INTRODUCTION

character ip of the unipotent group

TVA/N/C,

gives a Fourier coefficient

/

JN

Lis

TT)

^{n)E(s,gn)dn = (p(s,g,ip)-

JNA/Nk L ( l + S,7r)

where (/?(s, g, ip) ^ 0 for 5 £ iR. This in turn implies that

L(l + it,7r) ^ 0 ,

for all real values of t. An exposition of these ideas, including a brief introduction

to the powerful Langlands-Shahidi method is given in Chapter VI. In addition,

we have included a discussion of a little known result on the non-vanishing of ((s)

based on elementary Hilbert space techniques. The reader will also find here a

brief discussion to the generalized Ramanujan conjecture, a very significant

problem, full of implications for the future development of analytic number the-

ory. Some exciting new results have been obtained in this area by Iwaniec and

Sarnak and their collaborators ([155]) by using methods akin to those presented in

Chapter IV.

There is another application of the theory of Eisenstein series on metaplec-

tic groups to non-vanishing theorems for L-functions, which is not presented in

this book, but because of its non-classical nature deserves to be mentioned in this

introductory paragraph. In its simplest manifestation, it arises from the Fourier

expansion of Eisenstein series of half integral weight on the Hecke group

TQ(4:N).

The Fourier coefficients turn out to be essentially Dirichlet L-functions associated

to quadratic extensions of the field of rational numbers. By a sieving procedure,

average information about these coefficients can be extracted from knowledge of the

singularities of the relevant Eisenstein series. In the special case of automorphic

L-functions

L(S1TT)

on GL(2), a key fundamental idea is Waldspurger's character-

ization of the values of

L(~,TT

® \) m terms of lifts

TT

to the metaplectic cover of

GL(2). In one of the most striking applications of these ideas to number theory, it

has been possible to show the non-vanishing of the twisted Hasse-Weil zeta function

L(s, E ® x) °f elliptic curves at the point s — 1, (the central point of the critical

line) for infinitely many quadratic characters x,

a s w e

^

a s f°r

the derivatives. In

contrast to the classical situation, this type of non-vanishing has direct applica-

tions to problems of diophantine analysis, a situation that is controlled by the well

known Birch and Swinnerton-Dyer conjecture. For an excellent exposition of the

ideas surrounding this type of non-vanishing, the reader should consult the survey

article by Bump, Friedberg, and Hoffstein in [24].

Appendix. The functional equation of an Artin L-function, which relates its

values at 5 and at 1 — 5, contains an arithmetic factor, known as the root number,

whose behavior resembles that of a quadratic Gauss sum

V ^

e-27rin2/N

=

l

+

l

. ^

^ 1 + i

neZ/NZ

Hasse, who knew well the factorization of the Gaussian sum as a product of sim-

ilar sums with the integer T V replaced by its prime power divisors, suggested that

the root number appearing in the functional equation of Artin L-functions should

have analogous factorizations into locally defined root numbers. Tate's harmonic