14 GAUTAM CHINTA, SOLOMON FRIEDBERG, AND JEFFREY HOFFSTEIN

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

X

d')-

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

factors.

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

q~s

and

q~w

(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)

7r(e1e2)|e1rs|e2|1-2s.

eie2\di

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].)