26 kg

'This lovely book offers an easy-pace account of the Riemann Hypothesis (RH) for function fields.' Anton Deitmar, Mathematical Reviews

This book provides a lucid exposition of the connections between non-commutative geometry and the famous Riemann Hypothesis, focusing on the theory of one-dimensional varieties over a finite field. The reader will encounter many important aspects of the theory, such as Bombieri's proof of the Riemann Hypothesis for function fields, along with an explanation of the connections with Nevanlinna theory and non-commutative geometry. The connection with non-commutative geometry is given special attention, with a complete determination of the Weil terms in the explicit formula for the point counting function as a trace of a shift operator on the additive space, and a discussion of how to obtain the explicit formula from the action of the idele class group on the space of adele classes. The exposition is accessible at the graduate level and above, and provides a wealth of motivation for further research in this area.

List of illustrations
Preface
Introduction
1. Valuations
2. The local theory
3. The zeta function
4. Weil positivity
5. The Frobenius flow
6. Shift operators
7. Epilogue
References
Notation
Index.

Subject Areas: Differential & Riemannian geometry [PBMP], Geometry [PBM], Mathematics [PB]