Spectral Decomposition of B m

Z

R^d−1

bp(ξ)X

c∈o^∗

|N c|^α+βλ_ξ(c)SK(m, n;c)[h]_`(x(m, n;c);α)dξ

= 1

(2π)^d−1 (2π)^dp

|N(mn)|α+β+1Z

· · · Z

R^d−1

bp(ξ)λ_ξ(mn)S_m,n(α, β, ξ; [h]_`)dξ

(3.2.30)

where

S_m,n(α, β, ξ; [h]_`) = X

c∈o^∗

|N c|⁻¹S_K(m, n;c)[[h]]_`(x(m, n;c);α, β;ξ) (3.2.31) with

[[h]]_`(x;α, β;ξ) =

N x 2^d

−(α+β)−1

λ−ξ(x)[h]_`(x;α). (3.2.32) For α and β satisfying the conditions (3.2.17) and (3.2.27), by shifting the contour we obtain the estimate

[[h]]`(x;α, β;ξ) Tr(1 +|x|)−µ

·







|N x|^−12C0 if |N x| ≥1,

|N x| if |N x|<1

(3.2.33) withµ >0 as in the bound of [h]`. Furthermore the derivatives of[[h]]` are of rapid decay if at least one variable tends to0+ or+∞.

3.2.3 Spectral Decomposition of Bm

The sumSm,n(α, β, ξ; [h]_`)as defined in (3.2.31) has the right form to apply the geometric sum formula given in Lemma 3.1.5. Recall thatm0 and thusη_V(m) =η_V(1).

The definitions of [h]` and [[h]]` imply that [[h]]`(x;α, β, ξ) and its derivatives are smooth and by (3.2.33) of rapid decay if at least one variable tends to0+ or +∞. Therefore Lemma 3.1.5 can safely

be applied toS_m,n(α, β, ξ; [h]_`), provided

|Reα|+ Reβ <−¹₂C₁. (3.2.34) For instance chooseReβ negative and large, and |Reα|relatively small.

Lemma 3.1.5 yields the decomposition S_m,n(α, β, ξ; [h]_`) = X

V

a_Vη_V(1)t_V((m))η_V(n)t_V((n))B_[n][[h]]_`(κ_V;α, β, ξ)

+Ce

X

ν∈Λ^d−1

λ−ν(mn)

∞

Z

−∞

σ2it(m,2ν)σ2it(n,2ν)

|N(mn)|^it|ζ_K(1 + 2it,2ν)|²B_[n][[h]]_`(t+ν;α, β, ξ)dt

= : {S^c_m,n+ S^e_m,n}(α, β, ξ; [h]_`) wheret+ν = (t+ν₁, . . . , t+νd−1, t−Tr(ν)).

The uniformity of the convergence follows by inserting the bounds for all involved parameters stated in the previous sections. More precisely, we use the estimate (3.1.44) to bound the functionsB_[n][[h]]_`. Then Lemma 3.1.4 gives the convergence for the first term, and Lemma 3.1.1 combined with trivial estimates for the divisor functions does it for the second term.

We insert the decomposition into (3.2.30) and (3.2.26), and factorize n ∈ o^∗ into n·ε where ε runs over the units, and n∈o^∗modP()².

Then the contribution of S_m,n^c equals (2π)^dβ

i(2π²)^dζK(1−β)(N m)^12(α+β+1)×

× X

n∈o^∗modP()²

σ−α(n) (N n)^{12(1−α−β)}

sgn^`[n]X

V

a_Vη_V(1)t_V((m))η_V(n)t_V((n))×

× X

ν∈Z^d−1

Z

· · · Z

R^d−1

e^iTr(νξ)bp(ξ)λ_ξ(mn)B_[n][[h]]_`(κ_V;α, β;ξ)dξ.

The functionB_[n][[h]]`is smooth inξand is bounded by (3.1.44) withf = [[h]]`andCf =C1. Thus the integral decays sufficiently fast in ν andκ_V, and the triple sum converges absolutely. Using Poisson’s summation formula, we see that only the term with ξ=ν = 0 survives. Hence the expression changes to

(2π)^dβ

i(2π²)^dζ_K(1−β)(N m)^12(α+β+1) X

n∈o^∗modP()²

σ−α(n) (N n)^{12(1−α−β)}

sgn^`[n]×

×X

V

a_Vη_V(1)t_V((m))η_V(n)t_V((n))₂d−1¹ B_[n][[h]]_`(κ_V;α, β; 0).

