Voronoi summation formulae on GL(n)

Journal of Number Theory 162 (2016) 483–495

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Journal of Number Theory www.elsevier.com/locate/jnt

Voronoi summation formulae on GL(n) Fan Zhou Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA

a r t i c l e

i n f o

Article history: Received 3 February 2015 Received in revised form 22 September 2015 Accepted 5 October 2015 Available online 10 December 2015 Communicated by David Goss

a b s t r a c t We discover new Voronoi formulae for automorphic forms on GL(n) for n ≥ 4. There are [n/2] different Voronoi formulae on GL(n), which are Poisson summation formulae weighted by Fourier coefficients of the automorphic form with twists by some hyper-Kloosterman sums. © 2015 Elsevier Inc. All rights reserved.

MSC: primary 11F30 secondary 11F55 Keywords: Maass form Automorphic form Voronoi formula Functional equation Kloosterman sum

1. Introduction A Voronoi summation formula of an automorphic form is a Poisson summation formula weighted by Fourier coefficients of the automorphic form with twists by some additive arithmetic functions. The Voronoi formula of an automorphic form on GL(2) is well known and has a lot of applications. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jnt.2015.10.012 0022-314X/© 2015 Elsevier Inc. All rights reserved.

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A Voronoi formula of an automorphic form on GL(3) was first proved by Miller and Schmid in [MiSc1]. Later, a Voronoi formula was established for GL(n) for n ≥ 4 in [MiSc2,GoLi1,GoLi2], with [MiSc2] being more general and earlier than [GoLi2] (see the addendum to [GoLi2]). An adelic version was established in [IcTe]. The Voronoi formula has important applications, such as a nontrivial bound on the exponential sum in [Mi] and subconvexity bounds in [Li1]. Later, Xiaoqing Li and Stephen Miller established a new Voronoi formula on GL(4) (Theorem 1.2). Their formula is an equality between a weighted sum of Fourier coefficients of an automorphic form on GL(4) and a dual sum, both twisted by classical Kloosterman sums. Inspired by the new formula on GL(4) of Li and Miller, we discover that such a new formula shall not be isolated to GL(4). For an automorphic form on GL(n), we have [n/2] different Voronoi formulae. For an integer k ≥ 1, define the (k − 1)-dimensional hyper-Kloosterman sum by 

Klk (m, q) =

x1 ,...,xk−1 mod q (x1 ,q)=···=(xk−1 ,q)=1

 e

x1 + · · · + xk−1 + mx1 . . . xk−1 q



for a prime number q. Our formula on GL(n) is an equality between a sum of Fourier coefficients of an automorphic form twisted by hyper-Kloosterman sums of dimension (k − 1) and a dual sum twisted by hyper-Kloosterman sums of dimension (n − k − 1). Theorem 1.1. Let π be a Hecke–Maass cusp form for SL(n, Z) of the spectral parameter (λ1 , . . . , λn ) ∈ Cn and n ≥ 2. Let A(m1 , . . . , mn−1 ) be the Fourier coefficient of π. Let ω ∈ Cc∞ (0, ∞) be a test function with ω ˜ its Mellin transform and let q be a prime number, a and a two integers with aa ≡ 1 mod q. For 1 ≤ k ≤ n − 1, we have ∞ 

A(1, . . . , 1, m)Klk (am, q)ω(m)

m=1

+

∞  

position l

l+k l−1

(−1)

q

  

A(1, . . . , 1, q, 1, . . . , 1, m)ω mq l

2≤l≤k m=1 ∞

=

q k  A(m, 1, . . . , 1) 2 m=1 m      m m + Kln−k (−¯ × Kln−k (¯ am, q) (Ω+ + Ω− ) am, q) (Ω+ − Ω− ) n q qn position l

