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

Machiel van Frankenhuijsen

Jan 2014 · London Mathematical Society Student Texts Ibhuku elingu-80 · Cambridge University Press

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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.

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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.