An explicit inversion formula for the $p$-adic Whittaker transform on $\text{GL}_n(\mathbb{Q}_p)$
João Guerreiro



Whittaker functions on non-archimedean fields were first introduced in the work of Jacquet, and they were characterized explicitly by Shintani. We obtain an explicit inversion formula and a Plancherel formula for the p-adic Whittaker transform on GL_n(ℚ_p). As an application, integral representations are obtained for the local factors of certain symmetric power L-functions.



Asynchronous Incremental Stochastic  Dual Descent Algorithm for Network Resource Allocation
    Amrit Singh Bedi, Student Member, IEEE and Ketan Rajawat, Member, IEEE The	authors are with the Department of Electrical Engineering, IIT Kanpur, Kanpur
			(UP), India 208016 (email: amritbd, ketan@iitk.ac.in).

    
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§ INTRODUCTION


Whittaker functions on local non-archimedean fields (such as ℚ_p) have been studied by several authors since they were introduced in 1967 by Jacquet <cit.>. Shintani <cit.> obtained an explicit formula for these Whittaker functions, which was then generalized in different directions in other works (see <cit.>, <cit.>). 

Let G= GL_n(ℚ_p) for which we have the Iwasawa decomposition G = U T K where U = U(ℚ_p) is the unipotent radical of the standard Borel subgroup, T = T(ℚ_p) is the torus of diagonal matrices and K = GL_n(ℤ_p) is the maximal compact subgroup.
Let Z be the center of G. Let ψ be a character on U induced from a character ψ' on ℚ_p via

    ψ(u) = ψ'(∑_i=1^n-1 u_i,i+1),     ( where u = (u_i,j) ∈ U).



Define the spherical Hecke algebra ^K ℋ^K to be the set of locally constant, compactly supported functions f : GL_n(ℚ_p) → satisfying

    f( k_1 g k_2) = f(g)

for k_1, k_2 ∈ K, g ∈GL_n(ℚ_p).


On GL_n(ℚ_p), Whittaker functions transform my the action of a character under the unipotent radical and satisfy an integral property relative to elements of the spherical Hecke algebra, as below.


A Whittaker function on GL_n(ℚ_p) is a K-finite smooth function of moderate growth W : GL_n(ℚ_p) →ℂ that satisfies

    W (u g) = ψ(u) W(g)

for some fixed character ψ as defined above and any u ∈ U, and

    ∫_G W(g x) ϕ(x) d x = λ(ϕ) W(g)

for each algebra homomorphism λ :   ^Kℋ^K →, ϕ∈  ^Kℋ^K and g ∈GL_n(ℚ_p).


Let h:Z(ℚ_p)\ T(ℚ_p) →ℂ be a smooth function. The p-adic Whittaker transform of h is defined to be

    ∫_Z(ℚ_p)\ T(ℚ_p) h(t) W(t) d^× t,

where W is a Whittaker function on GL_n(ℚ_p). 


The main goal of the present paper is to obtain an inversion formula for the p-adic Whittaker transform. The precise inversion formula for the Whittaker transform is given in Theorem <ref>.
On archimedean local fields Whittaker functions have been thoroughly studied and an inversion formula for the Whittaker transform in known due to the works of Wallach <cit.> and Goldfeld-Kontorovich <cit.>. The approach presented in this paper is inspired by the latter. A crucial step in our approach is the computation of an integral of a product of two p-adic Whittaker functions.
One can view this computation as a non-archimedean analogue of Stade's formula <cit.>. As a corollary of the inversion formula we also obtain a Plancherel formula for the p-adic Whittaker transform. 

In the last section, we use the inversion formula to write integral representations of local L-factors associated to a symmetric square and symmetric cube lifts.

We remark that the inversion formula had already been obtained in the setting of p-adic reductive groups by Delorme <cit.>. However, the approach presented here provides a more explicit presentation of the result in this particular setting and its simple derivation solely relies on complex analysis.




