Multiple Dirichlet Series and Automorphic
Forms, I
Solomon Friedberg
July 20, 2005
Abstract
This is the first of a series of three lectures concerning multiple Dirichlet
series arising from sums of twisted automorphic L-functions. This lecture
begins with an historical overview, explaining how such series arise from
Rankin-Selberg constructions. Then more recent work, using Hartogs’s continuation principle in place of such constructions, is described. Applications
to the nonvanishing of L-functions and to other problems are also discussed.
I wish to thank Steven J. Miller, who took TEX notes on my conference
lecture as I was delivering it. His file was very valuable in preparing thesenotes.—Sol Friedberg.

11Multiple Dirichlet Series and Automorphic Forms,
I (Lecture by Sol Friedberg)

This is the first of a three part mini-course, with the remaining two lectures to bepresented by Gautam Chinta. I will start with an overview and historically-based
introduction.

1.1

The Family of Twisted L-Functions

Fix an n ≥ 2 and let F be a global field containing n nth roots of unity. (Though
the reader may focus on number fields, later, in one of Gautam Chinta’s talks,
we’ll see examples in the function field case.)
A basic problem is the following: let π be an automorphic representation ofGL(r) over the field F , with standard L-function
X
L(s, π) =c(n)||n||−s .
(In this lecture I will write the finite part of the L-function as L(s, π) for convenience.) We want to study the family of twisted L-functionsX
L(s, π × χ) =c(n)χ(n)||n||−swhere we fix π and vary the twist by a character χ; χ will range over the idèle
class characters of order exactly n. We may also wish to modify the problem alittle bit, and suppose instead that χ ranges over the subset of idèle class charactersof order exactly n with χv specified at a finite number of places. (We will also use
the notation L-function L(s, π, χ) for the twisted L-function.)

1.2

Questions

What kind of questions can we ask? The first is non-vanishing.
1.2.1

Non-vanishing

1. Non-vanishing: given a point in the critical strip s0 (with 0 < Re(s0 ) < 1),
can one show there exist infinitely many χ as above with L(s0 , π × χ) 6= 0?
Goes back to Shimura, Rohrlich, and Waldspurger. A natural question: if
we had perfect knowledge about RH then the interesting choice is s0 = 21 ,2where sometimes the L-function is zero and sometimes not (for example,
elliptic curves of rank greater than zero where when you twist you get rank0).
2. If n = 2 (quadratic twists) and π = πe (self-dual) and if ( 12 , π ×χ) = −1 for
all twists χ under consideration, can one show there exist infinitely many χsuch that L0 ( 12 , π×χ) 6= 0? Note that under these hypotheses, the functional
equation guarantees a zero of odd order for each twisted L-function at thecenter of the critical strip.
Note: In these questions, we are not assuming π is cuspidal – L(s, π) could be
a product. Then the first question becomes that of establishing a simultaneous
non-vanishing theorem. For example, take two independent GL(2) holomorphic
modular forms. It isn’t know if there is a twist such that both do not vanish at
the center of the critical strip (our L-functions are normalized so that s → 1 − s).
Using Multiple Dirichlet Series, one can establish simultaneous non-vanishingfor points s0 in the critical strip but sufficiently far from the center of the strip(Chinta-Friedberg-Hoffstein).
1.2.2

Distribution

1. Study the distribution of L(s, π × χ) as we vary χ as above. For example,
we studyX
L(s, π × χ)k a(s, π, d) ∼ what?
(1)cond(χ)<X

Here a(s, π, d) is some weight factor.
One approach that has been fruitful is the Multiple Dirichlet Series approach,and becomes a nice way to introduce the whole field. If these are the objects wewant to study as we vary χ, why not construct a function of two variables that
adds them up. For example,
Z(s, w) =X L(s, π × χd )a(s, π, d)
|d|w

d

.

