Non-holomorphic Eisenstein series of weight 0

[Fal11] if 𝑐 is infinite for the case of weight 2, which easily generalizes to the present case of weight 0. So 𝐸_𝑐^hyp(𝜏, 𝑠) defines a real analytic function in 𝜏, and a holomorphic function in𝑠 forRe(𝑠)>1. Moreover, the hyperbolic Eisenstein series is by construction modular of level 𝑁 and weight 0 in𝜏, and it satisfies the differential equation

Δ₀𝐸_𝑐^hyp(𝜏, 𝑠) = 𝑠(1−𝑠)𝐸_𝑐^hyp(𝜏, 𝑠) +𝑠²𝐸_𝑐^hyp(𝜏, 𝑠+ 2).

(2.6.4)

In contrast to the differential equation of the parabolic Eisenstein series given in (2.6.2), we discover the additional shifted term 𝑠²𝐸_𝑐^hyp(𝜏, 𝑠+ 2). In particular, the hyperbolic Eisenstein series is not an eigenfunction of the hyperbolic Laplace operator Δ₀.

Furthermore, if 𝑐 is a closed geodesic it has been shown in [JKP10] using spectral theory that the hyperbolic Eisenstein series 𝐸_𝑐^hyp(𝜏, 𝑠) has a meromorphic continuation in 𝑠 to the whole complex plane, and that this continuation has a double zero at the distinguished point 𝑠 = 0. Therefore, if 𝑐 is closed the associated hyperbolic Eisenstein series has the Laurent expansion

𝐸_𝑐^hyp(𝜏, 𝑠) = 𝑂(𝑠²) (2.6.5)

at𝑠= 0. However, if the given geodesic 𝑐is infinite, [Fal11] only proves the meromorphic continuation of the corresponding hyperbolic Eisenstein series to the half-plane given by Re(𝑠)>1/2. Indeed, the hyperbolic Eisenstein series associated to some infinite geodesic is not square-integrable, which is why the techniques from [JKP10] cannot be directly applied in this case.

2.6.2 Elliptic Eisenstein series

In 2004 Jorgenson and Kramer investigated elliptic analogs of the above hyperbolic Eisen-stein series in their unpublished work [JK04] (see also [JK11]). These are non-holomorphic Eisenstein series, which are associated to elliptic (and more general arbitrary) points in the upper half-plane, instead of being associated to geodesics or cusps. Elliptic Eisen-stein series were later studied in detail for arbitrary Fuchsian groups of the first kind by Kramer’s student von Pippich in [Pip10] and [Pip16].

Definition 2.6.2. Given 𝜔 ∈ H, we define the elliptic Eisenstein series of level 𝑁 associated to 𝜔 by

𝐸_𝜔^ell(𝜏, 𝑠) = ∑︁

𝑀∈(Γ0(𝑁))𝜔∖Γ0(𝑁)

sinh(𝑑_hyp(𝑀 𝜏, 𝜔))^−𝑠

for 𝜏 ∈H∖Γ₀(𝑁)𝜔 and 𝑠 ∈Cwith Re(𝑠)>1.

The sum defining the elliptic Eisenstein series converges absolutely and locally uni-formly in 𝜏 and 𝑠, and thus defines a real analytic function in 𝜏 and a holomorphic function in 𝑠 for Re(𝑠) >1. We note that it is indeed not necessary to assume that the point 𝜔 ∈ H is elliptic. Moreover, the elliptic Eisenstein series satisfies the differential equation

Δ₀𝐸_𝜔^ell(𝜏, 𝑠) =𝑠(1−𝑠)𝐸_𝜔^ell(𝜏, 𝑠)−𝑠²𝐸_𝜔^ell(𝜏, 𝑠+ 2), (2.6.6)

which agrees with the differential equation of the hyperbolic Eisenstein series given in (2.6.4) up to a sign, and the elliptic Eisenstein series also has a meromorphic continuation in𝑠 to the whole complex plane which was proven in [Pip10]. Furthermore, von Pippich shows in [Pip16] that the elliptic Eisenstein series admits a Kronecker limit type formula as in (2.6.3), which for 𝑁 = 1 and 𝜔=𝑖 or𝜔 =𝜌=𝑒^𝜋𝑖/3 is given by