+

 2≤l≤n−k

     m n−k+l k−1 A(m, 1, . . . , 1, q, 1, . . . , 1) , Ω+ (−1) q m q n−l m=1 ∞ 

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where Ω± is defined as

Ω± (x) =

1 2πi



s=−σ

ω ˜ (s)i−nδ π −n(1/2−s)

n j=1

 Γ

1 + δ − s − λj 2

  −1 s + δ − λj Γ xs ds. 2

For even Maass forms we define δ = 0 for Ω+ and δ = 1 for Ω− , and for odd Maass forms we define δ = 1 for Ω+ and δ = 0 for Ω− . Proof. This is literally the sum of Theorem 3.5 and Theorem 4.2. 2 Let k in Theorem 1.1 take values k = 1, 2, . . . , [n/2], and then we have the aforementioned [n/2] different Voronoi formulae on GL(n). The formula of [MiSc1,MiSc2,GoLi1, GoLi2,IcTe] overlaps our formulae in the case of k = 1. The new formula on GL(4) of Li and Miller overlaps our formulae in the case of n = 4 and k = 2. Therefore, our formulae are new, for n ≥ 5 and k ≥ 2. Our proof is based on the functional equations of the twisted L-functions of π by (multiplicative) Dirichlet characters. The relationship between the Voronoi formulae and the functional equations of L-functions is known to experts. The same method was used in Duke–Iwaniec [DuIw] and Section 4 of [GoLi1], and in proving Lemma 1.3 in [BuKh] for a special case on GL(3). Our proof does not depend on automorphicity but depends only on the family of the functional equations of GL(1)-twisted L-functions. Functional equations of GL(1)-twists are consequences of automorphicity but they do not imply automorphicity on GL(n) when n ≥ 3. In view of the converse theorem on GL(n) of Cogdell and Piatetski-Shapiro [CoPS], a proof based on GL(1)-twisted functional equations is stronger than a proof based on automorphicity. Our theorem and proof can be easily applied to some conjectural functorial lifts. For example, although the Rankin–Selberg convolutions on GL(m) ×GL(n) are not generally known to be automorphic on GL(m × n), except the few cases of GL(2) × GL(2) and GL(2) × GL(3), the functional equations of GL(m) × GL(n) are well known. Thus, we have the Voronoi formulae for the Rankin–Selberg convolutions on GL(m) × GL(n). Considering the Langlands–Shahidi method and the general Rankin–Selberg method, there are numerous functorial cases to have Voronoi formulae without being known to be automorphic. The obvious weakness of our result is that the denominator q in the hyper-Kloosterman sums has to be a prime number. More general formulae with arbitrary integer q awaits exploration. It is pointed out by the unpublished work of Xiaoqing Li and Stephen Miller that the formula in Theorem 1.1 and more general ones can be proved by taking finite sums of the Voronoi summation formula in [MiSc2]. It also points out that the crucial change needed to prove the general formula is to modify the chain of T ∗ operators used in (4.15)

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of [MiSc2]. To end the introduction, we record Li and Miller’s balanced Voronoi formula on GL(4), without proof. Theorem 1.2 (Li and Miller’s balanced Voronoi formula on GL(4)). Let π be a cusp form for SL(4, Z). We follow the notations in Theorem 1.1. Let S( , ; ) be the classical Kloosterman sum. For an integer N > 1 we have the identity ∞   m=−∞ d|N

= N2

  N ω(md2 ) dA(1, d, m)S m, 1; d

   ∞   A(m, d, 1)  N md2 , S m, 1; Ω md d N4 m=−∞ d|N

where Ω is defined by the representation of π at the Archimedean place Ω(x) =

1 2πi

 ω ˜ (s) s=−σ

L∞ (1 − s, π ˜ )∞ (s, π) s x ds. L∞ (s, π)

The work of Li and Miller also shows that, by some adjustment in the proof, a more general formula of the summation of A(a, d, m) for some fixed a instead of A(1, d, m) on the left of the formula in Theorem 1.2 can be obtained. 2. Background Let π be a Hecke–Maass cusp form for SL(n, Z) with the spectral parameter (λ1 , . . . , λn ) ∈ Cn . Let A(∗, . . . , ∗) be the Fourier–Whittaker coefficient of π. We have A(m1 , . . . , mn−1 ) = A(mn−1 , . . . , m1 ), and A(1, . . . , 1) = 1. We refer to [Go] for the definitions and the basic results of Maass forms for SL(n, Z). The dual (contragredient) Maass form of π is denoted by π ˜ . Let B(∗, . . . , ∗) be the Fourier–Whittaker coefficient of π ˜ and we have A(m1 , . . . , mn−1 ) = B(mn−1 , . . . , m1 ). Define the ratio of Gamma factors −nδ −n(1/2−s)

G± (s) := i

π

n j=1

 Γ

δ + 1 − s − λj 2

  −1 δ + s − λj Γ , 2

where for even Maass forms, we define δ = 0 in G+ and δ = 1 in G− and for odd Maass forms, we define δ = 1 in G+ and δ = 0 in G− . We refer to Section 9.2 of [Go] for the definition of even and odd Maass forms.