§ INVERSION FORMULA


In this section we provide all the necessary definitions and results that allow us to state the main theorem <ref>, which gives the inverse formula for the p-adic Whittaker transform.

We start by noting Whittaker functions on GL_n(ℚ_p) arise naturally from certain irreducible representation of this group.



Let V be a complex vector space and let (π,V) be an irreducible generic representation of G. Then the space 

    Hom_U(π,ψ) = { f : V →ℂ linear: f(π(u) v) = ψ(u) f(v), ∀ v ∈ V, u ∈ U }

is one-dimensionanal generated by an element λ. For v ∈ V a newform, define a Whittaker function W associated to π as

    W(g) = λ(π(g)v).




As the space of newforms in V is one-dimensional then the Whittaker function associated to a representation π is well-defined up to a constant. Furthermore, a Whittaker function associated to an irreducible generic representation π of GL_n(ℚ_p) is a nonzero Whittaker function on GL_n(ℚ_p). For proofs of these statements see <cit.> and <cit.>.


In order to explicitly describe Whittaker functions associated to certain representations of GL_n(ℚ_p) we proceed to quote a theorem of Shintani <cit.>. For g = z u t k ∈GL_n(ℚ_p) define the Iwasawa coordinates


    g = z u ( [ t_1 ⋯ t_n-1                                    ;                       ⋱                        ;                                 t_1            ;                                               1 ]) k

where z ∈ Z, u ∈ U, t ∈ T and k ∈ K.



Let π be an irreducible unramified generic representation of GL_n(ℚ_p) and W_π the Whittaker function associated to a representation π (normalized such that W_π(id) = 1 where id denotes the n × n identity matrix) as in Definition <ref>. We can write the L-function associated to π as

    L(s,π) = ∏_i=1^n (1-α_i p^-s)^-1

where α_i ∈ℂ are all nonzero and α_1 ⋯α_n = 1. 
Then, for

    t = ( [ t_1 ⋯ t_n-1                                    ;                       ⋱                        ;                                 t_1            ;                                               1 ])

we have

    W_π(t) = δ^1/2(t) s_-log|t|_p(α),     t_i ∈ℤ_p   for  i=1,⋯,n-1, 
     0,

where 

    δ(t) = ∏_k=1^n-1 |t_k|_p^k(n-k),



    log|t|_p = (log|t_1|_p, ⋯, log|t_n-1|_p )   for  log = log_p,

and

    s_λ(α) = | [ α_1^n-1+λ_1+λ_2+⋯+λ_n-1 α_2^n-1+λ_1+λ_2+⋯+λ_n-1                       ⋯ α_n^n-1+λ_1+λ_2+⋯+λ_n-1; α_1^n-2+λ_1+λ_2+⋯+λ_n-2 α_2^n-2+λ_1+λ_2+⋯+λ_n-2                       ⋯ α_n^n-2+λ_1+λ_2+⋯+λ_n-2;                       ⋮                       ⋮                       ⋱                       ⋮;               α_1^1+λ_1              α_2^1 +λ_1                       ⋯              α_n^1 +λ_1;                       1                       1                       ⋯                       1 ]|/| [ α_1^n-1 α_2^n-1       ⋯ α_n^n-1; α_1^n-2 α_2^n-2       ⋯ α_n^n-2;       ⋮       ⋮       ⋱       ⋮;     α_1     α_2       ⋯     α_n;       1       1       ⋯       1 ]|

is the Schur polynomial, where λ = (λ_1,⋯,λ_n) and λ_i ∈ℤ.



See <cit.>. For the relation between the L-function L(s,π) and the Whittaker function W_π see Subsection 3.1.3 <cit.>.


Throughout this paper we will always use log to denote the base p logarithm.
We will also use the notation W_α, for α∈ (ℂ\{0 })^n, to denote the function

    W_α(t) = δ^1/2(t) s_-log|t|_p(α),     t_i ∈ℤ_p   for  i=1,⋯,n-1, 
     0,    .
finite
This coincides with the definition of W_π when α = (α_1,⋯, α_n) are the Langlands parameters associated to π.