The factors sgn[n]andη_V(n) depend only on the sign ofn, in other words they depend on the unitε such thatn=n⁰εwithn⁰0. Thus we can rewrite the sum overnas

2(2π)^dβ

i(2π)^2dζK(1−β)(N m)^12(α+β+1)X

V

aVηV(1)tV((m))×

× X

εmodP()²

sgn<sup>`</sup>[ε]η<sub>V</sub>(ε)B<sub>[ε]</sub>[[h]]<sub>`</sub>(κ_V;α, β; 0)·X

n

t_V(n) σ−α(n) (Nn)^{12(1−α−β)}

.

Then the sum overn equalsH_V(¹₂(1−α−β))H_V(¹₂(1 +α−β)). We denote the sum overεby B^V,`. Hence we have

S_m,n^c = 2(2π)^dβ

i(2π)^2d(N m)^12(α+β+1)X

V

a_Vη_V(1)t_V((m))H_V(¹₂(1−α−β))H_V(¹₂(1 +α−β))B^V,`. This sum converges absolutely by Lemma 3.1.6 and (3.1.44).

The contribution coming from S_m,n^e can be treated in a similar way. We get 2(2π)^dβ

i(2π)^2dC_e(N m)^12(α+β+1) X

ν∈Λ^d−1

λ−ν(m)

∞

Z

−∞

σ_2it(m,2ν)

(N m)^it|ζ_K(1 + 2it,2ν)|²×

×ζ_K ¹₂(1 +α−β)−it, ν

ζ_K ¹₂(1 +α−β) +it,−ν

ζ_K ¹₂(1−α−β)−it, ν

×

×ζ_K ¹₂(1−α−β) +it,−ν

B<sup>Eν,`</sup>[[h]]<sub>`</sub>(t+ν;α, β,0)dt

where t+ν = (t+ν₁, . . . , t+νd−1, t−Tr(ν)). Lemma 3.1.1, (3.1.44) and trivial estimates for the factorsζK(σ+it,±ν), as σ >1, show the convergence.

Thus we have a spectral decomposition for Bm, as long asα and β satisfy our assumptions (3.2.17), (3.2.27) and (3.2.34). We have

Bm(α, β;h) = 2^d|D_K|^−α2+14|N m|eh₀(1)ζ_K(1−α)ζ_K(1−β)

ζ_K(2−α−β) σα+β−1(m) (3.2.35)

+ 2^d|D_K|^α2−14|N m|^1+αeh0(1 +α)ζK(1 +α)ζK(1−β)

ζ_K(2 +α−β) σ−α+β−1(m)

+X

`

h2(2π)^dβ

i(2π)^2d(N m)^12(α+β+1)X

V

a_Vη_V(1)t_V((m))H_V(¹₂(1−α−β))H_V(¹₂(1 +α−β))B^V,`

+2(2π)^dβ

i(2π)^2dC_e(N m)^12(α+β+1) X

ν∈Λ^d−1

λ−ν(m)

∞

Z

−∞

σ_2it(m,2ν)

(N m)^it|ζ_K(1 + 2it,2ν)|²×

×ζ_K ¹₂(1 +α−β)−it, ν

ζ_K ¹₂(1 +α−β) +it,−ν

ζ_K ¹₂(1−α−β)−it, ν

×

×ζ_K ¹₂(1−α−β) +it,−ν

B<sup>Eν,`</sup>[[h]]<sub>`</sub>(t+ν;α, β,0)dti .

But the domain coming from these assumptions does not contain the point (α, β) = (0,0) we need for the application toZ₂(g, ω;K). Thus, we have to continue the decomposition to a neighborhood of (α, β) = (0,0). To this end we consider the transform of hgiven by

Φ∗(r;α, β;h) = 1 i

X

`

X

εmodP()²

sgn<sup>`</sup>[ε]η∗(ε)B<sub>[ε]</sub>[[h]]<sub>`</sub>(r;α, β,0)

where∗=V or Eν andr =κ_V or t+ν= (t+ν₁, . . . , t+νd−1, t−Tr(ν)) respectively.

We use the definitions of[[h]]` in (3.2.32), [h]` in (3.2.25),eh` in (3.2.15) and Γ` in Lemma 3.2.1.

We obtain

where b is the contour from above, and ε is a unit. Stirling’s formula and (3.2.17) imply that the integral converges absolutely.