𝐸_𝜔^ell(𝜏, 𝑠) =−log(︀

|𝑗(𝜏)−𝑗(𝜔)|^2/ord(𝜔))︀

·𝑠+𝑂(𝑠²) (2.6.7)

as 𝑠 → 0. Here 𝑗(𝜏) := 𝐸4,∞(𝜏)³/Δ(𝜏) is the well-known modular 𝑗-function, which is the unique weakly holomorphic modular form of weight 0 and level 1 whose Fourier expansion at ∞ is of the form 𝑗(𝜏) =𝑒(−𝜏) + 744 +𝑂(𝑒(𝜏)).

2.6.3 Non-holomorphic Zagier cusp forms of weight 0

Using the parabolic, hyperbolic and elliptic Eisenstein series defined above, we may now restate Proposition 2.5.6 in the case of weight0 in a simpler way:

Corollary 2.6.3. Let 𝑄∈ 𝒬with 𝑄̸≡0. Then

𝑓_0,𝑄(𝜏, 𝑠) =

⎧

⎪⎪

⎪⎨

⎪⎪

⎪⎩

Δ(𝑄)^−𝑠/2𝐸_𝑐^hyp

𝑄 (𝜏, 𝑠), if Δ(𝑄)>0,

|𝜆_𝑄|^−𝑠𝐸_𝑝^par

𝑄(𝜏, 𝑠), if Δ(𝑄) = 0,

|Δ(𝑄)|^−𝑠/2𝐸_𝜏^ell_𝑄(𝜏, 𝑠), if Δ(𝑄)<0, for 𝜏 ∈H∖𝐻_𝑄 and 𝑠∈C with Re(𝑠)>1.

Thus we may understand 𝑓_0,𝑄(𝜏, 𝑠) as a general type of non-holomorphic Eisenstein series of weight 0. This is clear in the parabolic case. Moreover, in the hyperbolic case this identification can for example be found in [Fal07]: Comparing our equation (2.5.11) with the first line of the proof of Proposition 1.1 in [Fal07], we find that for𝑄∈ 𝒬 with Δ(𝑄)>0 the functions 𝑓_1,𝑄(𝜏, 𝑠) of weight 2 defined in this work can indeed be seen as scalar valued analogs of the form valued hyperbolic Eisenstein series defined in [Fal07].

We further remark that by Corollary 2.3.2 every hyperbolic Eisensteins series (in the sense of Definition 2.6.1) can be written in the form 𝐸_𝑐^hyp(𝜏, 𝑠) = Δ(𝑄)^𝑠/2𝑓_0,𝑄(𝜏, 𝑠) for some quadratic form 𝑄. In particular, the convergence of the sum defining the non-holomorphic functions 𝑓_0,𝑄(𝜏, 𝑠) (see Definition 2.5.2) also implies the convergence of the sum defining the hyperbolic Eisenstein series 𝐸_𝑐^hyp_𝑄 (𝜏, 𝑠), independent of whether the geodesic 𝑐_𝑄 is closed or infinite. On the other hand, not every elliptic Eisenstein series can be written in the form 𝐸_𝜏^ell

𝑄(𝜏, 𝑠) = Δ(𝑄)^𝑠/2𝑓_0,𝑄(𝜏, 𝑠), since not every point in the upper half-plane is a Heegner point associated to some integral binary quadratic form.

At this point we also want to mention the work [IS09], where hyperbolic, parabolic and elliptic Poincaré series are recalled and studied. These are meromorphic modular forms which for parameter 𝑚 = 0 become Eisenstein series, and they essentially agree with the meromorphic modular forms 𝑓𝑘,𝑄(𝜏)associated to quadratic forms defined in Section 2.4.2 of the present work, where the sign of the discriminant Δ(𝑄) determines whether the form 𝑓_𝑘,𝑄(𝜏) corresponds to a hyperbolic, parabolic or elliptic Poincaré series in the sense of [IS09] (see for example Proposition 9 and 10 in [IS09] for the hyperbolic case).