We are now able to define the p-adic Whittaker transform precisely.



Let h:Z(ℚ_p) \ T(ℚ_p) →ℂ. Let 𝒮_η be the annulus { z ∈ : p^-η < |z|_ < p^η}, for some η >0. Define the Whittaker transform h^♯ : 𝒮_η^n→ℂ by

    h^♯(α) = ∫_T(ℤ_p) \ T(ℚ_p) h(t) W_α(t) d^×t,

provided the integral converges absolutely,
where

    d^×t = ∏_k=1^n-1 |t_k|_p^-k(n-k)dt_k/|t_k|_p.





A sufficient condition for the absolute convergence of the above integral is that h satisfies the decay condition

    | h(t) |_≪δ^1/2(t) | t_1^n-1 t_2^n-2⋯ t_n-1|_p^η + ε

for any ε > 0.




Let H : 𝒮_η^n →ℂ be a holomorphic symmetric function, for some η >0. Let

    W_α(t) = δ^1/2(t) s_-log|t|_p(α),     t_i ∈ℤ_p   for  i=1,⋯,n-1, 
     0,

where α = (α_1,…,α_n). Define the inverse Whittaker transform of H, H^♭: Z(ℚ_p) \ T(ℚ_p) →ℂ by

    H^♭(t) = 1/n! (2π i)^n-1∫_𝕋 H(β) W_1/β(t) ∏_i, j = 1 
     i ≠ j^n (β_i - β_j ) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1

where  𝕋 = { (β_1,⋯,β_n-1) ∈ℂ^n-1 : |β_i|_ℂ = 1 },
1/β = (1/β_1,⋯, 1/β_n) and β_n = 1/β_1 ⋯β_n-1.



Note that H^♭ is supported on (ℤ_p \{0})^n-1, if we identify Z(ℚ_p) \ T(ℚ_p) with (ℚ_p^×)^n-1 as in Theorem <ref>. Moreover, H^♭ is invariant under the action of (ℤ_p^×)^n-1.


The inverse transform is given in the following theorem.



Let H : 𝒮_η^n →ℂ be a holomorphic symmetric function, for some η >0. Assume (H^♭)^♯ converges absolutely on 𝒮_η^n. Then,

    (H^♭)^♯(α) = H(α)

for α∈𝒮_η^n with α_1 ⋯α_n = 1.


By studying the image of the map H ↦ H^♭ we obtain the following corollary.










Let h : Z(ℚ_p) \ T(ℚ_p) →ℂ. For

    t = ( [ t_1 ⋯ t_n-1                                    ;                       ⋱                        ;                                 t_1            ;                                               1 ])

 assume that h(t) is supported on the region { t_i ∈ℤ_p \{ 0 } : i=1,⋯,n-1 }, that the integral which defines h^♯ (in Definition <ref>) is absolutely convergent, and that h(t) = f(-|t_1|_p,⋯,-|t_n-1|_p) for some function f : ℤ_≥ 0^n-1→ℂ. Then

    (h^♯)^♭ = h.



We postpone the proofs of Theorem <ref> and Corollary <ref> to the next section. As a consequence of Corollary <ref> we have the Plancherel formula for the p-adic Whittaker transform.


Let h_1, h_2 be two functions that satisfy the conditions of Corollary <ref>. We have,

    ⟨ h_1, h_2 ⟩ = ⟨ h_1^♯, h_2^♯⟩

where

    ⟨ h_1, h_2 ⟩   ∫_T(ℤ_p) \ T(ℚ_p) h_1(t) h_2(t)  d^×t, 
    ⟨ h_1^♯, h_2^♯⟩   1/n! (2π i)^n-1∫_𝕋 h_1^♯(β) h_2^♯(β)∏_i, j = 1 
     i ≠ j^n (β_i - β_j ) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1.