(2)

√
Above we have Res, Rew > 1 and χd corresponds to F ( n d)/F , and is given by
an nth power residue symbol. We have put in a weight function a(s, π, d). We willsay quite a bit more about this weight factor later.
3

The goal, the approach, is to construct this function. We understand its behavior in s as it is a sum of GL(r) L-functions; we want to obtain analytic informationin the new variable w. The s and w information gets smeared together (more than
you might think). You might think we just sum over extensions (as d corresponds
to certain cyclical extensions), but will see this isn’t the whole story.
Similarly we could look at a sum over more variables:
X L(s1 , π1 , χd )L(s2 , π2 , χd ) · · ·
|d|w

1.3

.

(3)

A First Example

Why is this series a reasonable thing to construct? Goes back to Siegel and half
integral weight Eisenstein
Let j(γ, z) be the series. 1/2
 theta multiplier: it is (in−1 ccstandard notation) d d (cz + d) . Note that d is a (quadratic) Kronecker
symbol. We can buildXe s) =E(z,j(γ, z)−1 Im(γz)s .
(4)γ∈Γ∞ \Γ0 (4)e s) is essentially
Maass showed (1937) that the mth Fourier coefficient of E(z,equal to L(2s, χm ) where χm is a quadratic character given by a Legendre symbol.
What does essentially equal mean? It means it isn’t exactly the right formula,but correct up to 2-factors (something happens at p = 2), archimedean factors(suppressing from the notation) and most importantly correction factors that take
into account that m might not be square-free. This factor will be a product ofpolynomials in ||v||−s at the places v such that ordv (m) ≥ 2.
Siegel took Maass’ observation and saidZ ∞X L(2s, χm )eE(y, s) − const term y w d× y ≈
.
(5)
mw
0
m
Goldfeld-Hoffstein in 1984 used this to get asymptotics forX
L(2s, χm ).

(6)

0<m<X

Similarly can do for m negative. Later Goldfeld-Hoffstein-Patterson used these
Eisenstein series over an imaginary quadratic field together with the Asai integral
4

to get similar results for L-functions attached to CM elliptic curves, and thenHoffstein and Rosen used the method over the rational function field (Fq (T )).
Goldfeld and Hoffstein anticipated the difficulty of settling a similar question
for automorphic forms in the higher rank case. They write:
At present, however, we cannot obtain mean value theorems for quadratic
twists of an arbitrary L-function associated to an automorphic form...
These appear to be difficult problems and their solution may ultimately involve the analytic number theory of GL(n).

1.4

Other Examples of MDS’s Arising From Rankin-Selberg
Integrals

There are other examples of Rankin-Selberg integrals that give rise to multiple
Dirichlet series (a more modern point of view to look at Siegel’s work this way).
A number of interesting examples can be understood as follows: in the previous
Section, we saw that the Mellin transform, which gives a standard L-function if
applied to something of integral weight, gives a Multiple Dirichlet Series of thedesired type when applied to an Eisenstein series of half-integral weight. Notethat the integral is no longer an Euler product in that case. In a similar way we canlook at other integrals that give Euler products (Rankin-Selberg type integrals)
when applied to an automorphic form. Replacing the automorphic form by aemetaplectic Eisenstein series (like the half-integral weight Eisenstein series E),
one can hope that the resulting object is an interesting multiple Dirichlet series.
1.4.1

Examples1. Let π be a GL(2) automorphic form. Then Bump-Friedberg-Hoffstein construct a half-integral weight Eisenstein series on GSp4 . Take an integral(represents a spin L-function, called Novodvorsky integral, when appliedin the non-metaplectic case) and get a similar type construction for a general GL(2) automorphic form, that is, a sum of quadratic twists of a GL(2)standard L-function, i.e. a function Z(s, w) as above.
(There is also another construction of Friedberg-Hoffstein that obtains sucha sum of twisted GL(2) without using GSp4 .)2. Let π be a GL(3) automorphic form. Work of Bump-Friedberg-HoffsteinGinzburg (never published) obtains the double Dirichlet series as an integralof an Eisenstein series on the double cover of GSp6 .
5

3. Suzuki, Banks-Bump-Lieman, generalizing earlier work of Bump-Hoffstein:
there is a metaplectic Eisenstein series on the n-fold cover of GL(n) (induced from the theta function on the n-fold cover of GL(n − 1)) whose
Whittaker coefficients are n-th order twists of a given GL(1) L-series. One
can then take an integral and get a sum of twists of GL(1):X L(s, ξχ(n) )
d

|d|w

d

,

(7)

where ξ is on GL(1) and is fixed. One is then able to control such sums, atleast modulo technical difficulties, discussed below. (This series has beenstudied by Friedberg-Hoffstein-Lieman, using a different method to be explained below.)
4. Similarly, working with n-th order twists,
X |L(s, χm )|2
mw

.