Now we applyBewith respect to xto this expression, and use the equations (3.1.45) – (3.1.47) to get a new expression forΦ∗, namely pass through a pole, we choose a small semicircle around it. Under the assumption (3.2.34) one may use the contour Resj = b, with the bounds ¹₂(Reα + Reβ + 1) < b < min{0,−Reα}. Then we move the contour appropriately to get (3.2.36). As the right hand side exists on a larger domain, the representation (3.2.36) gives an analytic continuation ofΦ∗.

In the next steps, we reformulate a factor of Φ∗ and state some estimates for it.

Consider the expression

separately, and reformulate the expression in every case. Combining these results we obtain for any space ∗=V and Eν withη∗(1)6= 0 the equality

(3.2.2) Lemma

Assume that (3.2.17) and (3.2.27) hold for α, β and h, and let V, the eigenvalue κV = (κ1, . . . , κd), and q_V = (q₁, . . . , q_d) be as in (3.1.28) and Lemma 3.1.3. Then Φ_V(κ_V;α, β;h) is regular in κ_V and bounded by

ΦV(κV;α, β;h) 1 +|κ₁|+· · ·+|κ_d|−^C₂⁰

uniformly in V and α, β, provided

|Reα|+ Reβ <2 min

j

|Imκj+ ¹₂|+δj (3.2.38) with

δj =







0, if qj = 0, 1, if q_j 6= 0.

LikewiseΦ_Eν is regular in tandν. Let t+ν = (t+ν1, . . . , t+νd−1, t−Tr(ν)). ThenΦ_Eν(t+ν;α, β;h) is bounded by

Φ_Eν(t+ν;α, β;h) 1 +|t|+|ν₁|+· · ·+|ν_d−1|−^C0

2 , (3.2.39)

provided

|Reα|+ Reβ <1−2|Imt|. (3.2.40)

Proof.

First we consider the caseqj 6= 0for all j. Theniκj =lj−¹₂, and we modify theΓ-factors of (3.2.36) using (3.1.48). We have

ΦV(κV;α, β;h) = (−1)^{l1+···+ld−d} d

X

e

X

ε

ηV(ε)×

× Z

· · · Z

(−i∞,i∞)^d

eh(s; e)

d

Y

j=1

(1 +ejsgn(εj)) cos(^π₂α)−(1−ejsgn(εj)) cos(π(sj−¹₂α))

×

×Γ(1−sj)Γ(1 +α−sj)Γ(lj −1 +sj−¹₂(α+β)) Γ(l_j + 1−s_j+¹₂(α+β))

ds

where the contours can be drawn under the assumption (3.2.38). Thus, the regularity of Φ_V follows.

To obtain the decay of ΦV we shift the contours appropriately to the left, and use Stirling’s formula and (3.2.17), as well as trivial estimates for the trigonometric functions.

Ifq_j = 0for somej, we do not need to rewrite the integral, and the shift of contour is not needed for thej-th integral. After reformulating the integrals withqj 6= 0the regularity follows and one proceeds as above with Stirling’s formula to get the estimates on Φ∗, under the assumption (3.2.38). For the spaces Eν the proof is analogously toq_j = 0, under the assumption (3.2.40).

Now we can state a first explicit formula for Bm in a neighborhood of(α, β) = (0,0):

(3.2.3) Theorem

Let Bm(α, β;h) be defined by (3.2.6), and assume (3.2.17). Letα, β be such that

−1<Re(±α+β)< ³₅. (3.2.41)

Then we have the spectral decomposition Bm(α, β;h) =

B^(r)m +B^(c)m +B^(e)m (α, β;h) where

B^(r)m (α, β;h) =|D_K|^−α2+14ζ_K(1−α)ζ_K(1−β)

ζK(2−α−β) (N m)σα+β−1(m)¨h(0,0) +|D_K|^α2−14ζK(1 +α)ζK(1−β)

ζ_K(2 +α−β) (N m)^1+ασ−α+β−1(m)¨h(α,0) + 2^2d

dπ^dRK|D_K|^β+12ζK(1−α)ζK(1 +β)

ζK(2−α+β) (N m)^1+βσα−β−1(m)¨h(0, β) + 2^2d

dπ^dR_K|D_K|^β+12ζ_K(1 +α)ζ_K(1 +β)

ζ_K(2 +α+β) (N m)^1+α+βσ−α−β−1(m)¨h(α, β), B^(c)_m(α, β;h) = 2(2π)^dβ

(2π)^2d (N m)^12(α+β+1)X

V

aVηV(1)tV((m))HV(¹₂(1−α−β))

×H_V(¹₂(1 +α−β))Φ_V(κ_V;α, β;h), B^(e)_m (α, β;h) = 2(2π)^dβ

(2π)^2d C_e X

ν∈Λ^d−1

λ−ν(m)

∞

Z

−∞

σ_2it(m,2ν)

(N m)^it|ζ_K(1 + 2it,2ν)|²×

×ζ_K ¹₂(1 +α−β)−it, ν

ζ_K ¹₂(1 +α−β) +it,−ν

ζ_K ¹₂(1−α−β)−it, ν

×

×ζ_K ¹₂(1−α−β) +it,−ν

Φ_Eν(t+ν;α, β;h)dt.

