The orthogonal space of signature (2, 2)

Finally, we state the theta lift from Definition 4.2.6 in the situation of the current lattice 𝐿. Let 𝑘 be a non-negative integer. Identifying H with H1 via the map 𝑧 ↦→𝑧𝑒₃, and using the identity from (4.3.5) and the fact that 𝑞(Im(𝑧𝑒₃)) = 𝑁Im(𝑧)² for 𝑧 ∈ H we can write Shintani’s theta function for the lattice 𝐿 as

Θ_𝐿,𝑘(𝜏, 𝑧) = 𝑁^−𝑘Im(𝜏)^1/2Im(𝑧)^−2𝑘 ∑︁

𝛽∈𝐿^′/𝐿

∑︁

𝜆∈𝐿+𝛽

𝑄_𝜆(𝑧,1)^𝑘𝑒(𝜏 𝑞(𝜆_𝑧) +𝜏 𝑞(𝜆_𝑧^⊥))e_𝛽 (4.3.6)

for 𝜏, 𝑧 ∈ H. Here the quantities 𝑞(𝜆_𝑧) and 𝑞(𝜆_𝑧^⊥) are given as in Lemma 4.3.1. By Corollary 4.2.3 the theta functionΘ_𝐿,𝑘(𝜏, 𝑧)is modular of weight𝑘+ 1/2with respect to 𝜌_𝐿 in 𝜏, and in the variable 𝑧 it transforms as

Θ_𝐿,𝑘(𝜏, 𝛾𝑧) = 𝑗(𝛾, 𝑧)^2𝑘Θ_𝐿,𝑘(𝜏, 𝑧)

for 𝜏, 𝑧 ∈ H and 𝛾 ∈ Γ₀(𝑁). Accordingly, the (regularized) theta lift of a real analytic function 𝐹:H→C[𝐿^′/𝐿] modular of weight 𝑘+ 1/2 with respect to𝜌_𝐿 is given by

Φ^𝐿_𝑘(𝑧;𝐹) = (𝐹,Θ_𝐿,𝑘(·, 𝑧))^reg

for 𝑧 ∈ H, and assuming that the (regularized) inner product exists the lift Φ^𝐿_𝑘(𝑧;𝐹) of 𝐹 is modular of weight 2𝑘 and level 𝑁.

For an integer 𝑘 ≥ 2 the theta lift Φ^𝐿_𝑘(𝑧; ·) is essentially the classical Shimura theta lift, mapping (vector valued) holomorphic cusp forms of weight 𝑘+ 1/2 to holomorphic cusp forms of weight 2𝑘. It was introduced by Shimura in [Shi73] and later expressed as a theta lift by Niwa in [Niw75]. In particular, the Shimura lift of a holomorphic Poincaré series is up to some constant the corresponding Zagier cusp form introduced in Section 2.4.2 (see for example [Oda77] or [KZ81]). More precisely, let𝛽 ∈𝐿^′/𝐿and 𝑚∈Z+𝑞(𝛽) with 𝑚 >0. Then an easy unfolding argument shows that

Φ^𝐿_𝑘(𝑧;𝑃_{𝑘+1/2,𝛽,𝑚}) = 2 (𝑘−1)!

𝜋^𝑘 𝑓_{𝑘,𝛽,4𝑁 𝑚}(𝑧) (4.3.7)

for 𝑘 ∈ Z with 𝑘 ≥ 2 and 𝑧 ∈ H. Here 𝑓_{𝑘,𝛽,4𝑁 𝑚}(𝑧) is the meromorphic modular form introduced in Definition 2.4.6. Note that we can ignore the regularization of the inner product defining the theta lift in this case since 𝑃_{𝑘+1/2,𝛽,𝑚}(𝜏) is a cusp form for 𝑚 >0.

for 𝑋 ∈ 𝑉. Then (𝑉, 𝑞) is a quadratic space of signature (2,2) with associated bilinear form given by (𝑋, 𝑌) = −tr(𝑋𝑌^#) for 𝑋, 𝑌 ∈ 𝑉 where (︀_{𝑎 𝑏}

𝑐 𝑑

)︀#

= (︀ _{𝑑 −𝑏}

−𝑐 𝑎

)︀. Given a positive integer 𝑁 we consider the lattice

𝐿:=

{︂(︂ 𝑎 𝑏 𝑁 𝑐 𝑑

)︂

: 𝑎, 𝑏, 𝑐, 𝑑∈Z }︂

in𝑉. Then𝐿 is an even lattice of level 𝑁 with dual lattice given by 𝐿^′ =

{︂(︂𝑎 𝑏/𝑁

𝑐 𝑑

)︂

: 𝑎, 𝑏, 𝑐, 𝑑∈Z }︂

,

and the discriminant group𝐿^′/𝐿is isomorphic to the quotient(Z/𝑁Z)². It is well-known that the action of the identity component SO⁺(𝑉) of the orthogonal group of 𝑉 can be realized by the action of the group SL₂(R)×SL₂(R)on 𝑉(R)via

(𝜎, 𝜎^′).𝑋 :=𝜎𝑋𝜎^′−1

for 𝜎, 𝜎^′ ∈ SL₂(R) and 𝑋 ∈ 𝑉(R). Here the four elements (±𝜎,±𝜎^′) define the same element inSO⁺(𝑉). Further, one easily checks that the discrete subgroupΓ₀(𝑁)×Γ₀(𝑁) of SL2(R)×SL2(R) acts on 𝐿^′, and that its subgroup Γ(𝑁)×Γ(𝑁) fixes the classes of the discriminant group 𝐿^′/𝐿, i.e., we have

Γ(𝑁)×Γ(𝑁)⊆Γ(𝐿) and Γ₀(𝑁)×Γ₀(𝑁)⊆SO⁺(𝐿), (4.4.1)

under the above identification of SO⁺(𝑉)with the productSL₂(R)×SL₂(R). HereΓ(𝑁) is the usual principal congruence subgroup given by

Γ(𝑁) :=

{︂(︂𝑎 𝑏 𝑐 𝑑

)︂

∈SL2(Z) :

(︂𝑎 𝑏 𝑐 𝑑

)︂

≡ (︂1 *

0 1 )︂

mod 𝑁 }︂

, which clearly is a subgroup of Γ0(𝑁).