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The standard L-function of π is L(s, π) =

∞  A(1, . . . , 1, m) . ms m=1

It satisfies the functional equation L(s, π) = G+ (s)L(1 − s, π ˜ ).

(1)

Let q be a prime number. Let ψ be a nontrivial Dirichlet character mod q and τ (ψ) its Gauss sum. The ψ-twisted L-function is L(s, π × ψ) =

∞  ψ(m)A(1, . . . , 1, m) . ms m=1

Let k be an integer with 1 ≤ k ≤ n − 1. In the case of ψ(−1) = 1, the twisted L-function satisfies the functional equation ¯ L(s, π × ψ) = τ (ψ)n q −ns G+ (s)L(1 − s, π ˜ × ψ), which can be rewritten as ¯ k L(s, π × ψ) = τ (ψ)n−k q k−ns G+ (s)L(1 − s, π ¯ τ (ψ) ˜ × ψ).

(2)

In the case of ψ(−1) = −1, the twisted L-function satisfies the functional equation ¯ L(s, π × ψ) = τ (ψ)n q −ns G− (s)L(1 − s, π ˜ × ψ), which can be rewritten as ¯ k L(s, π × ψ) = τ (ψ)n−k q k−ns G− (s)L(1 − s, π ¯ τ (ψ) ˜ × ψ).

(3)

2.1. Hyper-Kloosterman sums Let q be a prime number. For abbreviation, we define e(x) = exp(2πix). Let k be a positive integer. The (k − 1)-dimensional hyper-Kloosterman sum is defined as 

Klk (m, q) =

x1 ,...,xk−1 mod q (x1 ,q)=···=(xk−1 ,q)=1

 e

x1 + · · · + xk−1 + mx1 . . . xk−1 q



and we shall note that if q|m, Klk (m, q) = (−1)k−1 .

(4)

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In the degenerate case of k = 1, we have the additive character  Kl1 (m, q) = e

m q

 .

When k = 2, the 1-dimensional hyper-Kloosterman sum is identical to the classical Kloosterman sum.

∗ We define as the summation over nontrivial Dirichlet characters ψ of mod q. ψ mod q It is known  q−1 k  if (m, q) = 1 ∗ k¯ 2 (Klk (m, q) + Klk (−m, q)) − (−1) , (5) τ (ψ) ψ(m) = 0, otherwise. ψ mod q ψ(−1)=1

It is also known 

∗

¯ = τ (ψ)k ψ(m)

ψ mod q ψ(−1)=−1

q−1 (Klk (m, q) − Klk (−m, q)) . 2

(6)

We will use the family of (multiplicative) Dirichlet characters to recover the additive characters and the hyper-Kloosterman sums by the two identities above. 3. Even part Let q be a prime number, a and a two integers with aa ≡ 1 mod q. Let k be an integer with 1 ≤ k ≤ n − 1. Define two Dirichlet series Lq (s, π) =

 q|m,m>0

A(1, . . . , 1, m) ms

and Lq (s, π ˜) =

 q|m,m>0

A(m, 1, . . . , 1) . ms

Define two Dirichlet series  ∗ ¯ k ψ(a)L(s, π × ψ) + (−1)k (L(s, π) − qLq (s, π)) Z(s, q) = τ (ψ) ψ mod q ψ(−1)=1

and ˜ q) = Z(s,



∗

ψ mod q ψ(−1)=1

¯ + (−1)n−k (L(s, π τ (ψ)n−k ψ(a)L(s, π ˜ × ψ) ˜ ) − qLq (s, π ˜ )) .