Here V, κ_V, t_V, η_V, a_V, H_V are as defined in Section 3.1.2, Φ∗ is given by (3.2.36) and as before the vector t+ν = (t+ν1, . . . , t+νd−1, t−Tr(ν)). The function ¨h is defined by

¨h(η₁, η₂) = Z

· · · Z

R^d

h(u)|N u|^η1 |1 +u₁| · · · |1 +u_d|η2

du.

Remark that h(u) is of rapid decay if u_j tends to infinity, 0 or −1 for at least onej. The expressions for B^(r)m ,B^(c)m andB^(e)m are regular in the domain (3.2.41).

Proof.

The spectral decomposition holds in the domain (3.2.34). Thus we have to prove that all terms can be continued to the domain (3.2.41).

Consider the cuspidal contribution. The sum is absolutely and uniformly convergent in the domain

|Reα|+ Reβ <2 min

V min

j

|Imκ_j+¹₂

+δ_j , by Lemma 3.1.6 and Lemma 3.2.2, (3.2.42) and it is regular there. By the possible values of κ_j this domain contains the domain (3.2.41). Thus B^(c)m is of the form stated in the Theorem.

Now we consider the contribution of the Eisenstein series. In the domain (3.2.34) this is of the form stated in the Theorem, but the domain (3.2.41) is not contained in (3.2.34). Hence we have to extend the domain for the contribution of the Eisenstein series and pay attention to possible singularities.

However, for the terms withν6= 0the integrand is regular in (3.2.41), and the sumP

ν∈Λ^d−1 converges absolutely by Lemma 3.2.2.

Thus, we exclude this part from our consideration and restrict our attention to the termν = 0, namely 2(2π)^dβ

(2π)^2d C_e

∞

Z

−∞

σ_2it(m)

(N m)^it|ζ_K(1 + 2it)|²ζ_K ¹₂(1 +α−β)−it

ζ_K ¹₂(1 +α−β) +it

×

×ζ_K ¹₂(1−α−β)−it

ζ_K ¹₂(1−α−β) +it

Ψ_t(α, β;h)dt.

(3.2.43)

Here(α, β) is in the domain (3.2.34), and

Ψ_t(α, β;h) = Φ_E0(t, . . . , t

| {z }

d−times

;α, β;h). (3.2.44)

The integral is regular when |Reα|+ Reβ < −1, as in this domain Re(¹₂(1±α −β)) > 1. We consider the subdomain−⁵₄ <Re(±α+β)<−1. We move the contour toImt= ¹₈ by noting (3.2.39) and (3.2.40). To avoid poles on the contour, we can choose an appropriate broken line instead of a horizontal line.

The resulting integral is regular for−⁵₄ <Re(±α+β)<−³₄. Now we restrict this domain to

−1<Re(±α+β)<−3

4, (3.2.45)

and shift thet-contour back to Imt= 0.

As the new integral is regular in (3.2.41),B^(e)m continues to (3.2.41).

The first shift of the contour gives us poles att=−¹₂(1±α+β)i, and those from the factorζ_K⁻¹(1+2it).

By the second one we encounter poles fromζ_K⁻¹(1 + 2it) and at the pointst= +¹₂(1±α+β)i.

The residual terms coming from ζ_K⁻¹ cancel, and we are left with the terms coming from the residues at t=±¹₂(1±α+β)i, given by

(2π)^dβ (2π²)^d

RK

p|D_K|Ce·(N m)^1+βσα−β−1(m)ζK(−β)ζ_K(1−α) ζ_K(2−α+β) Ψ¹

2(1−α+β)i(α, β;h) +(2π)^dβ

(2π²)^d R_K

p|D_K|C_e·(N m)^1+α+βσ_{−α−β−1}(m)ζ_K(−β)ζ_K(1 +α) ζK(2 +α+β) Ψ1

2(1+α+β)i(α, β;h).