We now choose vectors 𝑒1 :=

(︂0 1 0 0

)︂

, 𝑒2 :=

(︂0 0 1 0

)︂

, 𝑒3 :=

(︂1 0 0 0

)︂

, 𝑒4 :=

(︂0 0 0 −1

)︂

.

Then𝑒₁ ∈𝐿is primitive and isotropic and𝑒₂ ∈𝐿^′ is isotropic with(𝑒₁, 𝑒₂) = 1. Moreover, the hyperbolic plane spanned by 𝑒₁ and 𝑒₂ is orthogonal to the plane spanned by the isotropic vectors 𝑒₃ and 𝑒₄ where (𝑒₃, 𝑒₄) = 1. We define the sublattice 𝐾 =𝐿∩𝑒^⊥₁ ∩𝑒^⊥₂, the corresponding quadratic space𝑈 =𝐾⊗Qof signature(1,1)and the induced complex planeℋ={𝑋+𝑖𝑌 ∈𝑈(C) :𝑞(𝑌)>0}. Then𝐾 ={(︀_𝑎0

0 𝑑

)︀:𝑎, 𝑑 ∈Z},𝑈 ={(︀_𝑎0

0𝑑

)︀: 𝑎, 𝑑∈ Q} and thus

ℋ =

{︂(︂𝑧 0 0 𝑧^′

)︂

: 𝑧, 𝑧^′ ∈C with Im(𝑧) Im(𝑧^′)<0 }︂

.

Hence by choosing H² as the connected component H𝑒₃⊕H𝑒₄ of ℋ we can identify the generalized upper half-planeH2 with the productH×H. Let𝒦be the complex manifold defined in (4.1.1) associated to the space 𝑉(C). We choose𝒦⁺ to be the image ofH2 in 𝒦 under the map 𝑍 ↦→[𝑍_𝐿] where

𝑍_𝐿 =𝑍 −(𝑞(𝑍) +𝑞(𝑒₂))𝑒₁+𝑒₂ =

(︂𝑧 −𝑧𝑧^′ 1 −𝑧^′

)︂

for 𝑍 =𝑧𝑒₃+𝑧^′𝑒₄ ∈H². Conversly, given 𝑧, 𝑧^′ ∈H with 𝑧=𝑥+𝑖𝑦, 𝑧^′ =𝑥^′ +𝑖𝑦^′ we set 𝑋_𝐿(𝑧, 𝑧^′) :=

(︂𝑥 𝑦𝑦^′−𝑥𝑥^′ 1 −𝑥^′

)︂

and 𝑌_𝐿(𝑧, 𝑧^′) :=

(︂𝑦 −𝑥𝑦^′−𝑥^′𝑦 0 −𝑦^′

)︂

such that𝑋_𝐿(𝑧, 𝑧^′) = Re(𝑍_𝐿),𝑌_𝐿(𝑧, 𝑧^′) = Im(𝑍_𝐿)for𝑍 =𝑧𝑒₃+𝑧^′𝑒₄. Then𝑞(𝑋_𝐿(𝑧, 𝑧^′)) = 𝑞(𝑌_𝐿(𝑧, 𝑧^′)) =𝑦𝑦^′, and the identification ofH2 with the Grassmannian Gr(𝑉) is given by the map 𝑧𝑒₃+𝑧^′𝑒₄ ↦→R𝑋_𝐿(𝑧, 𝑧^′)⊕R𝑌_𝐿(𝑧, 𝑧^′).

In order to understand the action of the groupSL2(R)×SL2(R) onH² we compute 𝑗((𝜎, 𝜎^′), 𝑍) = ((𝜎, 𝜎^′).𝑍𝐿, 𝑒1) =𝑗(𝜎, 𝑧)𝑗(𝜎^′, 𝑧^′)

(4.4.2)

for 𝜎, 𝜎^′ ∈ SL₂(R) and 𝑍 = 𝑧𝑒₃ +𝑧^′𝑒₄ ∈ H2. Here the automorphy factors on the right are the classical ones defined at the beginning of Section 2.4. Hence we find

((𝜎, 𝜎^′).𝑍)_𝐿=𝑗((𝜎, 𝜎^′), 𝑍)⁻¹(𝜎, 𝜎^′).𝑍_𝐿=

(︂𝜎𝑧 −𝜎𝑧·𝜎^′𝑧^′ 1 −𝜎^′𝑧^′

)︂

= (𝜎𝑧 𝑒₃ +𝜎^′𝑧^′𝑒₄)_𝐿 (4.4.3)

for 𝜎, 𝜎^′ ∈ SL₂(R) and 𝑍 = 𝑧𝑒₃ +𝑧^′𝑒₄ ∈ H2, i.e., (𝜎, 𝜎^′).𝑍 = 𝜎𝑧 𝑒₃+𝜎^′𝑧^′𝑒₄. Therefore, the identification of H2 with the product H×H is SL₂(R)×SL₂(R)-equivariant where the latter group naturally acts on H×H via(𝜎, 𝜎^′).(𝑧, 𝑧^′) = (𝜎𝑧, 𝜎^′𝑧^′).