(8)

can be obtained as a Rankin-Selberg integral of metaplectic Eisenstein series. (This series has been studied by A. Diaconu, again using the new
method.)The point is these multiple series come out in natural ways from constructions
like this.
1.4.2

Obstructions

Why isn’t the rest of the talk just doing these integrals? These integrals involvesome difficult things:
1. truncation (can be handled, general theory of Arthur, though complicated to
do in practice; needed as these Eisenstein integrals are not convergent);
2. bad primes (bad primes are difficult to handle in Rankin-Selberg type integrals, unlike Langlands-Shahidi method, and this is particularly true inthe case of integrals involving metaplectic automorphic forms, where theprimes dividing n present additional complications);
3. integrals of archimedean Whittaker functions (the general theory here is not
as well developed).
6

Since many properties of L-functions are already known, one might hope that
one can write down and study multiple Dirichlet series without Rankin-Selberg
integrals. Remarkably, this is possible in many cases, and it is one main goal ofthis conference to explain how. However, we note that such integrals do play arole in the study of higher twists, as we shall explain at the end of this lecture.

1.5

Conceptual Overview

Question: why should these be well-behaved in w? There are lots of cases
where we think they should be better behaved then we can prove.
1.5.1

Heuristic: BFH 1996

Let F = Q, n = 2 (quadratic twists only!). We start with an L-function
X
L(s, π) =c(n)n−s .

(9)nThe family of objects of interest is L(s, π × χm ):
m
X
c(n)
n−s ,
L(s, π, χm ) =
n
n

(10)

though this is not quite right (problem if m, n not both square-free). We set
Z(s, w) =X L(s, π, χm )
m

mw

.

We have, using the definition of the L-function, thatmXX
Z(s, w) =c(n)n−s m−w .
nmn
We reverse the order of summation, to obtain
X
Z(s, w) ≈c(n)L(w, χn )n−s .

(11)

(12)

(13)

n

Note that we started with a sum of L(s, π, χm ), that is, a sum of GL(r) Lfunctions, and we obtain a sum of L(w, χn ), that is, a sum of GL(1) L-functions!!
7



n
For the moment, this is only a heuristic, as it assumes m= m
and all numbersnsquare-free and prime to the conductor.
We need to keep track of how functional equations work while we twist. We
have a functional equation sending1

L(s, π × χm ) → |m|r( 2 −s) L(1 − s, πe × χm );(14)

this assumes m is square-free. Thus Z(s, w) satisfies two types of functional
equations:1. First we have the functional equations from s → 1 − s, but because of thepower of m we have w → w + r(s − 21 ). Thus there is a link between thetwo:(s, w) → (1 − s, w + r(s − 1/2)).
(15)
2. The second, coming from w → 1 − w in the equality (13), is(s, w) → (s + w − 1/2, 1 − w).

(16)

The observation is that these functional equations generate a finite group offunctional equations for GL(1), GL(2) and GL(3), but an infinite group for GL(4)(affine Weyl group) and higher. This suggests that a good Z(s, w) (here good deals
with all numbers including those that are not square-free) should continue to C2
for GL(1), GL(2) and GL(3) but to a proper subregion for GL(4) and higher.
What about poles? There should be a pole at w = 1 (since ζ(w) arises inequation (13) when n is 1). If this does have a pole at w = 1, then by (11) this
implies the non-vanishing of L(s0 , π×χm ) for infinitely many χm ! Similarly if all
epsilon factors are −1 then one gets a non-vanishing theorem for L0 (1/2, π × χm )∂from the pole of ∂sZ(s, w) at w = 1. (In fact there are several polar divisors
meeting at (1/2, 1) and one must check that they do not cancel.) Then standard
methods involving contour integrals give mean value theorems.
Consider the case of GL(4) and higher. There the group of functional equations is infinite. If we take this infinite group and move this line w = 1 around,the poles accumulate in what looks like a barrier to continuation. See BFH’s 1996
article in the Bulletin AMS. So we shouldn’t expect continuation to all of C2 .
However, if we could get continuation up to the conjectured barrier, that would bevery significant; we would get a tremendous amount of information (Lindelöf intwisted aspect, simultaneous non-vanishing at the center of the critical strip). At
the moment this problem seems difficult.
8