§ PROOF OF THEOREM <REF>

We start the section with the proof of two of the ingredients needed to prove the inverse transform theorem. Both results can also be found in Section 4 of <cit.>. They are included here as their proofs are concise and improve the exposition. The first of these is a version of Cauchy's identity.



Let s_λ be the Schur polynomial defined in the previous section. Let α_1,⋯,α_n,β_1,⋯,β_n ∈ℂ such that |α_i β_j| < 1 for every 1 ≤ i, j ≤ n. Then,

    ∑_m_1,⋯, m_n-1≥ 0 s_𝐦(α) s_𝐦(β) 
    
     = 1-α_1 ⋯α_n β_1 ⋯β_n/∏_i,j=1^n (1-α_i β_j )

 where 𝐦 = (m_1,⋯,m_n-1)



This proof follows <cit.> and <cit.>.
We start by stating the Cauchy determinant identity (lemma 7.4.18 in <cit.>)

    | [ 1/1-α_1 β_1           ⋯ 1/1-α_n β_1;           ⋮           ⋱           ⋮; 1/1-α_1 β_n           ⋯ 1/1-α_n β_n;             ]| = ∏_1 ≤ i < j ≤ n (α_i - α_j) ∏_1 ≤ i < j ≤ n (β_i - β_j)/∏_i,j = 1^n (1 - α_i β_j).

Expanding each term 1/1-α_i β_j as 1 + α_i β_j + α_i^2 β_j^2 + ⋯, we obtain


    |  [ 1/1-α_1 β_1           ⋯ 1/1-α_n β_1;           ⋮           ⋱           ⋮; 1/1-α_1 β_n           ⋯ 1/1-α_n β_n;             ]| = ∑_l_1,⋯,l_n ≥ 0 ∑_σ∈ S_n(σ) α_σ(1)^l_1⋯α_σ(n)^l_nβ_1^l_1⋯β_n^l_n.


Notice that if the l_i are not all distinct then the sum over S_n vanishes. Therefore

    | [ 1/1-α_1 β_1           ⋯ 1/1-α_n β_1;           ⋮           ⋱           ⋮; 1/1-α_1 β_n           ⋯ 1/1-α_n β_n;             ]|     = ∑_l_1 > ⋯ > l_n ≥ 0 
    σ, τ∈ S_n(σ) (τ) α_σ(1)^l_1⋯α_σ(n)^l_nβ_τ(1)^l_1⋯β_τ(n)^l_n
        = ∑_l_1 > ⋯ > l_n ≥ 0| [ α_1^l_1       ⋯ α_n^l_1;       ⋮       ⋱       ⋮; α_1^l_n       ⋯ α_n^l_n;         ]| | [ β_1^l_1       ⋯ β_n^l_1;       ⋮       ⋱       ⋮; β_1^l_n       ⋯ β_n^l_n;         ]|.

By making the substitution (l_1,⋯,l_n) = (m_0+⋯+m_n-1+(n-1), m_0+⋯+m_n-2+(n-2), ⋯, m_0) and dividing by ∏_1 ≤ i < j ≤ n (α_i - α_j) ∏_1 ≤ i < j ≤ n (β_i - β_j) we obtain

    1/∏_i,j=1^n (1-α_i β_j ) = ∑_m_0, m_1,⋯, m_n-1≥ 0 s_m(α) s_m(β) (α_1 ⋯α_n β_1 ⋯β_n)^m_0,

where right hand side simplifies to

    1/1-α_1 ⋯α_n β_1 ⋯β_n∑_m_1,⋯, m_n-1≥ 0 s_m(α) s_m(β).

Multiplying out by 1-α_1 ⋯α_n β_1 ⋯β_n concludes the proof.


The second necessary ingredient is an identity for the integral of a product of two Whittaker function, as in <cit.>.