In particular, for 𝑘 ∈ Z a function 𝐹: H² → C is modular of weight 𝑘 with respect to the group Γ(𝑁)×Γ(𝑁) in the sense of Definition 4.1.2 if and only if the function 𝑓(𝑧, 𝑧^′) = 𝐹(𝑧𝑒₃ +𝑧^′𝑒₄) for 𝑧, 𝑧^′ ∈ H is modular of weight 𝑘 for the group Γ(𝑁) in the classical sense in both variables, i.e., if

𝑓(𝛾𝑧, 𝛾^′𝑧^′) = 𝑗(𝛾, 𝑧)^𝑘𝑗(𝛾^′, 𝑧^′)^𝑘𝑓(𝑧, 𝑧^′) for all 𝛾, 𝛾^′ ∈Γ(𝑁) and 𝑧, 𝑧^′ ∈H.

For𝜆∈𝑉 and 𝑧, 𝑧^′ ∈H we denote the orthogonal projection of 𝜆 onto the positive or negative definite subspace R𝑋𝐿(𝑧, 𝑧^′)⊕R𝑌𝐿(𝑧, 𝑧^′) or𝑋𝐿(𝑧, 𝑧^′)^⊥∩𝑌𝐿(𝑧, 𝑧^′)^⊥ in 𝑉(R) by 𝜆_𝑧,𝑧^′ or𝜆_(𝑧,𝑧^′₎^⊥, respectively. As 𝑞(Im(𝑍)) = Im(𝑧) Im(𝑧^′)we can restate equation (4.1.2) in the present setting as

𝑞(𝜆𝑧,𝑧^′) = |(𝜆, 𝑍_𝐿)|² 4 Im(𝑧) Im(𝑧^′) (4.4.4)

for 𝜆 ∈ 𝑉 and 𝑧, 𝑧^′ ∈ H with 𝑍 = 𝑧𝑒₃ +𝑧^′𝑒₄. Further, given 𝑧, 𝑧^′ ∈ H we denote the majorant of 𝑞 associated to the pair (𝑧, 𝑧^′) as defined in (4.1.3) by

𝑞_𝑧,𝑧^′(𝜆) =𝑞(𝜆_𝑧,𝑧^′)−𝑞(𝜆_(𝑧,𝑧^′₎^⊥), and using (4.4.4) we thus find

𝑞_𝑧,𝑧^′(𝜆) = |(𝜆, 𝑍𝐿)|²

2 Im(𝑧) Im(𝑧^′)−𝑞(𝜆) (4.4.5)

for 𝜆∈𝑉 and 𝑧, 𝑧^′ ∈H with 𝑍 =𝑧𝑒3+𝑧^′𝑒4. 86

Eventually, we present the theta liftΦ^𝐿_𝑘(𝑍;𝐹)from Section 4.2 in the case of𝐿being our current lattice of signature(2,2). Recall that𝑞(Im(𝑍)) = Im(𝑧) Im(𝑧^′)for 𝑍 =𝑧𝑒₃+𝑧^′𝑒₄ with 𝑧, 𝑧^′ ∈ H. Let now 𝑘 be a non-negative integer. Identifying H2 with the product H×H we can write Shintani’s theta function for the lattice𝐿 as

Θ_𝐿,𝑘(𝜏, 𝑧, 𝑧^′) = 𝑣 Im(𝑧)^−𝑘Im(𝑧^′)^−𝑘 ∑︁

𝛽∈𝐿^′/𝐿

∑︁

𝜆∈𝐿+𝛽

(𝜆, 𝑍_𝐿)^𝑘𝑒(︁

𝑢𝑞(𝜆) +𝑖𝑣𝑞_𝑧,𝑧^′(𝜆))︁

e_𝛽 (4.4.6)

for𝜏, 𝑧, 𝑧^′ ∈Hwith 𝜏 =𝑢+𝑖𝑣. The theta functionΘ𝐿,𝑘(𝜏, 𝑧)is modular of weight𝑘 with respect to 𝜌_𝐿 in𝜏, and in the variables 𝑧 and 𝑧^′ it transforms as

Θ_𝐿,𝑘(𝜏, 𝛾𝑧, 𝛾^′𝑧^′) = 𝑗(𝛾, 𝑧)^𝑘𝑗(𝛾^′, 𝑧^′)^𝑘Θ_𝐿,𝑘(𝜏, 𝑧, 𝑧^′)

for 𝜏, 𝑧, 𝑧^′ ∈H and 𝛾, 𝛾^′ ∈Γ(𝑁). Therefore, the (regularized) theta lift of a real analytic function 𝐹:H→C[𝐿^′/𝐿] modular of weight 𝑘 with respect to𝜌_𝐿 is given by

Φ^𝐿_𝑘(𝑧, 𝑧^′;𝐹) = (𝐹,Θ𝐿,𝑘(·, 𝑧, 𝑧^′))^reg

for 𝑧, 𝑧^′ ∈H, and the lift Φ^𝐿_𝑘(𝑧, 𝑧^′;𝐹) of 𝐹 is modular of weight 𝑘 for the group Γ(𝑁)in both variables 𝑧 and 𝑧^′ whenever the (regularized) inner product exists. In this setting, the lift Φ^𝐿_𝑘(𝑧, 𝑧^′;𝐹) is sometimes called the Doi-Naganuma lift of 𝐹, which classically is a lift from scalar valued modular forms to Hilbert modular forms (see [DN67, DN69]).