38

Therefore, the hyperbolic, parabolic and elliptic Eisenstein series of weight 0 defined in this section can be seen as non-holomorphic weight0analogs of the mentioned hyperbolic, parabolic and elliptic Poincaré series for parameter 𝑚= 0.

Finally, we want to write the functions 𝑓_0,𝛽,Δ(𝜏, 𝑠) introduced in Definition 2.5.2 as averaged versions of non-holomorphic Eisenstein series of weight 0:

Corollary 2.6.4. Let 𝛽 ∈Z/2𝑁Z and Δ∈Z with Δ≡𝛽² mod 4𝑁. Then

𝑓_0,𝛽,Δ(𝜏, 𝑠) =

⎧

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎪

⎪⎪

⎩

Δ^−𝑠/2 ∑︁

𝑄∈𝒬_𝛽,Δ/Γ0(𝑁)

𝐸_𝑐^hyp_𝑄 (𝜏, 𝑠), if Δ>0,

∑︁

𝑝∈𝐶(Γ₀(𝑁))

𝜆_𝛽,𝑝(𝑠)𝐸_𝑝^par(𝜏, 𝑠), if Δ = 0,

|Δ|^−𝑠/2 ∑︁

𝑄∈𝒬𝛽,Δ/Γ0(𝑁)

𝐸_𝜏^ell

𝑄(𝜏, 𝑠), if Δ<0, (2.6.8)

for 𝜏 ∈H∖𝐻_𝛽,Δ and 𝑠∈C with Re(𝑠)>1. Here 𝜆𝛽,𝑝(𝑠) := ∑︁

𝑄∈(𝒬_𝛽,0∖{0})/Γ0(𝑁) 𝑝𝑄=𝑝

|𝜆𝑄|^−𝑠

for 𝑝∈𝐶(Γ0(𝑁)) and 𝑠 ∈C with Re(𝑠)>1.

Proof. The casesΔ>0and Δ<0follow directly from the identity (2.5.7) and Corollary (2.6.3). For the case Δ = 0 we additionally note that every 𝑄 ∈ 𝒬_𝛽,0 with 𝑄 ̸≡ 0 corresponds to a cusp𝑝_𝑄.

We note that the three sums given in equation (2.6.8) are all finite, running over the finitely many cusps of the modular curve 𝑋₀(𝑁) if Δ = 0, or over the finitely many Heegner geodesics or Heegner points of class𝛽 and discriminantΔ>0orΔ<0modulo Γ0(𝑁), respectively.

IfΔ = 0the sum defining the factor𝜆_𝛽,𝑝(𝑠)given in Corollary 2.6.4 is either trivial (in case there is no𝑄∈ 𝒬_𝛽,0 with𝑝_𝑄 =𝑝) or infinite. Assuming that the level𝑁 is squarefree 𝛽 = 0 is the only class allowing Δ = 0, and by Lemma 2.3.4 we have 𝜆0,𝑝(𝑠) = 2𝜁(𝑠) independent of the cusp 𝑝, giving

𝑓_0,0,0(𝜏, 𝑠) = 2𝜁(𝑠) ∑︁

𝑝∈𝐶(Γ₀(𝑁))

𝐸_𝑝^par(𝜏, 𝑠) (2.6.9)

for 𝜏 ∈H and 𝑠 ∈C with Re(𝑠)>1. This is a special case of Corollary 2.5.7.

2.6.4 The hyperbolic kernel function

We have seen in the Corollaries 2.6.3 and 2.6.4 that hyperbolic, parabolic and elliptic Eisenstein series are somehow deeply connected, even though their ad hoc definitions look quite different. We further support this idea by introducing the so-called hyperbolic kernel function for the group Γ0(𝑁).

Definition 2.6.5. The hyperbolic kernel function of level 𝑁 is defined by 𝐾(𝜏, 𝜔, 𝑠) = ∑︁

𝑀∈Γ0(𝑁)

cosh(𝑑hyp(𝑀 𝜏, 𝜔))^−𝑠 for 𝜏, 𝜔 ∈H and 𝑠∈C with Re(𝑠)>1.