(3.2.46)

We used thatΨt is an even function in t, and that the residue ofζK(s) ats= 1 is equal to ²√^d−1RK

|D_K|. Hence, we are left with computing the factorsΨ1

2(1∓α+β)i. Letrj = ¹₂(1−α+β)i. Then

X

εmodP()²

η∗(ε)

d

Y

j=1

∆^sgn(εej ^j)(sj, r;α, β) = 2^2d

d

Y

j=1

ej

sin(π(sj−α)) cos ¹₂π((1−ej)sj +ejβ)

asλ_E0 = 1.

We get Ψ¹

2(1−α+β)i(α, β;h) = 2^d i^dd

X

e∈{±1}^d

e₁· · ·e_d Z

· · · Z

(−i∞,i∞)^d

eh(s; e)×

×

d

Y

j=1

cos ¹₂π((1−e_j)s_j+e_jβ)

Γ(s_j−1−β)Γ(1−s_j) ds.

Thesj-contour separates the poles ofΓ(sj−1−β)andΓ(1−sj)to the left and the right, respectively, and by the restriction to the domain (3.2.45) can be drawn.

Now we let r_j = ¹₂(1 +α+β)i. Using the same arguments as above, we get Ψ1

2(1+α+β)i = 2^d i^dd

X

e∈{±1}^d

e₁· · ·e_d Z

· · · Z

(−i∞,i∞)^d

eh(s+α; e)×

×

d

Y

j=1

cos ¹₂π((1−ej)sj+ejβ)

Γ(sj−1−β)Γ(1−sj) ds

wheres+α= (s1+α, . . . , sd+α).

We have to compute the multiple integrals over s. We consider the second one in detail, as it is a bit more complicated than the first. The first can be done following these steps.

We use the definition ofeh given in (3.2.13) and apply partial integration in each variable.

Thus, the integral equals

e1· · ·e_d Z

· · · Z

(0,∞)^d

h_d(eu)

d

Y

j=1

R(uj, ej)du (3.2.47)

whereh_d(u) = (∂_u1. . . ∂_udh)(u),

R(u_j, e_j) =

i∞

Z

−i∞

cos ¹₂π((1−e_j)s_j+e_jβ)

Γ(s_j−1−β)Γ(1−s_j)u^s_j^j+α ds_j

sj+α. (3.2.48) By (3.2.45) we may take for instanceRes= ¹₂ as contour. In particular, we can assume that the pole s=−α is on the left of the contour.

We shift the contour in (3.2.48) to Res = +∞ or to Res = −∞, depending on whether u_j < 1 or u_j >1.

For uj < 1 the shift of contour to Res = +∞ collects poles of Γ(1−s), and the resulting integral vanishes. We get

R(u_j, e_j) = 2πie_jcos(¹₂πβ)Γ(−β)

uj

Z

0

u^α(1 +e_ju)^βdu.

Ifu_j >1 we collect the poles ofΓ(s−1−β)and the pole s=−α. We get

R(uj, ej) =−2πiejcos(¹₂πβ)Γ(−β)

∞

Z

uj

u^α(|1 +eju|^β−u^β)du

+ 2πiejcos(¹₂πβ)Γ(−β) u^1+α+β_j 1 +α+β + 2πicos ¹₂π((e_j−1)α+e_jβ)

Γ(1 +α)Γ(−1−α−β).

Hence, for u_j >0,u_j 6= 1, we have

∂_ujR(u_j, e_j) = 2πie_jcos(¹₂πβ)Γ(−β)u^α_j|1 +e_ju_j|^β.

Applying partial integration to (3.2.47) and inserting this expression for ∂_ujRwe obtain Ψ¹

2(1+α+β)i(α, β;h) = (2πi)^d cos(¹₂πβ)Γ(−β)d¨h(α, β) with¨h as defined in the Theorem.

Analogously, we have Ψ¹

2(1−α+β)i(α, β;h) = (2³π)^d

d cos(¹₂πβ)Γ(−β)dh(0, β).¨

We insert this into (3.2.46) and use the functional equation for the Dedekind ζ-function to see that (3.2.46) changes to

Ce2^d

dπ^d RK|D_K|^βζK(1−α)ζK(1 +β)

ζ_K(2−α+β) (N m)^1+βσα−β−1(m)¨h(0, β) +C_e2^d

dπ^d R_K|D_K|^βζ_K(1 +α)ζ_K(1 +β)

ζK(2 +α+β) (N m)^1+α+βσ−α−β−1(m)¨h(α, β).

(3.2.49)

Finally we consider the residual contribution. Taking the first two summands of the decomposition of Bm given in (3.2.35) and the terms (3.2.49) coming fromB^(e)m we see that they coincides withB^(r)m .

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