The situation for GL(1), GL(2) and GL(3) is different. There we can makethe heuristic rigorous and thereby prove continuation to C2 without using RankinSelberg integrals. Applications (non-vanishing, mean-value theorems) then follow. The key point is to take advantage of the finite group of functional equations,and Hartogs’s Continuation Principle.

1.6

Hartogs’s Continuation Principle

Definition 1.1 (Tube Domain). An open set Ω ⊂ Cm is called a tube domain if
there is an open set ω ∈ Rm such that Ω = {s ∈ Cm : Re(s) ∈ w}. We writeΩ = T (ω) to denote this relation.
Definition 1.2 (Convex Hull). If R ⊂ Rm or Cm and m ≥ 2, let R̂ be the convex
hull of R.
It is easy to see that if Ω = T (ω) then Ω̂ = T (ω̂).
With this as background, the relevant result is
Theorem 1.3. If Ω is a connected tube domain, then any holomorphic function inΩ can be extended to a holomorphic function on Ω̂.
When we continue complex functions, we get to take convex hulls for free!

1.7

Continuation of Z(s, w) to C2 for GL(r) if r ≤ 3

We can now sketch the continuation of Z(s, w). First we introduce some weightfunctions so that the interchange of summations is actually valid. What we meanby this is that interchange of summation implicitly assumed everything was squarefree, which is not the case. In the work on Rankin-Selberg integral representations
for Z(s, w), these weight factors arise from Fourier coefficients; remarkably, they
are exactly the ones needed to make the interchange of summation work.
Thus we look at
X
L(s, π × χm )a(s, π, m)ξ(m)m−w ,(17)where ξ is on GL(1)), and we write this as a sum of seriesX
L(w, ξχn )b(w, ξ, π, n)n−s .

(18)This comes (for the correct choice of weight factors a(·), b(·)) from interchanging
sums and using the Law of Quadratic Reciprocity.
9The sum of L-functions in s is also a sum of L-functions in w. We can use
the convexity bounds on each of these L-functions to extend the regions of convergence, and, on these enlarged regions, to prove the corresponding functional
equations. Thus the heuristic arguments can be made rigorous, though we need to
study how the -factors change under twisting. As Fisher and Friedberg show, by
a congruence sieving one can arrange it so that the sign of the -factor does notvary. So the interchange and functional equation gives another multiplie Dirichlet
series that is basically of the same form. (As we will explain later, for higher order
twists something more complicated happens!)
We iterate this procedure until we get a region whose convex hull is C2 , andthen use Hartogs. There are finitely many poles, but we can remove these bymultiplying by a finite product of linear terms.
1.7.1

Example: Quadratic Twists of GL(3)Everything hasn’t been done in the full generality it could be, but this will show
many key features. Recall if π 0 is a cuspidal automorphic representation of GL(2)then there is the Gelbart-Jacquet lift Ad2 (π 0 ) which is an automorphic representation of GL(3). (This was referred to in other conference talks as the symmetricsquare lift.)
Theorem 1.4 (Bump-Friedberg-Hoffstein: Shalika Volume). Let π 0 be on GL2 (AQ ).
Let M be a finite set of places including 2, ∞, primes dividing cond(π 0 ). Then
there exist infinitely many quadratic characters χd such that d falls in a given
quadratic residue class mod v for all v ∈ M (mod 8 if v = 2) and such thatL( 12 , Ad2 (π 0 ) × χd ) 6= 0.
Remarks
1. Q is not essential. Moreover, with a little more work one could specify χv
for all places v ∈ M . And one should be able to do non-lifts with extra
work.
2. If π is on GL(3) this method gives a new proof that the symmetric squareL(s, π, sym2 ) is holomorphic (more precisely, one sees that the productζ(3s − 1)L(s, π, sym2 ) is holomorphic excpet at s = 1, 2/3). We shallsee why the symmetric square arises presently.
Sketch of the proof.