5 Realizing non-holomorphic

Eisenstein series as theta lifts

In this chapter we realize the non-holomorphic Eisenstein series introduced in Section 2.6 as regularized theta lifts in two different ways. On the one hand we obtain averaged versions of these Eisenstein series as the theta lift of Selberg’s Poincaré series of the first kind using the lattice of signature (2,1)from Section 4.3. Remarkably, we indeed obtain averaged versions of hyperbolic, parabolic an elliptic Eisenstein series as the theta lift of a single type of Poincaré series.

On the other hand we realize the hyperbolic kernel function defined in Section 2.6.4 as the theta lift of a more elementary Poincaré series introduced in Section 3.6, using the lattice of signature (2,2) from Section 4.4. We can then use this representation of the hyperbolic kernel function to obtain individual hyperbolic, parabolic and elliptic Eisenstein series as variations of this theta lift.

5.1 Regularized theta lifts of non-holomorphic Poincaré series

Let (𝑉, 𝑞) be a quadratic space of signature (2, 𝑛) and let 𝐿 be an even lattice in 𝑉. We further fix vectors 𝑒₁ ∈ 𝐿 and 𝑒₂ ∈ 𝐿^′ such that 𝑒₁ is primitive with 𝑞(𝑒₁) = 0 and (𝑒₁, 𝑒₂) = 1 as in Section 4.1, and we let H𝑛 be the induced generalized upper half-plane.

For𝛽 ∈𝐿^′/𝐿 and 𝑚∈Z+𝑞(𝛽) we set

𝐿_𝛽,𝑚:={𝜆 ∈𝐿+𝛽: 𝑞(𝜆) = 𝑚}, and we define the subset 𝐻_𝛽,𝑚^𝐿 of H^𝑛 by

𝐻_𝛽,𝑚^𝐿 := ⋃︁

𝜆∈𝐿_𝛽,𝑚 𝜆̸=0

{𝑍 ∈H^𝑛: (𝜆, 𝑍𝐿) = 0}.

(5.1.1)

Note that𝐻_𝛽,𝑚^𝐿 =∅if𝑚≥0since𝜆∈𝐿_𝛽,𝑚can only be orthogonal to some𝑍_𝐿=𝑋_𝐿+𝑖𝑌_𝐿 with 𝑍 ∈H𝑛 if𝜆= 0 or𝑞(𝜆)<0as R𝑋_𝐿⊕R𝑌_𝐿 defines a2-dimensional positive definite subspace of 𝑉(R).

Let𝑘 be a non-negative integer and set𝜅:= 1 +𝑘−𝑛/2. Then 𝜅satisfies the condition given in (3.4.1). In the present section we study the regularized theta lift of two of the non-holomorphic Poincaré series given in Definition 3.4.3, namely Selberg’s Poincaré series of the first kind 𝑈_{𝜅,𝛽,𝑚}^𝐿 (𝜏, 𝑠)and the simpler, non-standard Poincaré series 𝑄^𝐿_{𝜅,𝛽,𝑚}(𝜏, 𝑠).

Definition 5.1.1. Let𝑘 ∈Z with 𝑘≥0, 𝜅= 1 +𝑘−𝑛/2,𝛽 ∈𝐿^′/𝐿 and 𝑚∈Z+𝑞(𝛽).

(a) We define the regularized theta lift of Selberg’s Poincaré series of the first kind 𝑈_{𝜅,𝛽,𝑚}^𝐿 (𝜏, 𝑠) by

Φ^Sel,𝐿_{𝑘,𝛽,𝑚}(𝑍, 𝑠) = Φ^𝐿_𝑘(𝑍;𝑈_{𝜅,𝛽,𝑚}^𝐿 (·, 𝑠)) for 𝑍 ∈H𝑛∖𝐻_𝛽,𝑚^𝐿 and 𝑠 ∈Cwith Re(𝑠)>1 +𝑛/2.

(b) We define the regularized theta lift of the Poincaré series 𝑄^𝐿_{𝜅,𝛽,𝑚}(𝜏, 𝑠)by Φ^Q,𝐿_{𝑘,𝛽,𝑚}(𝑍, 𝑠) = Φ^𝐿_𝑘(𝑍;𝑄^𝐿_{𝜅,𝛽,𝑚}(·, 𝑠))

for 𝑍 ∈H^𝑛 and 𝑠∈C with Re(𝑠)>1 +𝑛/2.

Here Φ^𝐿_𝑘(𝑍; ·) denotes the regularized theta lift from Definition 4.2.6. We will see in Theorem 5.1.3 that the above definition is indeed well-defined, i.e., that the corresponding (regularized) integrals exist in all the given cases. Thus,Φ^Sel,𝐿_{𝑘,𝛽,𝑚}(𝑍, 𝑠)and Φ^Q,𝐿_{𝑘,𝛽,𝑚}(𝑍, 𝑠) are modular of weight 𝑘 in the variable𝑍 with respect to the groupΓ(𝐿) defined in (4.1.4).

Before we start studying the above lifts, we quickly recall some technicalities. For 𝑍 ∈H^𝑛 the majorant of the present quadratic form𝑞 associated to𝑍 is given by𝑞𝑍(𝜆) = 𝑞(𝜆_𝑍)−𝑞(𝜆_𝑍^⊥)for𝜆 ∈𝑉 (see equation (4.1.3)). Here𝜆_𝑍 and 𝜆_𝑍^⊥ denote the orthogonal projections of 𝜆onto the positive or negative definite subspaces of𝑉 corresponding to 𝑍 and 𝑍^⊥, respectively. The majorant 𝑞𝑍 defines a positive definite quadratic form on 𝑉, and thus the Epstein zeta function associated to 𝑞_𝑍 and the lattice 𝐿^′ defined by

𝑍(𝑠;𝑞_𝑍, 𝐿^′) := ∑︁

𝜆∈𝐿^′∖{0}

𝑞_𝑍(𝜆)^−𝑠 (5.1.2)

converges absolutely and locally uniformly for 𝑠 ∈ C with Re(𝑠) > 1 +𝑛/2 (see for example [Eps03]).