The series converges absolutely and locally uniformly for Re(𝑠) > 1. Thus 𝐾(𝜏, 𝜔, 𝑠) is a real analytic function in the variables 𝜏 and 𝜔, and holomorphic in 𝑠 for Re(𝑠)>1.

Further, it is by definition modular of level 𝑁 and weight 0 in both variables 𝜏 and 𝜔, and it satisfies the differential equation

Δ₀𝐾(𝜏, 𝜔, 𝑠) =𝑠(1−𝑠)𝐾(𝜏, 𝜔, 𝑠) +𝑠(𝑠+ 1)𝐾(𝜏, 𝜔, 𝑠+ 2).

(2.6.10)

Also, the hyperbolic kernel function has a meromorphic continuation in𝑠 to all ofC (see for example Proposition 5.1.4 in [Pip10]).

The following proposition shows that the hyperbolic, parabolic and elliptic Eisenstein series can indeed all be expressed in terms of the hyperbolic kernel function 𝐾(𝜏, 𝜔, 𝑠).

However, in the hyperbolic case this relation only holds if the corresponding geodesic is closed.

Proposition 2.6.6.

(a) Let 𝑐 be a closed geodesic in H. Then 𝐸_𝑐^hyp(𝜏, 𝑠) = Γ(𝑠)

2^𝑠−1Γ(𝑠/2)²

∫︁

[𝑐]

𝐾(𝜏, 𝜔, 𝑠)𝑑𝑠(𝜔)

for 𝜏 ∈H and 𝑠∈C with Re(𝑠)>1. Here Γ(𝑧) denotes the usual Gamma function.

(b) Let 𝑝 be a cusp and let 𝑦 >1. Then 𝐸_𝑝^par(𝜏, 𝑠) = 2^𝑠(2𝑠−1)Γ(𝑠)²

4𝜋Γ(2𝑠) 𝑦^𝑠−1

∞

∑︁

𝑛=0

(𝑠/2)_𝑛(𝑠/2 + 1/2)_𝑛 𝑛! (𝑠+ 1/2)_𝑛

∫︁ 1 0

𝐾(𝜏, 𝜎_𝑝(𝑥+𝑖𝑦), 𝑠+2𝑛)𝑑𝑥 for 𝜏 ∈ H with Im(𝑀 𝜏) < 𝑦 for all 𝑀 ∈ Γ₀(𝑁) and 𝑠 ∈ C with Re(𝑠) > 1. Here 𝜎𝑝 ∈SL2(R)is a scaling matrix for the cusp𝑝, and the right-hand side of the equation is independent of 𝑦. Moreover, Γ(𝑧)denotes the Gamma function as in part (a), and

(𝑧)_𝑛 := Γ(𝑧+𝑛)

Γ(𝑧) =𝑧(𝑧+ 1)· · ·(𝑧+𝑛−1) (2.6.11)

denotes the Pochhammer symbol defined for 𝑧 ∈C and 𝑛 ∈N0. (c) Let 𝜔 ∈H. Then

𝐸_𝜔^ell(𝜏, 𝑠) = 1 ord(𝜔)

∞

∑︁

𝑛=0

(𝑠/2)𝑛

𝑛! 𝐾(𝜏, 𝜔, 𝑠+ 2𝑛)

for 𝜏 ∈H∖Γ₀(𝑁)𝜔 and 𝑠 ∈ C with Re(𝑠)>1. Here (𝑧)_𝑛 denotes the Pochhammer symbol as in part (b).

