# A Characterization of Planar Mixed Automorphic Forms

Authors: Allal Ghanmi
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

We characterize the functional space of the planar mixed automorphic forms with respect to an equivariant pair and given lattice as the image of the Landau automorphic forms (involving special multiplier)
by an appropriate isomorphic transform.


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## Body

## 1. Introduction

Mixed automorphic forms of type(ν,μ) arise naturally as holomorphic forms on elliptic varieties [1] and appear essentially in the context of number theory and algebraic geometry. Roughly speaking, they are a class of functions defined on a given (Hermitian symmetric) space X and satisfying a functional equation of type(1.1)F(γ⋅x)=jν(γ,x)jμ(ρ(γ),τ(x))F(x),
for every x∈X and g∈Γ. Here, jα(γ,x); α∈ℝ is an automorphic factor associated to an appropriate action of a group Γ on X, and (ρ,τ) is an equivariant pair for the data (Γ,X). Such notion was introduced by Stiller [2] and extensively studied by Lee in the case of X being the upper half-plane. They include the classical ones as a special case. Nontrivial examples of them have been constructed in [3, 4]. We refer to [5] for an exhaustive list of references.In this paper, we are interested in the space of mixed automorphic formsℳΓ,τν,μ(ℂ) defined on the complex plane X=ℂ with respect to a given lattice Γ in ℂ and an equivariant pair (ρ,τ). We find that ℳΓ,τν,μ(ℂ) is isomorphic to the space of Landau automorphic forms [6],(1.2)F(z+γ)=χτ(γ)jBτν,μ(γ,z)F(z);z∈C,γ∈Γ,
of “weight” (1.3)Bτν,μ(z)=ν+μ(|∂τ∂z|2-|∂τ∂z¯|2)=Bτν,μ,
with respect to a special pseudocharacter χτ defined on Γ and given explicitly through (5.3) below. The crucial point in the proof is to observe that the quantity Bτν,μ is in fact a real constant independent of the complex variable z.The exact statement of our main result (Theorem5.1) is given and proved in Section 5. In Sections 2 and 3, we establish some useful facts that we need to introduce the space of planar mixed automorphic forms ℳΓ,τν,μ(ℂ). We have to give necessary and sufficient condition to ensure the nontriviality of such functional space. In Section 4, we introduce properly the function φτν,μ that serves to define the pseudocharacter χτ.


## 2. Group Action

Let(2.1)G=T⋊C={g:=[a,b]≡(ab01);a∈T,b∈C}
be the semidirect product group of the unitary group T={λ∈ℂ;|λ|=1} and the additive group (ℂ,+). G acts on the complex plane ℂ by the holomorphic mappings g·z=az+b; g=[a,b], z∈ℂ, so that ℂ can be realized as Hermitian symmetric space ℂ=G/T.By aG-equivariant pair (ρ,τ), we mean that ρ is a G-endomorphism and τ:ℂ→ℂ is a compatible mapping, that is,(2.2)τ(g⋅z)=ρ(g)⋅τ(z);g∈G,z∈C.
Now, for given real numbers ν,μ, and an equivariant pair (ρ,τ), we define Jρ,τν,μ to be the complex-valued mapping(2.3)Jρ,τν,μ(g,z):=jν(g,z)jμ(ρ(g),τ(z));(g,z)∈G×C,
where jα; α∈ℝ is the “automorphic factor” given by(2.4)jα(g,z)=e2iαI〈z,g-1⋅0〉.
Here and elsewhere, ℑz denotes the imaginary part of the complex number z and 〈z,w〉=zw¯ the usual Hermitian scalar product on ℂ. Thus, one can check the following.Proposition 2.1.
The mappingJρ,τν,μ satisfies the chain rule
(2.5)Jρ,τν,μ(gg′,z)=e2iϕρν,μ(g,g′)Jρ,τν,μ(g,g′⋅z)Jρ,τν,μ(g′,z),
where ϕρν,μ(g,g′) is the real-valued function defined on G×G by
(2.6)ϕρν,μ(g,g′):=I(ν〈g-1⋅0,g′⋅0〉+μ〈ρ(g-1)⋅0,ρ(g′)⋅0〉).Proof.
For everyg,g′∈G and z∈ℂ, we have
(2.7)Jρ,τν,μ(gg′,z)=jν(gg′,z)jμ(ρ(gg′),τ(z))=jν(gg′,z)jμ(ρ(g)ρ(g′),τ(z)).
Next, one can see that the automorphic factor jα(·,·) satisfies
(2.8)jα(hh′,w)=e2iαI〈h-1⋅0,h′⋅0〉jα(h,h′⋅w)jα(h′,w),
for every h,h′∈G and w∈ℂ. This gives rise to
(2.9)Jρ,τν,μ(gg′,z)=e2iϕρν,μ(g,g′)jν(g,g′⋅z)jμ(ρ(g),ρ(g′)⋅τ(z))Jρ,τν,μ(g′,z).
Finally, (2.5) follows by making use of the equivariant condition ρ(g′)·τ(z)=τ(g′·z).Remark 2.2.
According to Proposition2.1 above, the unitary transformations
(2.10)[Tgν,μf](z):=Jρ,τν,μ(g,z)̅f(g⋅z)
for varying g∈G define then a projective representation of the group G on the space of 𝒞∞ functions on ℂ.