Lemma 5.1.2. Let 𝛽 ∈𝐿^′/𝐿, 𝑚∈Z+𝑞(𝛽) and 𝑍 ∈H𝑛∖𝐻_𝛽,𝑚^𝐿 . Then the sum

∑︁

𝜆∈𝐿_𝛽,𝑚∖{0}

𝑞(𝜆𝑍)^−𝑠

converges absolutely and locally uniformly for 𝑠∈C with Re(𝑠)>1 +𝑛/2.

Proof. Firstly, we note that for 𝜆 ∈ 𝐿_𝛽,𝑚 we have 𝑞(𝜆_𝑍) = 0 if and only if (𝜆, 𝑍_𝐿) = 0 by (4.1.2), which is the case if and only if 𝜆 = 0 or 𝑍 ∈ 𝐻_𝛽,𝑚^𝐿 . Thus the given sum is well-defined. Next, we can choose a finite subset 𝐾 ⊂ 𝐿^′ such that 𝑞_𝑍(𝜆)≥ 2|𝑚| for all 𝜆∈𝐿^′∖𝐾 since 𝑞_𝑍 is positive definite. Hence we obtain

𝑞(𝜆𝑍) = 𝑞_𝑍(𝜆) +𝑚

2 = 𝑞_𝑍(𝜆)

4 + 𝑞_𝑍(𝜆) + 2𝑚

4 ≥ 𝑞_𝑍(𝜆) 4 for 𝜆∈𝐿𝛽,𝑚∖𝐾, giving

∑︁

𝜆∈𝐿_𝛽,𝑚 𝜆̸=0

⃒⃒𝑞(𝜆_𝑍)^−𝑠⃒

⃒≤ ∑︁

𝜆∈𝐿_𝛽,𝑚∩𝐾 𝜆̸=0

𝑞(𝜆_𝑍)^−Re(𝑠)+ 4^Re(𝑠) ∑︁

𝜆∈𝐿_𝛽,𝑚∖𝐾 𝜆̸=0

𝑞_𝑍(𝜆)^−Re(𝑠).

Here the first sum is finite and the second sum is dominated by the Epstein zeta function associated to𝑞_𝑍 and 𝐿^′ given in (5.1.2), which converges absolutely and locally uniformly for 𝑠∈C with Re(𝑠)>1 +𝑛/2.

90

We can now prove that the theta lifts given in Definition 5.1.1 are indeed well-defined.

Moreover, if 𝑘 >0 we already obtain a meromorphic continuation of the two lifts to the half-plane defined by Re(𝑠)>1 +𝑛/2−𝑘/2

Theorem 5.1.3. Let 𝑘 ∈Z with 𝑘 ≥0, 𝜅= 1 +𝑘−𝑛/2, 𝛽 ∈𝐿^′/𝐿 and 𝑚∈Z+𝑞(𝛽).

(a) The regularized theta lift of Selberg’s Poincaré series𝑈_{𝜅,𝛽,𝑚}^𝐿 (𝜏, 𝑠)defines a real analytic function in 𝑍 ∈ H𝑛∖𝐻_𝛽,𝑚^𝐿 and a holomorphic function in 𝑠 for Re(𝑠) > 1 +𝑛/2, which is given by

Φ^Sel,𝐿_{𝑘,𝛽,𝑚}(𝑍, 𝑠) = 2 Γ(𝑠+𝑘) 4^𝑠𝜋^𝑠+𝑘

∑︁

𝜆∈𝐿_𝛽,𝑚 𝜆̸=0

𝑞(𝜆_𝑍)^−𝑠(𝜆, 𝑍_𝐿)^−𝑘.

For 𝑘 >0 and 𝑍 ∈ H𝑛∖𝐻_𝛽,𝑚^𝐿 the sum on the right-hand side yields a holomorphic continuation of Φ^Sel,𝐿_{𝑘,𝛽,𝑚}(𝑍, 𝑠) in 𝑠 to the half-plane Re(𝑠)>1 +𝑛/2−𝑘/2.

(b) The regularized theta lift of the Poincaré series 𝑄^𝐿_{𝜅,𝛽,𝑚}(𝜏, 𝑠) defines a real analytic function in 𝑍 ∈ H^𝑛 and a holomorphic function in 𝑠 for Re(𝑠)> 1 +𝑛/2, which is given by

Φ^Q,𝐿_{𝑘,𝛽,𝑚}(𝑍, 𝑠) = 2 Γ(𝑠+𝑘)

(2𝜋)^𝑠+𝑘 𝑞(Im(𝑍))^−𝑘 ∑︁

𝜆∈𝐿𝛽,𝑚

𝜆̸=0

𝑞_𝑍(𝜆)^{−𝑠−𝑘}(𝜆, 𝑍_𝐿)^𝑘.

For 𝑘 >0 and 𝑍 ∈H𝑛 the sum on the right-hand side yields a holomorphic continu-ation of Φ^Q,𝐿_{𝑘,𝛽,𝑚}(𝑍, 𝑠) in 𝑠 to the half-plane Re(𝑠)>1 +𝑛/2−𝑘/2.

Proof. Since (a) and (b) are poven analogously, we only give a detailed proof for part (a), and afterwards comment on the necessary adaptions of the given proof in the case of (b).