Let α_1,⋯,α_n,β_1,⋯,β_n ∈ℂ such that |α_i β_j| ≤ 1 for every 1 ≤ i, j ≤ n, and ε > 0. Let W_α, W_β be Whittaker functions as defined in Definition <ref>. Then

    ∫_Z(ℚ_p) \ T(ℚ_p) W_α(t) W_β(t) ∏_k=1^n-1 |t_k|_p^ε(n-k)  d^×t = 1-α_1 ⋯α_n β_1 ⋯β_n/p^ε n/∏_i,j=1^n (1-α_i β_j/p^ε)




We start by applying Theorem <ref> to write down the Whittaker functions explicitly:

    ∫_Z(ℚ_p) \ T(ℚ_p) W_α(t) W_β(t) ∏_k=1^n-1 |t_k|_p^ε(n-k)  d^×t 
        = ∫_ℤ_p^n-1δ(t) s_-log|t|_p(α) s_-log|t|_p(β) ∏_k=1^n-1 |t_k|_p^-(k-ε)(n-k)dt_k/|t_k|_p
        = ∫_ℤ_p^n-1 s_-log|t|_p(α) s_-log|t|_p(β) ∏_k=1^n-1 |t_k|_p^ε(n-k)dt_k/|t_k|_p.

Breaking up the region of integration by absolute value we get

    ∫_Z(ℚ_p) \ T(ℚ_p) W_α(t) W_β(t) ∏_k=1^n-1 |t_k|_p^ε(n-k)  d^×t 
        = ∑_m_1,⋯, m_n-1≥ 0  ^n-1  times_p^m_kℤ_p^×  s_-log|t|_p(α) s_-log|t|_p(β) ∏_k=1^n-1 |t_k|_p^ε(n-k)dt_k/|t_k|_p
        = ∑_m_1,⋯, m_n-1≥ 0 s_𝐦(α) s_𝐦(β) p^-ε(n-1)m_1 -ε(n-2)m_2 + ⋯ -ε m_n-1
        = ∑_m_1,⋯, m_n-1≥ 0 s_𝐦(α/p^ε) s_𝐦(β) 
        = 1-α_1 ⋯α_n β_1 ⋯β_n/p^ε n/∏_i,j=1^n (1-α_i β_j/p^ε)

where the last equality follows from Cauchy's identity <ref>.


We are now ready to proceed to the proof of the Theorem <ref>.


Restate the theorem as 
    H(α) = ∫_Z(ℚ_p) \ T(ℚ_p) H^♭(t) W_α(t) d^×t.

Recall that α∈𝒮_η^n and α_1 ⋯α_n = 1. Further assume that |α_1|_ℂ = ⋯ = |α_n|_ℂ = 1. We can later remove this assumption as both sides of the equality are holomorphic functions. Note that H is invariant under permutations of (α_1,⋯,α_n) as the Whittaker function also has those symmetries.
Define

    H_ε(α) = ∫_Z(ℚ_p) \ T(ℚ_p)H^♭(t) W_α(t) ∏_k=1^n-1 |t_k|_p^ε(n-k) d^×t.

Then

    lim_ε→ 0 H_ε(α) = ∫_Z(ℚ_p) \ T(ℚ_p) H^♭(t) W_α(t) d^×t.

It suffices to show that lim_ε→ 0 H_ε(α) = H(α) as well. 

Replacing H^♭(t) by its explicit formula and swapping the order of integration we get

    H_ε(α) = 1/n!(2π i)^n-1∫_𝕋(  ∫_Z(ℚ_p) \ T(ℚ_p) W_α(t) W_1/β(t) ∏_k=1^n-1 |t_k|_p^ε(n-k)  d^×t ) ×
    
    H(β) ∏_i, j = 1 
     i ≠ j^n (β_i - β_j ) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1

We use Proposition <ref> to compute the innermost integrals and get

    H_ε(α) = 1/n!(2π i)^n-1∫_𝕋 H(β) (1-1/p^ε n)/∏_i,j=1^n (β_j-α_i/p^ε)∏_i, j = 1 
     i ≠ j^n (β_i - β_j ) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1

Now shift each variable β_i (i=1,⋯,n-1) to the contour given by the equation |z|_ℂ = 1/p^ε+ε', in order. For the sake of clearness, we shall assume that all the α_i are distinct.
When the contour of integration for β_1 is shifted one picks up several residues at

    β_1 = α_1/p^ε, ⋯, β_1 = α_n/p^ε.

