The Riemann Hypothesis for Function Fields

Frobenius Flow and Shift Operators

Nonfiction, Science & Nature, Mathematics, Number Theory, Geometry
Cover of the book The Riemann Hypothesis for Function Fields by Machiel van Frankenhuijsen, Cambridge University Press
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Author: Machiel van Frankenhuijsen ISBN: 9781107721081
Publisher: Cambridge University Press Publication: January 9, 2014
Imprint: Cambridge University Press Language: English
Author: Machiel van Frankenhuijsen
ISBN: 9781107721081
Publisher: Cambridge University Press
Publication: January 9, 2014
Imprint: Cambridge University Press
Language: English

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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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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