More applications of multiple
Dirichlet series
Gautam Chinta
Bretton Woods, NH
13 July 2005

Nonvanishing nth order twists of a GL2form
Theorem (Fearnley-Kisilevsky, David-F-K) Fixn prime. Let E be an
curve defined over elliptic
Q. Suppose Lalg 12 , E 6≡ 0 (mod n). Then
1Lalg 2 , E ⊗ χ is nonzero for infinitely many

Dirichlet characters of order n.
Proof Arithmetic of modular symbols and Cebotarev density theorem.
Arithmetic interpretation Let F be a cyclic
extension of Q of degree n. Then a Galois character χ of Gal(F/Q) corresponds to a Dirichlet
character of order n. The Birch/SwinnertonDyer conjecture predicts that
1L 2 , E ⊗ χ is nonzero

iffrank of E(F )χ is zero.
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Nonvanishing nth order twists, (cont.)
Theorem(Brubaker, Bucur, C, Frechette, Hoffstein) Let K be a number field containing thenth roots of unity. Let π be a cuspidal, self-dual
automorphic
 representation of GL2(AK ). Sup1 , π 6= 0. Then there exist infinitely
pose L 2many idele class characters χ of K of order ns.t. L 12 , π ⊗ χ 6= 0.
So, BBCFH + Modularity + BSD implies:
Let E/K be a rank zero elliptic curve. Thenthere exist infinitely many degree n cyclic extensions F of K s.t. #E(F ) < ∞.

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Nonvanishing nth order twists, (cont.)
Remarks(i) n = 2 (Waldspurger, Kohnen-Zagier) relates quadratic twists of L-series of π toFourier coefficients of a metaplectic cuspidal automorphic representation. Don’tknow how to do this for n > 2!
1, π(ii) Note necessity of assumption L 26=0 in both [FK] and [BBCFH]. Theorem
should be true without this assumption. Infact, “almost all” twists should be nonzero
when n > 2. (See e.g. [DFK] where a random matrix model is given for predicting
frequency of vanishing twists.)3

(iii) (Diaconu-Tian) Let p be a prime number,
F a totally real field of odd degree s.t.
[F (µp) : F ] = 2. Let Wδ be the twisted
Fermat curveWδ : xp + y p = δ.
Then there exist infinitely many δ ∈ F ×/F ×pfor which Wδ has no F -rational solutions.
The proof of this result is based on Zhang’s
extension of the Gross-Zagier formula tototally real fields and on Kolyvagin’s technique of Euler systems. Then, a double
Dirichlet series is used to show that a certain family of twisted L-series has nonvanishing central value infinitely often.

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Idea of proof of BBCFH
Consider the double Dirichlet series
Z(s, w; π) ==(n)X L s, π ⊗ χd|d|w

G(1, χd)

X amD(w, m)|m|swhere D(w, m) is the Gauss sum Dirichlet series of Kubota, which is essentially the mth
Fourier coefficient of Ẽ (n)(z, W ).
Some properties of Z(s, w; π) :• Two functional equations:(s, w) →7(1 − s, w + 2s − 1)
1 , 1 − w).
(s, w) →7(w + s − 2• Meromorphic continuation to C2.
1 + 1.
• simple pole at w = 2n
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Idea of proof of BBCFH, (cont.)
1 + 1 : is a conResidue of Z(s, w; π) at w = 2nstant multiple of1, π ⊗ θ(n) .
L s+2n

ThereforeXL

1 , π ⊗ χ(n)d2

G(1, χd)|d|<x1+11 + 1 −1(n)1∼ ∗L 2 +,π ⊗ θ
x 2 n + O(x 2 n ).
2n





1 , π ⊗ θ (n)How to show L 1+22nnonzero??

6

But nonvanishing assumption L 12, π



6 0 pro=vides another means of showing infinitely many
1 . Thennonvanishing central twists. Fix s = 2
we have functional equation
Z 12 , w; π
17 Z 2 , 1 − w; π .
→But a nonzero Dirichlet series with such a functional equation must have infinitely many nonzero
coefficients!
Recently an unconditional nonvanishing theorem for n = 3 has been established by [BrubakerFriedberg-Hoffstein]. It remains an open problem to prove such a statement for n ≥ 4.

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Determination of modular forms by twistsof critical values
Theorem (Luo-Ramakrishnan) Let f, g be two
Hecke newforms for a congruence subgroup ofSL2(Z). Suppose there exists a nonzero constant c s.t.
L 1
2 , f ⊗ χd= cL 1
2 , g ⊗ χd

for all quadratic characters χd. Then f = cg.
Theorem is more general than stated: applies
to any critical values, also to other families oftwists, ...