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Lemma 3.1. We have the identities ∞ q−1  A(1, . . . , 1, m) Z(s, q) = (Klk (am, q) + Klk (−am, q)) 2 m=1 ms

and ∞  A(m, 1, . . . , 1) ˜ q) = q − 1 (Kln−k (¯ am, q) + Kln−k (−¯ am, q)) . Z(s, 2 m=1 ms

Proof. This follows from (4) and (5). 2 For 1 ≤ l ≤ n − 1, define a Dirichlet polynomial position i

Hl (s, q) =

   (−1)n i A(1, . . . , 1, q, 1, . . . , 1) (−1) + ns , is q q

n−1  i=l

where q appears on the ith position from the right. We define another Dirichlet polynomial position i

˜ l (s, q) = H

   (−1)n i A(1, . . . , 1, q, 1, . . . , 1) (−1) + , q is q ns

n−1  i=l

where q appears on the ith position from the left. Lemma 3.2. We have the identities position l

   ∞  A(1, . . . , 1, q, 1, . . . , 1, m) l L(s, π)Hl (s, q) = (−1) q ls ms m=1

(7)

for 2 ≤ l ≤ n − 1 and L(s, π)H1 (s, q) = −Lq (s, π).

(8)

Proof. By the Hecke relation position i

   A(1, . . . , 1, q, 1, . . . , 1)A(1, . . . , 1, m) ⎧ position i+1 position i ⎪ ⎪     ⎪ ⎨A(1, . . . , 1, q, 1, .  . . , 1, m) + A(1, . . . , 1, q, 1, . . . , 1, m/q), = position i ⎪ ⎪    ⎪ ⎩A(1, . . . , 1, q, 1, . . . , 1, m),

if q|m otherwise,

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we have (7) for 2 ≤ l ≤ n − 2 after multiplying the Dirichlet series L(s, π)Hl (s) term by term. Similarly, by the Hecke relation

A(1, . . . , 1, q)A(1, . . . , 1, m) =

 A(1, . . . , 1, mq) + A(1, . . . , 1, q, m/q),

if q|m

A(1, . . . , 1, mq),

otherwise,

and

A(q, 1, . . . , 1)A(1, . . . , 1, m) =

 A(q, 1, . . . , 1, m) + A(1, . . . , 1, m/q), A(q, 1, . . . , 1, m),

we have (8), and (7) for l = n − 1, respectively.

if q|m otherwise,

2

By a similar proof, we have the following lemma for π ˜. Lemma 3.3. We have the identities position l

   ∞  A(m, 1, . . . , 1, q, 1, . . . , 1) l ˜ L(s, π ˜ )Hl (s, q) = (−1) q ls ms m=1 for 2 ≤ l ≤ n − 1 and ˜ 1 (s, q) = −Lq (s, π L(s, π ˜ )H ˜ ). Lemma 3.4. We have the following identity between Dirichlet polynomials  1 + H1 (s, q) + (q l−1 − q l−2 )Hl (s, q) q 2≤l≤k

⎛

1 ˜ 1 (1 − s, q) + = (−1)n q k−ns ⎝ + H q



⎞ ˜ l (1 − s, q)⎠ . (q l−1 − q l−2 )H

2≤l≤n−k

Proof. The left side equals position i

   (−1)n q k−1 1 i min{i,k}−1 A(1, . . . , 1, q, 1, . . . , 1) + (−1) q + . q i=1 q is q ns n−1 

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The right side equals position i

1 + q

n−1  i=1

   (−1)n q n−k−1 i min{i,n−k}−1 A(1, . . . , 1, q, 1, . . . , 1) (−1) q + i−is q q n−ns position i

   1 (−1)n q n−k−1 n−i min{n−i,n−k}−1 A(1, . . . , 1, q, 1, . . . , 1) = + (−1) q + (n−i)−(n−i)s q i=1 q n−ns q n−1 

position i

=

(−1)n q −k−1 q −ns

   1 n−i min{0,i−k}−1 A(1, . . . , 1, q, 1, . . . , 1) + (−1) q + . −(n−i)s q q i=1 n−1 

After multiplying (−1)n q k−ns , we match term by term with the left side. 2 3.1. Theorem 3.5 Let ω(x) ∈ Cc∞ (0, ∞) be a test function. Let ∞ ω(x)xs−1 dx

ω ˜ (s) = 0

be the Mellin transform of ω. Define 1 Ω+ (x) = 2πi

 ω ˜ (s)G+ (s)xs ds.