For which residue integral obtained in this manner one can shift the contour of integration for β_2 picking up n-1 residues in the process. Repeating this process for which β_i, i=3,⋯,n-1 one obtains

    H_ε(α) = 1/n!∑_σ∈ S_nℛ_ε^σ + ℐ_ε,ε'
this
where

    ℛ_ε^σ = H(α_σ(1)/p^ε,⋯,α_σ(n)/p^ε) (1-1/p^ε n) ∏_i=1^n-1( α_σ(i)/p^ε - p^(n-1)εα_σ(n))/(p^(n-1)εα_σ(n) - α_σ(n)/p^ε) ∏_i=1^n-1( α_σ(i)/p^ε - α_σ(n)/p^ε) ∏_i=1^n-1α_σ(i)/p^ε

and
ℐ_ε,ε' is a sum of residue integrals with each integrand bounded (in absolute value) by


    C_H (1 - 1/p^nε) p^ε(n^3+n)/1 - 1/p^ε'

for some constant C_H > 0 depending only on the function H.
Therefore,

    lim_ε→ 0ℛ_ε^σ 
    =    lim_ε→ 0H(α_σ(1)/p^ε,⋯,α_σ(n)/p^ε)/∏_i=1^nα_σ(i)/p^ε(1-1/p^ε n)/(p^(n-1)ε - 1/p^ε)∏_i=1^n-1( α_σ(i)/p^ε - p^(n-1)εα_σ(n))/∏_i=1^n-1( α_σ(i)/p^ε - α_σ(n)/p^ε)
        = H(α_σ(1),⋯,α_σ(n))/∏_i=1^nα_σ(i)∏_i=1^n-1( α_σ(i) - α_σ(n))/∏_i=1^n-1( α_σ(i) - α_σ(n))
        = H(α_σ(1),⋯,α_σ(n)) 
        = H(α)

and

    lim_ε→ 0 |ℐ_ε,ε'| ≪_Hlim_ε→ 0(1 - 1/p^nε) p^ε(n^3+n)/1 - 1/p^ε' = 0.

In conclusion, we obtain

    lim_ε→ 0 H_ε(α) = 1/n!∑_σ∈ S_n H(α) = H(α)

as desired.
If the α_i are not all distinct then the original integrand would have higher order poles. This means that the residue sum would have a small number of residue but some residues would contribute to the sum with a multiple of H(α).


To prove the corollary one simply needs to compute the image of the inverse transform H ↦ H^♭ as any function h in the image of this mapping will satisfy (h^♯)^♭ = h. Simply choose H such that h = H^♭, and apply Theorem <ref> to H, to obtain (h^♯)^♭ = ((H^♭)^♯)^♭ = H^♭ = h.


We start by noting that any function h satisfying the assumptions in the corollary is a sum of scalar multiples of functions of the form:


    f_λ_1,⋯,λ_n-1(t) =  1,    log |t_i|_p = -λ_i  for  i=1,⋯,n-1, 
     0,

where λ_1,⋯,λ_n-1 are nonnegative integers. Therefore, it suffices to show that all such functions are in the image of the map of the inverse transform H ↦ H^♭.
Let H(β) = W_β(p^λ)/δ(p^λ) where

    p^λ = ( [ p^λ_1⋯ p^λ_n-1                                             ;                             ⋱                              ;                                        p^λ_1               ;                                                           1 ]).

