character ip of the unipotent group
gives a Fourier coefficient
^{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
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
on GL(2), a key fundamental idea is Waldspurger's character-
ization of the values of
® \) m terms of lifts
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 ^
. ^
^ 1 + i
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
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