The Riemann Hypothesis for Function Fields: Frobenius Flow and Shift Operators

by Machiel van Frankenhuijsen

Overview

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.

Product Details

ISBN-13:	9781107721081
Publisher:	Cambridge University Press
Publication date:	01/09/2014
Series:	London Mathematical Society Student Texts , #80
Sold by:	Barnes & Noble
Format:	eBook
File size:	12 MB
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About the Author

Machiel van Frankenhuijsen is an Associate Professor at Utah Valley University. His research interests lie in number theory, especially the abc conjecture and the connections between geometry and number theory.

Table of Contents

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.