## 3. The Space of Planar Mixed Automorphic Forms

LetΓ be a uniform lattice of the additive group (ℂ,+) that can be seen as a discrete subgroup of G by the identification (3.1)γ∈Γ⟼[1,γ]=(1γ01),
so that the action of Γ on ℂ is the one induced from this of G, that is,(3.2)γ⋅z=z+γ.
Associated to such Γ and given fixed data of ν,μ>0 and (ρ,τ) as above, we perform ℳΓ,τν,μ(ℂ) the vector space of smooth complex-valued functions F on ℂ satisfying the functional equation(3.3)F(γ⋅z)=Jρ,τν,μ(γ,z)F(z)=jν(γ,z)jμ(ρ(γ),τ(z))F(z),
for every γ∈Γ and z∈ℂ.Definition 3.1.
The spaceℳΓ,τν,μ(ℂ) is called the space of planar mixed automorphic forms of biweight (ν,μ) with respect to the equivariant pair (ρ,τ) and the lattice Γ.We assert the following.Proposition 3.2.
The functional spaceℳΓ,τν,μ(ℂ) is nontrivial if and only if the real-valued function (1/π)ϕρν,μ in (2.6) takes integral values on Γ×Γ.Proof.
The proof can be handled in a similar way as in [6] making use of (2.5) combined with the equivariant condition (2.2). Indeed, assume that ℳΓ,τν,μ(ℂ) is nontrivial, and let F be a nonzero function belonging to ℳΓ,τν,μ(ℂ). According to (2.5), we get
(3.4)F((γ+γ′)⋅z)=(3.2)F((γ′⋅γ)⋅z)=(3.3)Jρ,τν,μ(γ′⋅γ,z)F(z)=(2.5)e2iϕρν,μ(γ′,γ)Jρ,τν,μ(γ′,γ⋅z)Jρ,τν,μ(γ,z)F(z),
for every γ,γ′∈Γ. On the other hand, we can write
(3.5)F((γ+γ′)⋅z)=F(z+γ+γ′)=F(γ′⋅[z+γ])=F(γ′⋅[γ⋅z])=(3.3)Jρ,τν,μ(γ′,γ⋅z)F(γ⋅z)=(3.3)Jρ,τν,μ(γ′,γ⋅z)Jρ,τν,μ(γ,z)F(z).
Now, by equating the right hand sides of (3.4) and (3.5), keeping in mind that F is not identically zero, we get necessarily
(3.6)e2iϕρν,μ(γ′,γ)=1,∀γ,γ′∈Γ.
Conversely, by classical analysis, we pick an arbitrary nonzero𝒞∞ and compactly supported function ψ with support contained in a fundamental domain Λ(Γ) of the lattice Γ. Next, we consider the associated Poincaré series 𝒫Γν,μψ given by
(3.7)[PΓν,μψ](z)=∑γ∈ΓJρ,τν,μ(γ,z)¯ψ(γ⋅z).
Then, it can be shown that the function 𝒫Γν,μψ is 𝒞∞ and a nonzero function on ℂ for Γ being discrete and Suppψ⊂Λ(Γ). Indeed, for every z∈Suppψ, we have
(3.8)[PΓν,μψ](z)=ψ(z).
Furthermore, under the condition that (1/π)ϕρν,μ takes integral values on Γ×Γ, we see that 𝒫Γν,μψ belongs to ℳΓ,τν,μ(ℂ). In fact, for every γ∈Γ and z∈ℂn, we have
(3.9)[PΓν,μψ](γ⋅z)=∑k∈ΓJρ,τν,μ(k,γ⋅z)¯ψ(k⋅(γ⋅z))=∑k∈ΓJρ,τν,μ(k,γ⋅z)¯ψ((k+γ)⋅z)=h=γ+k∑h∈ΓJρ,τν,μ(h⋅γ-1,γ⋅z)¯ψ(h⋅z)=(2.5)∑h∈Γe-2iϕρν,μ(h,γ-1)Jρ,τν,μ(h,γ-1⋅(γ⋅z))¯Jρ,τν,μ(γ-1,γ⋅z)¯ψ(h⋅z).
Finally, since e2iϕρν,μ(γ,h)=1 (by hypothesis on ϕρν,μ), it follows that
(3.10)[PΓν,μψ](γ⋅z)=∑h∈ΓJρ,τν,μ(h,z)¯Jρ,τν,μ(γ-1,γ⋅z)¯ψ(h⋅z)=Jρ,τν,μ(γ-1,γ⋅z)¯∑h∈ΓJρ,τν,μ(h,z)¯ψ(h⋅z)=Jρ,τν,μ(γ,z)[PΓν,μψ](z).
The last equality, that is,
(3.11)Jρ,τν,μ(γ-1,γ⋅z)¯=Jρ,τν,μ(γ,z)
holds for every γ∈Γ using the chain rule (2.5) and taking into account the assumption made on ϕρν,μ. This completes the proof.Remark 3.3.
The condition involved in Proposition3.2 ensures that ℳΓ,τν,μ(ℂ) can be realized as the space of cross-sections on a line bundle over the complex torus ℂ/Γ.