10

1. In the paper it is shown that not only do weight factors exist, but they are
unique (complicated set of recursion relations). These weight factors allow
us to interchange summations.
2. Establish the functional equationsα(s, w)β(s, w)==(1 − s, w + 3s − 3/2)
(s + w − 1/2, 1 − w).

(19)(Note that α and β generate a dihedral group of order 12.)
3. Use α, β, α and get a continuation to a region whose convex hull is C2 . In
fact, one shows thatw(w−1)(3s+w−5/2)(3s+2w−3)(3s = w−3/2)×bad prime factor×Z(s, w)
has analytic continuation to C2 .

Theorem 1.5. Suppose π is automorphic on GL3 (AQ ) with trivial central character. Then for σ = ±1 we have11
LM ( , π, χσd )a( , π, σd)e−d/X = CX log X + C 0 X + C 00 + O(X −3/4 ),
2
2
d>0
(20)
where C is a non-zero multiple ofX

lim(s − 1/2)LM (2s, π, sym2 );

s→1

(21)

The term C arises by contour integration as the residue of the pole at w =1. Note that by equation (13), this residue arises
the summands indexed
P fromby n a perfect square, so it is approximatelyc(n2 )n−2s , which is related toL(2s, π, sym2 ).
To complete the proof of this Theorem, suppose that π = Ad2 (π 0 ). Then
L(s, π, sym2 ) = ζ(s)L(s, sym4 (π 0 ), χ2π0 ).

(22)Here χπ0 denotes the central character of π 0 . Using this equality, one can see that
L(s, π, sym2 ) has a simple pole at s = 1. The proof in our paper uses the KimShahidi result on the automorphicity of sym4 (π 0 ) as well as the Jacquet-Shalika
11nonvanishing theorem to conclude that the second term does not vanish at s = 1,
and hence that C 6= 0. Prof. Shahidi has kindly informed me that a simpler proof
that L(1, sym4 (π 0 ), χ2π0 ) 6= 0 is available in an older paper of his.
If we took something on GL(3) that isn’t a lift then C = 0; thus this provides
an analytic way to tell if something is a lift or not from the asymptotic behavior
of its quadratically-twisted L-functions.

1.8

Summary of the Quadratic Twist CaseFor GL(1), GL(2) and GL(3): quadratic twists continue to C2 (resp. C3 , C4 forthe multi-variable sums corresponding to GL(1) × GL(1) and GL(1) × GL(2),GL(1) × GL(1) × GL(1)). The weight factors needed to make the heuristic rigorous (i.e. to show that a sum of Euler products in s is also a sum of Euler productsin w) are unique.
Though the heuristics are easiest to explain over Q, we emphasize that themethod works over a general global field (Fisher-Friedberg). For example, over afunction field we get a rational function in q −s and q −w with a specified denominator; this comes from the functional equations. For example, to any algebraic curve
over a finite field and a conductor one gets a finite dimensional vector space of rational functions of two complex variables; see Fisher-Friedberg 2004 for details
and examples.

