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.
