MULTIPLE DIRICHLET SERIES AND AUTOMORPHIC FORMS 23
To study a sum of twists via Hartogs' principle/Bochner's theorem, the relationship
between e(s, 7r) and e(s, n x Xd ) *s needed. The quotient is a power of the conduc-
tor, which is essentially the square-free part of n, times the quotient at s 1/2.
This last factor is essentially the r-th power of a Gauss sum of order n, G(xYi that
has been normalized to have absolute value 1. (More precisely, this is true after
sieving, so it is more convenient to work with a finite dimensional vector space
of multiple Dirichlet series; see Fisher-Friedberg [31] for a discussion of this point
in a classical language and Brubaker-Bump [7] for a discussion which is adelic in
nature.)
Because of this crucial change, the heuristic that describes the quadratic twist
case is not useful. In fact, after a functional equation one obtains a new mutliple
Dirichlet series—not Z(s,w), but a series whose weight factors involve n-th order
Gauss sums. A similar situation occurs if one interchanges and then applies a func-
tional equation. Moreover, these two operations need not commute (even ignoring
scattering matrix and bad prime considerations)! To use the convexity methods of
Section 3.2, one is then led to consider several different families of multiple Dirichlet
series that are linked by functional equations. We discuss two cases in detail (n-fold
twists of GL(1) and cubic twists of GL(2). This is followed by a discussion of the
nonvanishing of n-th order twists of a GL(2) automorphic L-function for arbitrary
n. Though the sum of twisted L-functions Z(s, w) has not been continued to
C2,
a
variation on the method of double Dirichlet series gives an interesting result.
4.1. n-Fold Twists of GL(1). The study of the sum of the n-fold twists of a
given Hecke character was carried out by Friedberg, Hoffstein and Lieman [34]. One
obtains two different families of multiple Dirichlet series: the n-th order twists of the
original L-function ^2 L(s, £x^ )a(s, £,
f)|i|-ti;
and a multiple Dirichlet series built
up from infinite sums of n-th order Gauss sums. The second series is obtained from
the first by use of the functional equation for L(s, %Xd
f°ll°wed
by an interchange
of summation. But these latter sums arise as the Fourier coefficients of Eisenstein
series on the n-fold cover of GL(2), and they can thus be controlled by using the
theory of metaplectic Eisenstein series. In particular, they satisfy a functional
equation of their own, even though they are not Eulerian! To keep this paper to
manageable length, we do not give many details; we will supply them in the more
complicated case of GL(2) below. We remark that automorphic methods, which
could be for the most part avoided in the quadratic twist case, seem unavoidable
in many problems involving n-th order twists for n 2.
In the case at hand, the continuation of the two families of double Dirichlet
series to
C2
is established from Bochner's theorem. Note that earlier we mentioned
that such a sum could be approached by an integral of an Eisenstein series on
the n-fold cover of GL(n). Thus the Hartogs/Bochner-based method allows one to
replace the use of Eisenstein series on the n-fold cover of GL(n) with the use of
Eisenstein series on the n-fold cover of GL(2), which are considerably simpler. We
shall see a similar reduction to GL(2) in the work on Weyl group multiple Dirichlet
series that is discussed in [9].
Let us also note that Brubaker and Bump ([8], in this volume) have obtained the
double Dirichlet series discussed in this section as residues of Weyl group multiple
Dirichlet series, and have shown that their functional equations may be understood
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