Then

    H^♭(t)     = 1/δ(p^λ) n! (2π i)^n-1∫_𝕋 W_β(p^λ) W_1/β(t) ∏_i, j = 1 
     i ≠ j^n (β_i - β_j ) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1
        = δ^1/2(t)/δ^1/2(p^λ) n! (2π i)^n-1∫_𝕋 s_λ(β) s_-log|t|_p(1/β) ∏_i, j = 1 
     i ≠ j^n (β_i - β_j ) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1.

By the definition of Schur polynomials we further infer that

    H^♭(t)     = δ^1/2(t)/δ^1/2(p^λ) n! (2π i)^n-1∫_𝕋∑_σ, τ∈ S_n(β_σ(1)^(n-1)+λ_1+⋯+λ_n-1⋯β_σ(n-1)^1+λ_1) ×
       ( β_τ(1)^-(n-1)+log|t_1|_p+⋯+log|t_n-1|_p⋯β_τ(n-1)^-1+log|t_1|_p) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1
        = δ^1/2(t)/δ^1/2(p^λ) n! (2π i)^n-1×
       ∫_𝕋∑_σ∈ S_n(β_σ(1)^λ_1+⋯+λ_n-1+log|t_1|_p+⋯+log|t_n-1|_p⋯β_σ(n-1)^λ_1+log|t_1|_p) d β_1 ⋯ d β_n-1/β_1 ⋯β_n-1
        = δ^1/2(t)/δ^1/2(p^λ)  f_λ_1,⋯,λ_n-1(t) 
        = f_λ_1,⋯,λ_n-1(t)

where the last equality follows from the definition of f_λ_1,⋯,λ_n-1(t).




§ INTEGRAL REPRESENTATIONS OF LOCAL L-FACTORS


In this final section, we obtain integral representations of the local L-factors of symmetric dth power L-functions of GL(2) representations, with d ≤ 4.
Let L(s, Sym^d  π) be the symmetric dth power L-function associated to an irreducible automorphic representation π of GL(2,𝔸_). This L-function has local factors (at the unramified places p) given by

    L_p(s,Sym^d  π) = ∏_i=0^d (1-α^d-2ip^-s)^-1 =: h_s,p,d(α).


We shall obtain an integral representation for the L-factors L_p(s,π) using Theorem <ref>. 

Assume ℜ(s) > 1. We start by computing (h_s,p,d)^♭ explicitly. By definition,


    (h_s,p,d)^♭(t)     = 1/4π i∫_|β|_ = 1 h_s,p,d(β) W_1/β(t) (β - β^-1)(β^-1 - β) d β/β
        = | t_1 |_p^1/2/4π i∫_|β|_ = 1 h_s,p,d(β) ( β^λ+1 - β^-(λ+1)) (β^-1 - β) d β/β

with t = [ t_1   0;   0   1 ] and | t_1 |_p = p^-λ for λ≥ 0.

We now want to use the residue theorem to evaluate the integral above. To simplify the calculation of the residues of the integrand we first note that


    ∫_|β|_ = 1 h_s,p,d(β) β^-(λ+1)(β^-1 - β) d β/β  = - ∫_|β|_ = 1 h_s,p,d(β) ( β^λ+1) (β^-1 - β) d β/β


via the change of variable β↦1/β. Therefore,


    (h_s,p,d)^♭(t) = | t_1 |_p^1/2/2π i∫_|β|_ = 1 h_s,p,d(β)  (β^λ-1 - β^λ+1) d β.


Let i_s,p,d(β) be the integrand of the integral in Equation <ref>. By the definition of h_s,p,d, the function i_s,p,d(β) has possible poles inside the unit circle at


    β = e^2π i l/k p^-s/k


for 0 < k ≤ d, k ≡ d  2 and 0 ≤ l < k. We will compute the residues at these poles for 1 ≤ d ≤ 4, in order to compute the integral in Equation <ref>.



