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Non-vanishing of L-Functions and Applications (Paperback, 1997)
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Non-vanishing of L-Functions and Applications (Paperback, 1997)
Series: Modern Birkhauser Classics
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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.
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