1.9

Higher Twists

The most recent work is here, where some new phenomena occur. The heuristic
we wrote earlier goes into the trash bin. We cannot use it: the functional equations
involve Gauss sums – we cannot ignore them. We can use higher reciprocity, but
we are naturally in the world of Gauss sums. The basic idea of adopting theHartogs’s method is that we need to consider several different families of multiple
Dirichlet series that are linked by functional equations. The weight functions often
involve Gauss sums.
The basic fact is that on GL(r) the twist of an L-function by an n-th ordercharacter χm has an epsilon factor involving G(χm )r where G is a normalized
Gauss sum. Moreover, when one carries out the operations of “take the functional
equation" and “interchange summation", these two operations are not commuting
involutions (even ignoring scattering matrix, bad primes, etc.) in contrast to thequadratic-twist case. Instead, these operations give rise to the linked families of
MDS of the previous paragraph.
121.9.1n-Fold Twists of GL(1)This is work of Friedberg-Hoffstein-Liemann 2004. We end up with two different
families, the n-th order twists of the original L-function and a multiple Dirichlet
series built up from infinite sums of n-th order Gauss sums. These latter sums arise
as the Fourier coefficients of Eisenstein series on the n-fold cover of GL(2), andthey can thus be controlled by using the theory of metaplectic Eisenstein series.
Note that automorphic methods, which could be for the most part avoided in thequadratic twist case, seem unavoidable in many problems involving n-th order
twists for n > 2.
In the case at hand, the continuation of the MDS to C2 is obtained by using
the functional equation of the Fourier coefficients of Eisenstein series on the nfold cover of GL(2), which is inherited from the Eisenstein series themselves.
Note that earlier we mentioned such a sum could be approached by an integralof an Eisenstein series on the n-fold cover of GL(n). Thus the Hartogs-based
method allows one to replace the use of Eisenstein series on the n-fold cover ofGL(n) with the use of Eisenstein series on the n-fold cover of GL(2), which areconsiderably simpler. We shall see a similar reduction to GL(2) in the work of
Brubaker, Bump, and Friedberg on Weyl group multiple Dirichlet series that is the
subject of the other minicourse.
1.9.2

Cubic Twists of GL(2)This is work of Brubaker-Friedberg-Hoffsteinin 2005.
√
Let K = Q( −3). For d ∈ OK ,d ≡ 1 mod 3 let kdk denote the absolute(n)norm of d. Let χd (a) = ad n denote the Hecke character associated by class√field theory to the extension K( n d)/K, with n = 2, 3, 6. Let P (s; d) denote a
certain Dirichlet polynomialdefined in our paper. P (s; d) depends on π but wesuppressthis from the notation. P (s; d) is a complicated object, but has the propertiesthat if one factorsd = d1 d22 d33 with each di ≡ 1 mod 3, d1 square-free,d1 d22
cube-free, then P (s; d) = 1 if d3 = 1 and also for fixed d1 , d2 , the sumX
P (s; d1 d2 d3 )
d3 ≡1

2 3
3w
kd3 k

mod 3

converges absolutely for Rew > 1/2 and Res ≥ 1/2.
Then we prove
Theorem 1.6. Let π = ⊗πv be an automorphic representation of GL(2, AK ) suchthat L(s, π, χ) isentire for all Hecke characters χ such that χ3 = 1. Let S be a13

finite set of primes including the archimedean prime and the primes dividing 2, 3
and thelevel of π.Then, for any sufficiently large positive integer k, the asymptotic
formula

k
X
1 (3)
kdk
(3)
LS (s, π, χd1 d2 )P (s; d) 1 −∼c (s, π)X
2
X
k+1
kdk<X

holds for any s with Res ≥ 1/2. The constant c(3) (s, π) isnon-zero, and is givenbyYc(3) (s, π) = cS LS (3s, π, ∨3 )ζS (6s)ζS (6s+1)−1(1−γp3 kpk−3s−1 )(1−δp3 kpk−3s−1 ),p∈S/where ζS denotes the Dedekind zeta function of K with the Euler factors at theplaces in S removed, γp , δp are the Satake parameters of the representation πp ,and cS is a non-zero constant.
An immediate consequence of this, the convergence of the basic sum, and the(3)usualconvexity bound forL(1/2, π, χd1 d2 ) is2

Corollary 1.7. Let π be as in the Main Theorem. Then there exist infinitely many(3)
cube-free d such that L(1/2, π, χd ) 6= 0. More precisely, let N (X) denote thenumber of such d with kdk ≤ X. Then for any  > 0, N (X)  X 1/2− .
Sketch of the Proof: Define the multiple Dirichlet series(3)

LS (s, π, χd1 d2 )P (s; d)X

Z1 (s, w) =d≡1

2

mod 3,(d,S)=1

kdkw

.

