equations, and the method applies. An additional complication is the epsilon-
factors that arise in the functional equations for automorphic L-functions. As
shown in Fisher and Friedberg [31, 32], it is possible to sieve the efs using a finite
set of characters so that for dy d! in the same class,
e(l/2,7r x xd) = e(l/2,7r x
This is crucial, and allows one to apply the functional equation to the sum of L-
functions Z(s, w) and obtain an object that is a finite linear combination of similar
double Dirichlet series, rather than series with new weights coming from the epsilon
Since the base field may be general, one may study the functions Z(s^w) for
function fields. In that case, for TT on GL(r) with r 3, Z(s, w) reduces to a rational
function in
(where q is the cardinality of the field of constants) with
a specified denominator; this comes from the functional equations. For example,
given any algebraic curve over a finite field one gets a finite dimensional vector space
of rational functions of two complex variables; see [31] for details and examples,
and Section 5 below for a discussion of the rational function field case. It would be
intriguing to give a cohomological interpretation of these rational functions, but so
far no one has done so.
In the next two sections, we discuss the crucial ingredient in making the heuris-
tic rigorous—the interchange of summation—in more detail. Then in Sections 3.6,
3.7 we describe several applications of the method.
3.4. The interchange of summation: GL(1) and GL(2) cases. In this
section, we explain the interchange of summation that relates (3.7) and (3.8) when
IT is on GL(1) or GL(2) in more detail. For the moment, we simply exhibit the
weight factors a(s,7r, d), 6(u, £, 7r, m) directly. One might ask what conditions these
weight factors must satisfy if the method is to work, whether or not they are unique
(they are), and how they can be determined. These questions are taken up for IT on
GL(2) in the following section; the case of GL(1) is similar. The weight factors and
the interchange for 7r on GL(3), as well as the uniqueness of these weight factors,
is more complicated, and we refer the reader to [17] for details. (For GL(4) and
beyond the interchange, functional equation and Euler product properties are not
enough to force uniqueness; see [17].)
Throughout this section we will write sums without specifying the precise set we
are summing over. For convenience, the reader may imagine that we are summing
over positive rational integers prime to the conductor. Over a general number or
function field, one sums over a suitable set of ideals prime to a finite set 5, and
adjusts the definitions to be independent of units. We refer to [31], Section 1, or
to Brubaker and Bump [7] for details.
3.4.1. Sums of GL(1) quadratic twists. Let ix be an idele class character. Let
d dodf where do is square-free. We write Xd Xd0 for the character given by the
quadratic Kronecker symbol Xd(a) (j~) if (^5^o) 1? a n d extend this function
to take value 0 if (a, do) 1. Let a(s, 7r, d) be given by
(3.15) a(s,ir,d)= ^ /x(ei) Xd(ei)
Here /i(ei) is a Mobius function. (This factor arises in the Fourier expansion of the
half-integral Eisenstein series E(z,s/2) described in Section 2.2 above; see [39].)
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