as a consequence of this fact. They take n = 3 for convenience, but (as they explain)
one should have such a realization for all n 3.
4.2. Cubic Twists of GL(2).
4.2.1. The main result. The double Dirichlet series coming from cubic twists
of an automorphic representation on GL(2) was continued by Brubaker, Friedberg
and Hoffstein [12]. Let K = Q t v ^ ) . For d G OK, d = 1 mod 3 let \d\ denote
the absolute norm of d. Let P(s;d) denote a certain Dirichlet polynomial defined
in [12]; P(s;d) depends on TT but we suppress this from the notation. P(s;d) is a
complicated object, but has the properties that if one factors d = did^d^ with each
di = 1 mod 3, d\ square-free, d\d\ cube-free, then P(s;d) = 1 if d% 1. Also for
fixed d\, c?21 the sum
d3 = l mod 3 ' J |
converges absolutely for $lw 1/2 and 5fts 1/2.
The main theorem of [12] is:
THEOREM 4.1. Let n = S)7rv be an automorphic representation of GL(2,Ax)
such that L(s,7r, X) ^ entire for all Hecke characters \ such that
= 1. Let S be
a finite set of primes including the archimedean prime and the primes dividing 2, 3
and the level ofir. Then, for any sufficiently large positive integer k, the asymptotic
^Ls(s^,X^)P(s;d) ( l - J £ j ~ ^ ^ ( s ^ X
holds for any s with Jfts 1/2. The constant c^(s^n) is non-zero, and is given by
= C
( 3 ^ ^
where (5 denotes the Dedekind zeta function of K with the Euler factors at the
places in S removed, 7P,£P are the Satake parameters of the representation TTP, and
cs is a non-zero constant.
An immediate consequence of this, the convergence of the basic sum, and the
usual convexity bound for L(l/2,7r,x^ ^2) is
COROLLARY 4.2. Let 7r be as in (4-1) Then there exist infinitely many cube-free
d such that L(l/2,n, Xd ) 7^ 0- More precisely, let N(X) denote the number of such
d with \d\ X. Then for any e 0, N(X)
Xx 2~e.
We sketch the proof, which is somewhat involved. Define the multiple Dirichlet
Ls(s,7r,x^)P(s;d )
Z1(s,w)= —-^ .
d=l mod 3,(d,5)=l ' '
(Here the sum is over all d G OK with d = 1 mod 3 and ordv(d) 0 for all finite
v G S.) This series converges absolutely for 5ft(s), 9ft(w) 1. Our goal is to establish
the continuation of this function to a larger region. Let
Z*(s, w) = Zi(s, w) Cs(6s + 6w - 5) Cs(12s + 6w - 8) x
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