(Here the sum is over all d ∈ OK with d ≡ 1 mod 3 and ordv (d) = 0 for all finitev ∈ S.) This series convergesabsolutely for Re(s), Re(w) > 1. We establish thecontinuation of this function to a larger region. Let
Z ∗ (s, w) = Z1 (s, w) ζS (6s + 6w − 5) ζS (12s + 6w − 8)×Y(1 − γp3 kpk2−3s−3w )−1 (1 − δp3 kpk2−3s−3w )−1 ,p∈S/

where γp , δp are the Satake parameters of the representation πp .We show thatZ ∗ (s, w)
has a meromorphic continuation to the half plane Re(s + w) > 1/2 and is analytic
14

in this region except for polarlines at w = 1, w = 0, w = 5/3 − 2s, w = 3/2 − 2s,
w = 4/3−2s, w = 7/6−s, w = 1−s, w = 5/6−s. We alsoshow that theresidue
atw = 1 satisfies
Resw=1 Z∗ (s, w) = cS LS (3s, π, sym3 ) ζS (6s) ζS (12s − 2)and is an analytic function of s for Res > −1/2, except possibly at the pointss = 1/3, 1/4, 1/6, 0, which require a more detailed analysis. The properties ofthesymmetric cube L-series have been completely described by Kim and Shahidi.

The First Two Series and the First Functional Equation
This step is based on the exact functional equation for the cubically-twisted
L-series.Write d = d1 d22 d33 as above. Ignoring bad primes such asthose dividing(3)
the level of π and the infinite place,L(s, π, χd1 d2 ) has a functional equation of the
2
form
(3)
(3)
(3)
L(s, π, χd1 d2 ) → π G(χd1 d2 )2 L(1 − s, π̃, χ̄d1 d2 )kd1 d2 k1−2s .
222Here π̃ denotes the contragredient of π,π (the central value of the usual epsilon(3)factor for π) has absolute value 1 andG(χd ) is the usual Gauss sum associated(3)
to χd ,normalized to have absolute value 1. The crucial factor kd1 d2 k1−2s arises(3)
as part of the epsilon-factor of the twisted L-function since π ⊗ χd is ramified at
the primes dividing d1 d2 . This functional equationgives rise to a functional equation for the double Dirichlet series Z1 ,reflecting Z1 (s, w) into a second double
Dirichletseries
(3)

(3)

Z6 (s, w) =X LS (s, π̃, χ̄d1 d2 ) G(χd1 d2 )2 P (1 − s; d1 d22 d33 ) kd2 d33 k1−2s22

kd1 d22 d33 kw.

More precisely, the functional equation above induces a transformationrelating
Z1 (s, w) to Z6 (1 − s, w + 2s − 1).(The exact transformation is somewhat complicated due to bad primes.)
The Second Functional Equation Next we study the series Z6 (s, w) itself.The(3)

appearance of G(χd1 d2 )2 , the square of a cubic Gauss sum,introduces, via the2
Hasse-Davenport relation, a conjugate 6th order Gausssum. However, the Fourier15

coefficients of Eisenstein series on the 6-fold cover of GL(2) may be written assums of Gauss sums
X
G(6) (m, d)
,
kdkw
d≡1

mod 3,(d,S)=1

and accordingly series of this type possess a functional equation in w. We show,
using this functional equation, that Z6 (s, w) possesses a functional equation as(s, w) →(s + 2w − 1, 1 − w), transforming into itself.
The Third Series and the Third Functional Equation
We show that the order of summation in Z1 (s, w)written as a doubly-indexed
Dirichlet series can beinterchanged, leading to an expression of the form
(3)

Z1 (s, w) =X LS (w, χm1 m2 ) Q(w; m1 m22 m33 )2

km1 m22 m33 ks,

where Q is once again a specific Dirichlet polynomial depending on πand the Lseries on the right are Hecke L-series.Applying the functional equation in w for
the Hecke L-series weare led to introduce the third double Dirichlet series(3)

Z3 (s, w) =(3)

X LS (w, χ̄m1 m2 ) G(χm1 m2 ) Q(1 − w; m1 m22 m33 ) km2 m33 k1/2−w
2

2

km1 m22 m33 ks.