(9)

s=−σ

We have the Mellin inversion theorem 1 ω(x) = 2πi



x−s ω ˜ (s) ds.

(10)

s=σ

Theorem 3.5. Let π be a Hecke–Maass cusp form for SL(n, Z) with n ≥ 2. Let A(m1 , . . . , mn−1 ) be the Fourier coefficient of π. Let ω ∈ Cc∞ (0, ∞) be a test function and let q be a prime number, a and a two integers with aa ≡ 1 mod q. For 1 ≤ k ≤ n −1, we have ∞ 1  A(1, . . . , 1, m) (Klk (am, q) + Klk (−am, q)) ω(m) 2 m=1

+

∞   2≤l≤k m=1

position l

l+k l−1

(−1)

q

  

A(1, . . . , 1, q, 1, . . . , 1, m)ω mq l

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  ∞ q k  A(m, 1, . . . , 1) m (Kln−k (¯ = am, q) + Kln−k (−¯ am, q)) Ω+ 2 m=1 m qn position l

+

     m n−k+l k−1 A(m, 1, . . . , 1, q, 1, . . . , 1) , Ω+ (−1) q m q n−l m=1 ∞ 

 2≤l≤n−k

where Ω+ is defined in (9). Proof. By the functional equations (1) and (2) and Lemma 3.4, we have 

Z(s, q) + (−1)k q

(q l−1 − q l−2 )Hl (s, q)L(s, π)

2≤l≤k

=



∗

¯ k ψ(a)L(s, π × ψ) τ (ψ)

ψ mod q ψ(−1)=1

⎛

+ (−1)k ⎝L(s, π) − qLq (s, π) + q



⎞ (q l−1 − q l−2 )Hl (s, q)L(s, π)⎠

2≤l≤k

=



∗

¯ k ψ(a)L(s, π × ψ) τ (ψ)

ψ mod q ψ(−1)=1

⎛

⎞  1 + (−1)k q ⎝ + H1 (s, q) + (q l−1 − q l−2 )Hl (s, q)⎠ L(s, π) q 2≤l≤k

=



∗

¯ τ (ψ)n−k ψ(a)q k−ns G+ (s)L(1 − s, π ˜ × ψ)

ψ mod q ψ(−1)=1

⎛

1 ˜ 1 (1 − s, q) + + (−1)n−k q ⎝ + H q



⎞ ˜ l (1 − s, q)⎠ (q l−1 − q l−2 )H

2≤l≤n−k

× q k−ns G+ (s)L(1 − s, π ˜) ⎛ ˜ − s, q) + (−1)n−k q = q k−ns G+ (s) ⎝Z(1

×



⎞ ˜ l (1 − s, q)L(1 − s, π (q l−1 − q l−2 )H ˜ )⎠ .

2≤l≤n−k

Integrating on both ends with ω ˜ and shifting the integral on the left from s = −σ to s = σ, we have

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⎞  1 ⎝ Z(s, q) + (−1)k q l−1 Hl (s, q)L(s, π)⎠ ω ˜ (s) ds q−1 ⎛

 s=σ

2≤l≤k

 q k−ns G+ (s) ×

= s=−σ

⎛

˜ − s, q) + (−1)n−k ⎝ 1 Z(1 q−1



⎞ ˜ l (1 − s, q)L(1 − s, π q l−1 H ˜ )⎠ ω ˜ (s) ds.

2≤l≤n−k

Recalling Lemma 3.1, Lemma 3.2, and Lemma 3.3, we complete the proof with the Mellin inversion (10). 2 4. Odd part Define 

Y(s, q) =

∗

¯ k ψ(a)L(s, π × ψ) τ (ψ)

ψ mod q ψ(−1)=−1

and 

˜ q) = Y(s,

∗

¯ τ (ψ)n−k ψ(a)L(s, π ˜ × ψ).