40

Proof. Part (a) is given as Proposition 11 in [JPS16], and part (c) as Remark 3.3.9 in [Pip10]. We thus only comment on part (b). By the well-known relation between the parabolic Eisenstein series and the hyperbolic Green’s function we have

𝐸_𝑝^par(𝜏, 𝑠) = (2𝑠−1)𝑦^𝑠−1

∫︁ 1 0

𝐺_𝑠(𝜏, 𝜎_𝑝(𝑥+𝑖𝑦))𝑑𝑥, (2.6.12)

for 𝜏 ∈ H, 𝑦 ∈R with 𝑦 > Im(𝑀 𝜏) for all 𝑀 ∈ Γ₀(𝑁) and 𝑠 ∈C with Re(𝑠)>1. This identity can for example be found as Proposition 24 in [JPS16]. Here we also recall that the hyperbolic Green’s function of level𝑁 is given by

𝐺_𝑠(𝜏, 𝜔) = 1 2𝜋

∑︁

𝑀∈Γ₀(𝑁)

𝑄𝑠−1(cosh(𝑑_hyp(𝑀 𝜏, 𝜔)))

for 𝜏, 𝜔 ∈H with 𝜏 ̸≡𝜔 modulo Γ₀(𝑁) and 𝑠∈C with Re(𝑠)>1, where 𝑄_𝜈(𝑧) =𝑄⁰_𝜈(𝑧) is the associated Legendre function of the second kind (see for example [GR07], Section 8.7). By Lemma 7.1 in [Pip16] there is also a relation between the hyperbolic Green’s function and the hyperbolic kernel function𝐾(𝜏, 𝜔, 𝑠), namely

𝐺𝑠(𝜏, 𝜔) = 2^𝑠Γ(𝑠)² 4𝜋Γ(2𝑠)

∞

∑︁

𝑛=0

(𝑠/2)𝑛(𝑠/2 + 1/2)𝑛

𝑛!(𝑠+ 1/2)_𝑛 𝐾(𝜏, 𝜔, 𝑠+ 2𝑛) (2.6.13)

for 𝜏, 𝜔 ∈ H with 𝜏 ̸≡𝜔 modulo Γ₀(𝑁) and 𝑠 ∈C with Re(𝑠) >1. Combining (2.6.12) and (2.6.13), and interchanging summation and integration we obtain the identity claimed in part (b) of the proposition.

3 Vector valued modular forms

In the present chapter we introduce holomorphic and non-holomorphic vector valued modular forms for the Weil representation. In particular, we define different types of non-holomorphic Poincaré series in Section 3.6, whose theta lifts we study in the following chapters. Moreover, we present a translation of the spectral theory of automorphic forms given in [Roe66, Roe67] to the setting of vector valued modular forms in Section 3.7.

Our main references for this chapter are [Bru02] and [BF04], as well as the mentioned work of Roelcke.

3.1 The metaplectic group

The set of pairs Mp₂(R) :={︀

(𝑀, 𝜑) :𝑀 ∈SL₂(R), 𝜑: H→C holomorphic with 𝜑(𝜏)² =𝑗(𝑀, 𝜏)}︀

together with the operation (𝑀, 𝜑) ∘(𝑀^′, 𝜑^′) = (𝑀 𝑀^′, 𝜑(𝑀^′𝜏)𝜑^′(𝜏)) is the so-called metaplectic group, which is a double cover of SL₂(R). For 𝑀 ∈SL₂(R) we write

𝑀˜ := (𝑀, 𝜑_𝑀)

for the corresponding element in Mp₂(R) where 𝜑𝑀(𝜏) :=√

𝑐𝜏 +𝑑 for 𝑀 =(︀_{𝑎 𝑏}

𝑐 𝑑

)︀. Here 𝑧 ↦→√

𝑧 denotes the usual choice of the complex square root, i.e.,arg(√

𝑧)∈(−𝜋/2, 𝜋/2].

We further denote the inverse image of the modular group SL₂(Z) under the covering map (𝑀, 𝜑)↦→𝑀 by Mp₂(Z). One can check that it is generated by the two elements

𝑇˜:= (𝑇,1) and 𝑆˜:= (𝑆,√ 𝜏), where 𝑇 =(︀_{1 1}

0 1

)︀ and 𝑆 =(︀₀₋₁

1 0

)︀ are the usual generators of SL2(Z).

Im Dokument Realizing Hyperbolic and Elliptic Eisenstein Series as Regularized Theta Lifts (Seite 46-53)