The functional equation for the Hecke L-series induces a transformationrelating
Z1 (s, w) to Z3 (s + w − 1/2, 1 − w).
Continuing Z3 Once again, the series Z3 may be studied using metaplectic
Eisenstein series. Indeed, we showthat this series is a sum of cubictwists ofRankin-Selberg convolutions of π with the theta function on the3-fold cover ofGL(2). (Recall that this function is the residue of anEisenstein series on the 3fold cover of GL(2); see Patterson’s Crelle paper.) From the meromorphic continuation of the twisted Rankin-Selberg convolutionswe deduce a corresponding
continuation for Z3 .
Applying Hartogs’s Theorem We now apply Hartogs’s theorem to obtain thecontinuation of these 3 functions.The functions Z1 (s, w) and Z6 (s, w) have overlapping regions of absoluteconvergence. If the functional equation interchanging
Z1 (s, w) andZ6 (s, w) is used several times, the convexity principle for severalcomplex variables applied to the union of translates of these regions impliesan
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analytic continuation of Z1 (s, w) and Z6 (s, w) to the half plane Re(w + s) > 3/2.
The relations with Z3 (s, w) then imply an analyticcontinuation to the half plane
Re(w + s) > 1/2, which is what we require forour applications.
Remarks:
1. A furtherfunctional equation transforming Z3 (s, w) into itself as(s, w) →(1 − s, w + 4s − 2), can be proved. This then allows ananalytic continuation of all three double Dirichlet series to C2 . This alsogives rise to agroup of functional equations which is non-abelianand of order 384. These
computations have not been written down in detail.
2. As mentioned above, in the quadratic twist case the double Dirichletseries
for r = 1, 2, 3 can be identified, up to afinite number of places, with certain
integral transforms ofmetaplectic Eisenstein series. In the case at hand,
althoughthere is no known way to construct the double Dirichlet series asasimilar integral transform (or as a Rankin-Selberg convolution),there is anatural candidate attachedto the cubic cover of G2 , and it is possible that ourcomplicatedformulas reflect in a certain sense combinatorial issues arising
fromthat group.
3. One may also obtain a mean value result for the product of two Hecke Lfunctionsin different variables when they aresimultaneously twisted by cubic characters. This has been accomplishedby BEN BRUBAKER in his
Brown University doctoral dissertation.

1.10

Concluding Remarks

The Multiple Dirichlet Series that continue to a product of complex planes areready-made for establishing distribution results via contour integration. We emphasize that the MDS method applies over a general global field containing sufficiently many roots of unity; thus such mean value theorems may be establishedwithout 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, π, d).
There are many additional recent applications of Multiple Dirichlet Series toautomorphic forms and analytic number theory. These will be discussed by Chinta
is his lectures. For completeness, we note some of them now.

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1. A proof that one twist of a GL(2) L-function of order n (a prime) has
nonzero central value implies that infinitely many do (Brubaker, Bukur,
Chinta, Frechette, Hoffstein).
2. Generalization of the Luo-Ramakrishnan theorem (which characterizes aGL(2) modular form by the central values of its quadratically-twisted Lfunctions) to all number fields, due to my doctoral student Li Ji.
3. A generalization of the Luo-Ramakrishan theorem to GL(3) over Q by
Chinta and Diaconu.
4. The use of unweighted multiple Dirichlet series to prove simultaneous nonvanishing theorems inside the critical strip and also a distribution theorem
at s = 1, by Chinta, Friedberg, and Hoffstein. (Note: Unfortunately, Chinta
did not have time to discuss this in his talk. But I am hoping that he will
include it in his lecture notes.)
5. Chinta’s mean value theorem for biquadratic extensions of Q.
6. Work of Diaconu and Tian applying MDS to study twisted Fermat curves
over totally real fields.
7. Work of Diaconu, Goldfeld, and Hoffstein relating MDS to predictions
coming from random matrix theory.
An additional set of lectures by Bump and Brubaker describes the very new theoryof MDS’s attached to a reduced root system, which we have dubbed Weyl GroupMultiple Dirichlet Series. As this list shows, the theory of MDS seems quite rich.
In the rest of this conference these matters will be developed further.

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