For (a) fix 𝑍 ∈ H𝑛 ∖𝐻_𝛽,𝑚^𝐿 and 𝑠 ∈ C with Re(𝑠) > 1 + 𝑛/2. Evaluating the inner product ⟨︀

𝑈_{𝜅,𝛽,𝑚}^𝐿 (𝜏, 𝑠),Θ_𝐿,𝑘(𝜏, 𝑍)⟩︀

and splitting the sum over matrices 𝑀 ∈ ⟨𝑇⟩∖SL₂(Z) coming from the Poincaré series into matrices 𝑀 with lower left entry 𝑐̸= 0 and 𝑐= 0 we can write the theta lift Φ^Sel,𝐿_{𝑘,𝛽,𝑚}(𝑍, 𝑠) as

CT_𝑡=0 [︃

𝑇lim→∞

∫︁

ℱ_𝑇

∑︁

𝑀∈⟨𝑇⟩∖SL2(Z) 𝑀̸=±1

Im(𝑀 𝜏)^𝑠+𝜅𝑒(𝑚𝑀 𝜏)Θ_{𝐿,𝑘,𝛽}(𝑀 𝜏, 𝑍)𝑣^−𝑡𝑑𝜇(𝜏) ]︃

(5.1.3)

+ 2 CT_𝑡=0 [︂

𝑇lim→∞

∫︁

ℱ_𝑇

𝑣^{𝑠+𝜅−𝑡}𝑒(𝑚𝜏)Θ_{𝐿,𝑘,𝛽}(𝜏, 𝑍)𝑑𝜇(𝜏) ]︂

. Here we use the notation𝜏 =𝑢+𝑖𝑣,𝜅= 1 +𝑘−𝑛/2and Θ_{𝐿,𝑘,𝛽}(𝜏, 𝑍) = ⟨Θ_𝐿,𝑘(𝜏, 𝑍),e𝛽⟩.

Letℛ:={𝜏 ∈H: |Re(𝜏)| ≤1/2}. Then ℛ is a rectangular fundamental domain for the action of ⟨𝑇⟩ on H, and by Corollary 4.2.5 we have

∫︁

ℛ∖ℱ

⃒

⃒

⃒𝑣^𝑠+𝜅𝑒(𝑚𝜏)Θ_{𝐿,𝑘,𝛽}(𝜏, 𝑍)

⃒

⃒

⃒𝑑𝜇(𝜏)≤𝐶

∫︁ 1 0

𝑣Re(𝑠)−𝑛/2−2

𝑑𝑣 <∞ (5.1.4)

for some constant𝐶 > 0asRe(𝑠)>1 +𝑛/2. Hence we can plug in𝑡= 0 in the first term in (5.1.3), take the limit 𝑇 → ∞, interchange summation and integration and apply the usual unfolding trick to find

CT_𝑡=0 [︃

𝑇lim→∞

∫︁

ℱ𝑇

∑︁

𝑀∈⟨𝑇⟩∖SL2(Z) 𝑀̸=±1

Im(𝑀 𝜏)^𝑠+𝜅𝑒(𝑚𝑀 𝜏)Θ_{𝐿,𝑘,𝛽}(𝑀 𝜏, 𝑍)𝑣^−𝑡𝑑𝜇(𝜏) ]︃

= 2

∫︁

ℛ∖ℱ

𝑣^𝑠+𝜅𝑒(𝑚𝜏)Θ_{𝐿,𝑘,𝛽}(𝜏, 𝑍)𝑑𝜇(𝜏).

In the second term in (5.1.3) we split the integral at the horizontal line 𝑣 = 1, i.e., CT_𝑡=0

[︂

𝑇lim→∞

∫︁

ℱ𝑇

𝑣^{𝑠+𝜅−𝑡}𝑒(𝑚𝜏)Θ_{𝐿,𝑘,𝛽}(𝜏, 𝑍)𝑑𝜇(𝜏) ]︂

=

∫︁

ℱ₁

𝑣^𝑠+𝜅𝑒(𝑚𝜏)Θ𝐿,𝑘,𝛽(𝜏, 𝑍)𝑑𝜇(𝜏) + CT𝑡=0

[︃

𝑇lim→∞

∫︁

ℱ_𝑇∖ℱ₁

𝑣^{𝑠+𝜅−𝑡}𝑒(𝑚𝜏)Θ𝐿,𝑘,𝛽(𝜏, 𝑍)𝑑𝜇(𝜏) ]︃

. Here we can drop the regularization in the first term on the right-hand side as ℱ1 is compact. Putting everything back together we obtain

Φ^Sel,𝐿_{𝑘,𝛽,𝑚}(𝑍, 𝑠) = 2

∫︁ 1 0

∫︁ 1/2

−1/2

𝑣^𝑠+𝜅𝑒(𝑚𝜏)Θ_{𝐿,𝑘,𝛽}(𝜏, 𝑍)𝑑𝑢𝑑𝑣 𝑣² (5.1.5)

+ 2 CT_𝑡=0 [︃∫︁ ∞

1

∫︁ 1/2

−1/2

𝑣^{𝑠+𝜅−𝑡}𝑒(𝑚𝜏)Θ_{𝐿,𝑘,𝛽}(𝜏, 𝑍)𝑑𝑢𝑑𝑣 𝑣²

]︃

. In both terms the integral with respect to 𝑢 is simply given by

∫︁ 1/2

−1/2

𝑒(𝑚𝑢)Θ_{𝐿,𝑘,𝛽}(𝜏, 𝑍)𝑑𝑢.

