The Riemann Hypothesis for Function Fields

Frobenius Flow and Shift Operators

By: Machiel Van Frankenhuijsen

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Hardcover | 9 January 2014

At a Glance

Hardcover
166 Pages

Dimensions(cm)
22.86 x 15.24 x 1.27

Hardcover

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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.

Industry Reviews

'This charming book is an attempt to understand some modern approaches to the [Riemann Hypothesis] ... This intriguing material is recommended, e.g. for an advanced student seminar.' Nieuw Archief voor Wiskunde
'This lovely book offers an easy-pace account of the Riemann Hypothesis (RH) for function fields.' Anton Deitmar, Mathematical Reviews

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