Non-vanishing of L-functions and applications
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Bibliographic Information
Non-vanishing of L-functions and applications
（Modern Birkhäuser classics）
Birkhäuser, [2011?], c1997
- : pbk
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"Reprint of the 1st edition 1997 by Birkhäuser Verlag, Switzerland. Originally published as volume 157 in the Progress in mathematics series"--T.p. verso
"Ferran Sunyer i Balaguer Award winning monograph"--P. [1] of cover
Includes bibliographical references and indexes
Description and Table of Contents
Description
This monograph brings together a collection of results on the non-vanishing of- functions.Thepresentation,thoughbasedlargelyontheoriginalpapers,issuitable forindependentstudy.Anumberofexerciseshavealsobeenprovidedtoaidinthis endeavour. The exercises are of varying di?culty and those which require more e?ort have been marked with an asterisk. The authors would like to thank the Institut d'Estudis Catalans for their encouragementof thiswork throughtheFerranSunyeriBalaguerPrize.Wewould also like to thank the Institute for Advanced Study, Princeton for the excellent conditions which made this work possible, as well as NSERC, NSF and FCAR for funding. Princeton M. Ram Murty August, 1996 V. Kumar Murty xi Introduction Since the time of Dirichlet and Riemann, the analytic properties of L-functions have been used to establish theorems of a purely arithmetic nature. The dist- bution of prime numbers in arithmetic progressions is intimately connected with non-vanishing properties of various L-functions.
With the subsequent advent of the Tauberian theory as developed by Wiener and Ikehara, these arithmetical t- orems have been shown to be equivalent to the non-vanishing of these L-functions on the line Re(s)=1. In the 1950's, a new theme was introduced by Birch and Swinnerton-Dyer. Given an elliptic curve E over a number ?eld K of ?nite degree over Q,they associated an L-function to E and conjectured that this L-function extends to an entire function and has a zero at s = 1 of order equal to the Z-rank of the group of K-rational points of E. In particular, the L-function vanishes at s=1ifand only if E has in?nitely many K-rational points.
Table of Contents
Preface.- Introduction.- Chapter 1 The Prime Number Theorem and Generalizations.- Chapter 2 Artin L-functions.- Chapter 3 Equidistribution and L-functions.- Chapter 4 Modular Forms and Dirichlet Series.- Chapter 5 Dirichlet L-functions.- Chapter 6 Non-vanishing of Quadratic Twists of Modular L-functions.- Chapter 7 Selberg's Conjectures.- Chapter 8 Suggestions for further reading.- Author index.- Subject index.?
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