  * d=1:

In this case, the function i_s,p,1(β) only has a pole at β = p^-s and


    Res_β = p^-s  i_s,p,1(β) = p^-s λ,

which implies that 

    (h_s,p,1)^♭(t) = | t_1 |_p^s+1/2.



  * d=2:

In the case of a symmetric square L-function, the integrand i_s,p,2(β) has poles at β = ± p^-s/2. Their residues are given by

    Res_β = p^-s/2  i_s,p,2(β)  = p^-sλ/2/2(1-p^-2s),
       Res_β = -p^-s/2  i_s,p,2(β) = -p^-sλ/2/2(1-p^-2s).


Adding the two residues we infer that


    (h_s,p,2)^♭(t)  = | t_1 |_p^(s+1)/2/1-p^-2s,    λ, 
     0,    λ.



  * d=3:

For the symmetric cube L-function, the poles of the integrand i_s,p,3(β) are at β = p^-s and at β = e^2π i l/3 p^-s/3 for l = 0,1,2.
The residue at β = p^-s is given by

    Res_β = p^-s  i_s,p,3(β)  = -p^-s(λ+2)/(1-p^-2s)(1-p^-4s).

For clarity's sake we won't write down the values of the residues at the remaining three poles, but simply note that their sum is equal to

    1/(1-p^-2s)(1-p^-4s) 
    p^-sλ/3,    λ≡ 0  3, 
     
    p^-s(λ+8)/3,    λ≡ 1  3,
    
    p^-s(λ+4)/3,    λ≡ 2  3.

We remark that calculating these residues is a straightforward, albeit tedious, exercise.
Finally, we conclude that (h_s,p,3)^♭(t) is equal to

    1/(1-p^-2s)(1-p^-4s)| t_1 |_p^s/3+1/2-| t_1 |_p^s-3/2,    λ≡ 0  3, 
    | t_1 |_p^s/3-13/6-| t_1 |_p^s-3/2,    λ≡ 1  3,
    | t_1 |_p^s/3-5/6-| t_1 |_p^s-3/2,    λ≡ 2  3.



  * d=4:

In this case, the integrand i_s,p,4(β) has poles at β = ± p^-s/2 and at β = ± p^-s/4, ± i p^-s/4. The sum of the residues at the first two poles is

    1/(1-p^-2s)(1-p^-3s) 
    -p^-s(λ+2)/2,    λ≡ 0  2, 
     
    0,    λ≡ 1  2.

and the sum of the residues at the last four poles is

    1/(1-p^-2s)(1-p^-3s) 
    p^-sλ/4,    λ≡ 0  4, 
     
    p^-s(λ+6)/4,    λ≡ 2  4,
    
    0,


Adding up all the residues one obtains that (h_s,p,4)^♭(t) is equal to

    1/(1-p^-2s)(1-p^-3s)| t_1 |_p^s/4+1/2-| t_1 |_p^s/2-1/2,    λ≡ 0  4, 
    | t_1 |_p^s/4-1-| t_1 |_p^s/2-1/2,    λ≡ 2  4,
    
    0,





















For large enough ℜ(s), since the functions (h_s,p,d)^♭(t) satisfy the convergence condition in Remark <ref>, it follows from Theorem <ref> that


    L_p(s,Sym^d  π) = ∫_ℚ_p^× (h_s,p,d)^♭(t) W_α(t)  d^×t,

for 1 ≤ d ≤ 4, where the functions (h_s,p,d)^♭(t) are explicitly given by Equations <ref>-<ref>.

This approach should provide (conditionally on convergence issues) integral representations for L_p(s, Sym^d  π) for d > 4 as well. However, the residue calculations appear to become rather complicated. 



§ ACKNOWLEDGMENTS


The author would like to thank Dorian Goldfeld for helpful discussions, as well as for the comments on earlier drafts of this work. The author was partially supported by the FCT doctoral grant SFRH/BD/68772/2010.


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