ψ mod q ψ(−1)=−1

By the function equation (3), we have ˜ − s, q). Y(s, q) = q k−ns G− (s)Y(1

(11)

Lemma 4.1. We have the identities Y(s, q) =

∞ q−1  A(1, . . . , 1, m) (Klk (am, q) − Klk (−am, q)) 2 m=1 ms

and ∞  A(m, 1, . . . , 1) ˜ q) = q − 1 (Kln−k (¯ am, q) − Kln−k (−¯ am, q)) . Y(s, 2 m=1 ms

Proof. This follows from (3).

2

∞ Let ω(x) ∈ Cc∞ (0, ∞) be a test function. Let ω ˜ (s) = 0 ω(x)xs−1 dx be its Mellin transform. Define  1 Ω− (x) = ω ˜ (s)G− (s)xs ds. (12) 2πi s=−σ

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Integrating with ω ˜ on both sides of (11) and shifting the integral on the left to s = σ, we get 1 2πi

 Y(s, q)˜ ω (s) ds = s=σ

1 2πi

 ˜ − s, q)˜ q k−ns G− (s)Y(1 ω (s) ds, s=−σ

which, along with Lemma 4.1 and (10), gives us the following theorem. Theorem 4.2. Let π be a Hecke–Maass cusp form for SL(n, Z) with n ≥ 2. Let A(m1 , . . . , mn−1 ) be the Fourier coefficient of π. Let ω ∈ Cc∞ (0, ∞) be a test function and let q be a prime number, a and a two integers with aa ≡ 1 mod q. For 1 ≤ k ≤ n −1, we have ∞ 1  A(1, . . . , 1, m) (Klk (am, q) − Klk (−am, q)) ω(m) 2 m=1

=

  ∞ m q k  A(m, 1, . . . , 1) (Kln−k (¯ , am, q) − Kln−k (−¯ am, q)) Ω− 2 m=1 m qn

where Ω− is defined in (12). Acknowledgments The author wants to thank Stephen David Miller for instructive and illuminating comments and for his generosity of sharing his unpublished work joint with Xiaoqing Li. The author wants to thank Wenzhi Luo, Roman Holowinsky, and Eyal Kaplan for helpful discussion. The author wants to thank Dorian Goldfeld for introducing this topic to him. References [BuKh] Jack Buttcane, Rizwanur Khan, L4 -norms of Hecke newforms of large level, Math. Ann. 362 (3) (2015) 699–715. [CoPS] James W. Cogdell, Ilya I. Piatetski-Shapiro, Converse theorems, functoriality, and applications to number theory, in: Proceedings of the International Congress of Mathematicians, vol. II, Beijing, 2002, Higher Ed. Press, Beijing, 2002, pp. 119–128. [DuIw] William Duke, Henryk Iwaniec, Estimates for coefficients of L-functions. I, in: Automorphic Forms and Analytic Number Theory, 1990, pp. 43–47. [Go] Dorian Goldfeld, Automorphic Forms and L-Functions for the Group GL(n, R), Cambridge University Press, 2006. [GoLi1] Dorian Goldfeld, Xiaoqing Li, Voronoi formulas on GL(n), Int. Math. Res. Not. IMRN 2006 (2006) 86295. [GoLi2] Dorian Goldfeld, Xiaoqing Li, The Voronoi formula for GL(n, R), Int. Math. Res. Not. IMRN 2008 (2008), rnm144. [IcTe] Atsushi Ichino, Nicolas Templier, On the Voronoi formula for GL(n), Amer. J. Math. 135 (1) (2013) 65–101. [Li1] Xiaoqing Li, Bounds for GL(3) × GL(2) L-functions and GL(3) L-functions, Ann. of Math. 173 (2011) 301–336.

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Stephen David Miller, Cancellation in additively twisted sums on GL(n), Amer. J. Math. 128 (3) (2006) 699–729. [MiSc1] Stephen D. Miller, Wilfried Schmid, Automorphic distributions, L-functions, and Voronoi summation for GL(3), Ann. of Math. 164 (2006) 423–488. [MiSc2] Stephen D. Miller, Wilfried Schmid, A general Voronoi summation formula for GL(n, Z), in: Geometry and Analysis. No. 2, in: Adv. Lect. Math. (ALM), vol. 18, Int. Press, Somerville, MA, 2011, pp. 173–224.