The Riemann Hypothesis for Function Fields
Frobenius Flow and Shift Operators
Seiten
2014
Cambridge University Press (Verlag)
978-1-107-68531-4 (ISBN)
Cambridge University Press (Verlag)
978-1-107-68531-4 (ISBN)
A description of how non-commutative geometry could provide a means to attack the Riemann Hypothesis, one of the most important unsolved problems in mathematics. The book will be of interest to graduate students in analytic and algebraic number theory, and provides a strong foundation for further research in this area.
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.
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.
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.
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.
| Erscheint lt. Verlag | 9.1.2014 |
|---|---|
| Reihe/Serie | London Mathematical Society Student Texts |
| Zusatzinfo | 1 Tables, black and white; 12 Line drawings, unspecified |
| Verlagsort | Cambridge |
| Sprache | englisch |
| Maße | 152 x 228 mm |
| Gewicht | 260 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie |
| Mathematik / Informatik ► Mathematik ► Geometrie / Topologie | |
| ISBN-10 | 1-107-68531-1 / 1107685311 |
| ISBN-13 | 978-1-107-68531-4 / 9781107685314 |
| Zustand | Neuware |
| Haben Sie eine Frage zum Produkt? |
Mehr entdecken
aus dem Bereich
aus dem Bereich
Sieben ausgewählte Themenstellungen
Buch | Softcover (2024)
De Gruyter Oldenbourg (Verlag)
CHF 89,95
unlock your imagination with the narrative of numbers
Buch | Softcover (2024)
Advantage Media Group (Verlag)
CHF 27,90
Seltsame Mathematik - Enigmatische Zahlen - Zahlenzauber
Buch | Softcover (2024)
BoD – Books on Demand (Verlag)
CHF 27,95