Plugging in the definition of the 𝛽-th component of the theta function Θ𝐿,𝑘(𝜏, 𝑍) and using that

∑︁

𝜆∈𝐿+𝛽

∫︁ 1/2

−1/2

⃒

⃒

⃒𝑒(𝑚𝑢)(𝜆, 𝑍_𝐿)^𝑘𝑒(𝜏 𝑞(𝜆_𝑍) +𝜏 𝑞(𝜆_𝑍^⊥))

⃒

⃒

⃒𝑑𝑢= ∑︁

𝜆∈𝐿+𝛽

⃒⃒(𝜆, 𝑍_𝐿)^𝑘𝑒^{−2𝜋𝑣𝑞𝑍(𝜆)}⃒

⃒<∞ as the sum defining Θ_{𝐿,𝑘,𝛽}(𝜏, 𝑍)is absolutely convergent, we obtain

∫︁ 1/2

−1/2

𝑒(𝑚𝑢)Θ_{𝐿,𝑘,𝛽}(𝜏, 𝑍)𝑑𝑢 = 𝑣^𝑛/2 𝑞(Im(𝑍))^𝑘

∑︁

𝜆∈𝐿+𝛽

(𝜆, 𝑍_𝐿)^𝑘𝑒^{−2𝜋𝑣𝑞𝑍(𝜆)}

∫︁ 1/2

−1/2

𝑒(𝑢(𝑚−𝑞(𝜆))𝑑𝑢.

Here𝑞_𝑍(𝜆) = 𝑞(𝜆_𝑍)−𝑞(𝜆_𝑍^⊥) as in (4.1.3), and the integral on the right-hand side is1 if 𝑞(𝜆) = 𝑚 and vanishes otherwise. Therefore, we can write (5.1.5) as

Φ^Sel,𝐿_{𝑘,𝛽,𝑚}(𝑍, 𝑠) = 2 𝑞(Im(𝑍))^𝑘

(︃∫︁ 1 0

𝑣^{𝑠+𝑘−1} ∑︁

𝜆∈𝐿_𝛽,𝑚

(𝜆, 𝑍𝐿)^𝑘𝑒^{−4𝜋𝑣𝑞(𝜆𝑍)}𝑑𝑣

+ CT_𝑡=0 [︃∫︁ ∞

1

𝑣^{𝑠+𝑘−𝑡−1} ∑︁

𝜆∈𝐿_𝛽,𝑚

(𝜆, 𝑍_𝐿)^𝑘𝑒^{−4𝜋𝑣𝑞(𝜆𝑍)}𝑑𝑣 ]︃)︃

.

92

Here the contribution of the element 𝜆= 0 appearing only if 𝑚 =𝛽 = 0 is zero since

∫︁ 1 0

𝑣^{𝑠+𝑘−1}𝑑𝑣+ CT_𝑡=0

∫︁ ∞ 1

𝑣^{𝑠+𝑘−𝑡−1}𝑑𝑣= 1

𝑠+𝑘 −CT_𝑡=0

(︂ 1 𝑠+𝑘−𝑡

)︂

= 0.

Hence we can exclude the element 𝜆 = 0 in each of the two sums above. Moreover, we reunite the remaining terms, giving

Φ^Sel,𝐿_{𝑘,𝛽,𝑚}(𝑍, 𝑠) = 2

𝑞(Im(𝑍))^𝑘CT_𝑡=0 [︃∫︁ ∞

0

𝑣^{𝑠+𝑘−𝑡−1} ∑︁

𝜆∈𝐿_𝛽,𝑚 𝜆̸=0

(𝜆, 𝑍_𝐿)^𝑘𝑒^{−4𝜋𝑣𝑞(𝜆𝑍)}𝑑𝑣 ]︃

. (5.1.6)

Next we note that 𝑞(𝜆_𝑍)>0 and (𝜆, 𝑍_𝐿)̸= 0 for 𝜆∈𝐿_𝛽,𝑚 with 𝜆̸= 0 as𝑍 ∈H𝑛∖𝐻_𝛽,𝑚^𝐿 . Thus, using equation (4.1.2) we find that

∑︁

𝜆∈𝐿_𝛽,𝑚 𝜆̸=0

∫︁ ∞ 0

⃒

⃒

⃒𝑣^{𝑠+𝑘−1}(𝜆, 𝑍_𝐿)^𝑘𝑒^{−4𝜋𝑣𝑞(𝜆𝑍)}

⃒

⃒

⃒𝑑𝑣= Γ(Re(𝑠) +𝑘)𝑞(Im(𝑍))^𝑘/2 4^{Re(𝑠)+𝑘/2}𝜋^{Re(𝑠)+𝑘}

∑︁

𝜆∈𝐿_𝛽,𝑚 𝜆̸=0

𝑞(𝜆_𝑍)^{−Re(𝑠)−𝑘/2} (5.1.7)

forRe(𝑠)+𝑘 >0, and by Lemma 5.1.2 the sum on the right-hand side converges absolutely and locally uniformly for Re(𝑠)>1 +𝑛/2−𝑘/2. Therefore we can simply plug in 𝑡= 0 on the right-hand side of (5.1.6), interchange summation and integration and use again (4.1.2) to proof the statement given in part (a) of the theorem.