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Theorem of Luo and Ramakrishnan• An application to a question of Kohnen: letg1, g2 be two newforms in the Kohnen subspace S + 1 with Fourier coefficients b1(n), b2(n)k+ 2respectively. Suppose
2b2
1 (|D|) = b2 (|D|)

for almost all fundamental discriminants with(−1)k D > 0. Then g1 = ±g2, i.e. you can’t
just switch some of the signs of the coefficients and get another eigenform. Proofuses Waldspurger’s formula relating the squareof bj (|D|) to a suitable multiple of a twisted
central value.
• Similar theorem holds for central derivatives in the case of negative root number[L-R]. By the theorem of Gross-Zagier, thisallows one to determine an elliptic curve byheights of Heegner points.
9Extensions of Luo-Ramakrishnan• Ji Li, in his recent thesis, extends [LR]to π1, π2 cuspidal automorphic representations of GL2(AK ), for K an arbitrary number field.

• (C, Diaconu) extends [LR] to symmetricsquares of cusp forms on GL2(AQ).
All of these theorems are proved by considering
twisted averages of twists of central L-values.
The result of J. Li should also extend to cover
the case of determining π by twisted central
derivatives. Over a number field, the averaging method employed by [LR] (originating in
the work of Iwaniec and Murty-Murty) will notwork.
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Proof of Chinta-Diaconu
By contrast with J. Li’s result, the result of[CD] is valid only over Q. It relies on GL3 averaging techniques of Bump-Friedberg-Hoffsteinand Diaconu-Goldfeld-Hoffstein. It would beinteresting to see if the methods of Soundararajan will work here.
Idea: for r prime, consider the twisted averages
X
1
1L 2 , π1 ⊗ χd χr (d)Pd 2 , π1 ∼0<d<xar (π1)x log xX
1
1L 2 , π2 ⊗ χd χr (d)Pd 2 , π2 ∼


0<d<x

ar (π2)x log x
It follows that ar (π1) = ar (π2) provided wecan remove the weights Pd 12 , πi from the summation.
11Proof of Chinta-Diaconu (cont.)
In sieving to to remove the weights, we require• Lindelöf on average:

P|d|<x
1|L 2 , π ⊗ χd | <<x5/4+
• Work of Kim and Kim-Shahidi establishing
the automorphy of the symmetric fourth
power of a GL2 form
Of course,X
1|L 2 , π ⊗ χd | << x1+|d|<xis expected. Our proof of the bound above isvalid only over Q, as we appeal to a charactersum estimate of Heath-Brown. It would be ofgreat interest to see what types of bounds canbe proved over an arbitrary number field.
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Mean values of quadratic zeta functions
Recall result of Jutilla, Takhtadzjan- Vinogradov:
XL(1/2, χd) ∼ c1X log X + c2x + O(x1−).

0<±d<X

Multiplying the LHS by ζ(s), the sum will range
over Dedekind zeta functions of quadratic extensions of Q, ordered by conductor.
The quadratic multiple Dirichlet series associated to A5 will give an analogous mean value
result for zeta functions of biquadratic extensions.

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Dynkin diagrams and multiple Dirichlet
seriesGiven a simply-laced Dynkin diagram, verticesv1, · · · , vr . Construct a multiple Dirichlet series
which is roughly of the form:∞XQ

n1 ,n2 ,...,nr =1

j > i, vj adjacent to vi
sn11 · · · nsrrninj.

Functional equations:
σi : si 7→ 1 − si, sj 7→ sj + si − 1/2,if vj is adjacent to vi. The other sk are left
unchanged.
We have the Coxeter relations
σi2 = 1,(σiσj )(i,j) = 1,

where (i, j) = 3 if vi and vj are adjacent nodes
in the Dynkin diagram, and (i, j) = 2 if theyare not.
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The region of absolute convergence of the heuristic multiple Dirichlet series contains a Weylchamber. Translating this region by the groupof functional equations will yield the analytic
continuation of Z to Cr .
The multiple Dirichlet series constructed in thismanner are conjectured to coincide with theWhittaker coefficients of Eisenstein series on
the metaplectic double cover of the split simply
connected semisimple group associated withthis Dynkin diagram. (Bump-Hoffstein verifiedfor A2.)
See Brubaker-Bump-Chinta-Friedberg-Hoffstein15

Biquadratic zeta functionsThis heuristic can be precisely realized for A5.
The associated multiple Dirichlet series is roughlyof the form Z(s1, s2, s3, s4, s5) =X L(s1, χd )L(s3, χd d )L(s5, χd )
4
22 4.
s2 s4
d2 d4
d2 ,d4

Specialize to s1, s3, s5 = 1/2, analytically continue, and contour integration:
Theorem(C)X

1 , χ )L ( 1 , χ )L ( 1 , χa(d1, d2)L2( 22 2 d2
2 2 d1 d2 )d1

d1 ,d2 >0
d1 d2 <xodd∼3
ζ2 ( 32 )ζ2 (2)48

X log4 X,

as X → ∞.
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