In this result, the ground field is chosen to be Q solely for convenience; the
method works in general. Moreover, with a little more work one could specify \v
for all places v G M. One should also be able to establish a similar result for GL(3)
automorphic representations that are not lifts from GL(2) by a similar method.
Theorem 3.3 is proved by continuing a suitable double Dirichlet series.
Applying Tauberian techniques to the previous theorem one gets
THEOREM 3.4. Suppose n is automorphic on GLS(AQ) with trivial central char-
acter. Then for a ±1 we have
= CX\ogX+C X+C"
where C is a non-zero multiple of
lim (s
The term C arises by contour integration as the leading coefficient of the second
order pole at w = 1. Note that by equation (3.8), this residue arises from the
summands indexed by m a perfect square, when £ is trivial, so it is approximately
J]c(ra 2 )|ra| _ 2 s , which is related to L(2s,7r,sym2).
To complete the proof of Theorem 3.3, suppose that TT = Ad2(7r'). Then
= («)£(*, symV),X*')-
Here Xn' denotes the central character of 7r'. Using this equality, one can see
that L(s,7r,sym2) has a simple pole at s = 1. The proof in [17] uses the Kim-
Shahidi result on the automorphicity of
as well as the Jacquet-Shalika
nonvanishing theorem to conclude that the second term does not vanish at s 1,
and hence that C ^ 0. Prof. Shahidi has kindly informed us that a simpler proof
that L(l,
sym4(7r/), x2,)
^ 0 is available in an older paper of his.
If we take an automorphic representation on GL(3) that is not a lift then C 0.
Surprisingly, this thus gives an analytic way to tell if an automorphic representation
on GL(3) is or is not a lift from GL(2); the cases are separated by the asymptotic
behavior of their quadratically-twisted L-functions.
Returning to general n on GL(3), and looking at the residue of the series Z(s,w)
at w 1, one obtains a proof that for any n on GL(3), the symmetric square L-
function L(s,7r,sym2) (which is of degree 6) is holomorphic; more precisely, one
sees that the product ("(3s
is holomorphic except at s = 1, 2/3.
As the results of this section illustrate, the multiple Dirichlet series that con-
tinue to a product of complex planes are ready-made for establishing distribution
results via contour integration. Though some of the results above are stated over
Q, in fact the method of multiple Dirichlet series applies over a general global field
containing sufficiently many roots of unity; thus such mean value theorems may
be established without being constrained by the proliferation of Gamma factors
in higher degree extensions. The most natural theorems to prove involve sums of
L-functions times weighting factors a(s, 7r, d).
3.7. Determination of automorphic forms by twists of critical values.
An additional application of multiple Dirichlet series, reflecting the power of the
method, concerns the determination of an automorphic form by means of its twisted
L- values.
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