In order to also proof part (b) we need to modify the above proof of (a) as follows: As in equation (5.1.6) we obtain

Φ^Q,𝐿_{𝑘,𝛽,𝑚}(𝑍, 𝑠) = 2

𝑞(Im(𝑍))^𝑘CT_𝑡=0 [︃∫︁ ∞

0

𝑣^{𝑠+𝑘−𝑡−1} ∑︁

𝜆∈𝐿𝛽,𝑚

𝜆̸=0

(𝜆, 𝑍_𝐿)^𝑘𝑒^{−2𝜋𝑣𝑞𝑍(𝜆)}𝑑𝑣 ]︃

(5.1.8)

for 𝑍 ∈ H𝑛 (without the exclusion of 𝐻_𝛽,𝑚^𝐿 ) and 𝑠 ∈ C with Re(𝑠) >1 +𝑛/2, where in (5.1.6) we find the term 𝑒^{−4𝜋𝑣𝑞(𝜆𝑍)} = 𝑒^{−2𝜋𝑣𝑞𝑍(𝜆)}𝑒^{−2𝜋𝑣𝑚} instead of 𝑒^{−2𝜋𝑣𝑞𝑍(𝜆)} as in (5.1.8).

Here the missing factor, namely 𝑒^{−2𝜋𝑣𝑚}, is exactly the term by which the two building blocks 𝑣^𝑠𝑒(𝑚𝜏)e_𝛽 and 𝑣^𝑠𝑒(𝑚𝑢)e_𝛽 defining the Poincaré series𝑈_{𝜅,𝛽,𝑚}^𝐿 (𝜏, 𝑠)and𝑄^𝐿_{𝜅,𝛽,𝑚}(𝜏, 𝑠) differ. Now as in equation (5.1.7) we see that

∑︁

𝜆∈𝐿_𝛽,𝑚 𝜆̸=0

∫︁ ∞ 0

⃒

⃒

⃒𝑣^{𝑠+𝑘−1}(𝜆, 𝑍_𝐿)^𝑘𝑒^{−2𝜋𝑣𝑞𝑍(𝜆)}

⃒

⃒

⃒𝑑𝑣 (5.1.9)

= Γ(Re(𝑠) +𝑘)𝑞(Im(𝑍))^𝑘/2 2^Re(𝑠)𝜋^{Re(𝑠)+𝑘}

∑︁

𝜆∈𝐿_𝛽,𝑚 𝜆̸=0

𝑞(𝜆_𝑍)^𝑘/2𝑞_𝑍(𝜆)^{−Re(𝑠)−𝑘},

where we have used again the identity (4.1.2). As 𝑞(𝜆_𝑍)≤𝑞_𝑍(𝜆) = 𝑞(𝜆_𝑍) +|𝑞(𝜆_𝑍^⊥)|, and since the sum∑︀

𝜆∈𝐿^′∖{0}𝑞_𝑍(𝜆)^−𝑠 converges forRe(𝑠)>1 +𝑛/2as remarked in (5.1.2), we find that the right-hand side of (5.1.9) converges for Re(𝑠)> 1 +𝑛/2−𝑘/2. Therefore we can plug in 𝑡 = 0 on the right-hand side of (5.1.8), interchange summation and integration, and compute the remaining integral. This finishes the proof of part (b).

Remark 5.1.4.

(1) If𝛽 =−𝛽in𝐿^′/𝐿and2−𝑛−2𝜅≡2mod4then the Poincaré series𝑈_{𝜅,𝛽,𝑚}^𝐿 and𝑄^𝐿_{𝜅,𝛽,𝑚} vanish identically. This matches the two formulas given in the previous theorem since if 𝛽 = −𝛽 then 𝐿_𝛽,𝑚 = −𝐿_𝛽,𝑚 and thus the sums in (a) and (b) of Theorem 5.1.3 cancel completely if 𝑘 is odd as

𝑞((−𝜆)_𝑍) =𝑞(𝜆_𝑍), 𝑞_𝑍(−𝜆) =𝑞_𝑍(𝜆), (−𝜆, 𝑍_𝐿) = −(𝜆, 𝑍_𝐿)

for 𝜆∈𝑉 and 𝑍 ∈H𝑛. On the other hand, the congruence 2−𝑛−2𝜅≡2 mod4 is satisfied if and only if the corresponding non-negative integer𝑘(with𝜅= 1+𝑘−𝑛/2) is odd.

(2) If𝑚 = 0 then𝑈_𝜅,𝛽,0^𝐿 (𝜏, 𝑠) = 𝑄^𝐿_𝜅,𝛽,0(𝜏, 𝑠), and thus also the corresponding lifts need to agree, i.e., we have

Φ^Sel,𝐿_𝑘,𝛽,0(𝑍, 𝑠) = Φ^Q,𝐿_𝑘,𝛽,0(𝑍, 𝑠)

for 𝑍 ∈ H𝑛 and 𝑠 ∈ C with Re(𝑠) >1 +𝑛/2−𝑘/2. This agrees with the formulas given in Theorem 5.1.3 since

𝑞(𝜆𝑍) = 𝑞_𝑍(𝜆)

2 and (𝜆, 𝑍𝐿) = 2𝑞_𝑍(𝜆)𝑞(Im(𝑍)) (𝜆, 𝑍𝐿)

for 𝜆∈𝐿_𝛽,0 with 𝜆̸= 0 and 𝑍 ∈H𝑛. Here the first equality is clear as𝑞(𝜆) = 0, and the second equality follows from the identity in (4.1.2).

In the following two sections we now evaluate the above theta lifts for the special lattices of signature (2,1) and (2,2) from Section 4.3 and Section 4.4, respectively. Though we are mainly interested in the lift of Selberg’s Poincaré series in the case of signature(2,1), and in the lift of the Poincaré series 𝑄^𝐿_{𝜅,𝛽,𝑚}(𝜏, 𝑠) in the case of signature (2,2), we also present the opposite cases for the sake of completeness.

5.2 Averaged non-holomorphic Eisenstein series as

Im Dokument Realizing Hyperbolic and Elliptic Eisenstein Series as Regularized Theta Lifts (Seite 94-104)