## 4. On the Functionφτν,μ

In order to prove the main result of this paper, we need to introduce the functionφτν,μ.Proposition 4.1.
The first-order differential equation(4.1)∂φτν,μ̃∂z¯=-iμ((τ∂τ¯∂z¯-τ¯∂τ∂z¯)-(|∂τ∂z|2-|∂τ∂z¯|2)z)
admits a solution φτν,μ̃:ℂ→ℂ such that ℑφτν,μ̃ is constant.Proof.
By writing theG-endomorphism ρ:G→G=T⋊ℂ as ρ(g)=[χ(g),ψ(g)], and differentiating the equivariant condition
(4.2)τ(g⋅z)=ρ(g)⋅τ(z)=χ(g)τ(z)+ψ(g),
it follows that
(4.3)(∂g⋅z∂z)∂τ∂z(g⋅z)=χ(g)∂τ∂z(z),(∂g⋅z¯∂z¯)∂τ∂z¯(g⋅z)=χ(g)∂τ∂z¯(z).
Hence, for (∂g·z)/(∂z) and χ(g) being in T, we deduce that
(4.4)Bτν,μ(g⋅z)=ν+μ(|∂τ∂z(g⋅z)|2-|∂τ∂z¯(g⋅z)|2)=ν+μ(|∂τ∂z(z)|2-|∂τ∂z¯(z)|2)=Bτν,μ(z).
Therefore, z↦Bτν,μ(z) is a real-valued constant function (since the only G-invariant functions on ℂ are the constants). Now, by considering the differential 1-form
(4.5)θτν,μ(z):=i{ν(z¯dz-zdz¯)+μ(τ¯dτ-τdτ¯)},
one checks that dθτν,μ=dθBτν,μ, where θBτν,μ:=iBτν,μ(z¯dz-zdz¯). Therefore, there exists a function φτν,μ̃:ℂ→ℂ such that ℑφτν,μ̃=Constant and satisfying the first-order partial differential equation
(4.6)∂φτν,μ̃∂z¯=-i([ν-Bτν,μ]z+μ(τ∂τ¯∂z¯-τ¯∂τ∂z¯))=-iμ((τ∂τ¯∂z¯-τ¯∂τ∂z¯)-(|∂τ∂z|2-|∂τ∂z¯|2)z).
This completes the proof.Remark 4.2.
The partial differential equation (4.6) satisfied by φτν,μ̃ can be reduced further to the following:
(4.7)∂ψτν,μ∂z¯=τ¯∂τ∂z¯,∂ψτν,μ∂z=1μ(ν-Bτν,μ)z¯+τ¯∂τ∂z,
with φτν,μ̃(z)=i([Bτν,μ-ν]|z|2-μ|τ(z)|2)+2iμψτν,μ(z).


## 5. Main Result

Letφτν,μ be the real part of φτν,μ̃-φτν,μ̃(0), where φτν,μ̃ is a complex-valued function on ℂ as in Proposition 4.1. Define 𝒲τν,μ to be the special transformation given by(5.1)[Wτν,μ(f)](z):=eiφτν,μ(z)f(z).
We have the following.Theorem 5.1.
The image ofℳΓ,τν,μ(ℂ) by the transform (5.1) is the space of Landau (Γ,χτ)-automorphic functions. More exactly, one has
(5.2)Wτν,μ(MΓ,τν,μ(C))={F;C∞,F(z+γ)=χτ(γ)jBτν,μ(γ,z)F(z)},
with Bτν,μ=ν+μ(|∂τ/∂z|2-|∂τ/∂z¯|2)∈ℝ and χτ is the pseudocharacter defined on Γ by
(5.3)χτ(γ)=exp⁡⁡(2iφτν,μ(γ)-2iμI〈τ(0),ρ(γ)-1⋅0〉).For the proof, we begin with the following.Lemma 5.2.
The functionχτ̂ defined on ℂ×Γ by
(5.4)χτ̂(z;γ):=ei(φτν,μ(z+γ)-φτν,μ(z))e-2i([Bτν,μ-ν]I〈z,γ〉+μI〈τ(z),ρ(γ)-1⋅0〉)
is independent of the variable z.Proof.
Differentiation ofχτ̂(z;γ) with respect to the variable z gives
(5.5)∂χτ̂∂z=i(∂φτν,μ∂z(z+γ)-∂φτν,μ∂z(z))χτ̂-([Bτν,μ-ν]γ¯+μ[(ρ(γ-1)⋅0)¯∂τ∂z(z)-(ρ(γ-1)⋅0)∂τ¯∂z(z)])χτ̂.
On the other hand, using the equivariant condition τ(z+γ)=ρ(γ)·τ(z) and (4.6), one gets
(5.6)i(∂φτν,μ∂z(z+γ)-∂φτν,μ∂z(z))=Bτν,μγ¯+Sτν,μ¯(z)-Sτν,μ¯(z+γ)=[Bτν,μ-ν]γ¯+μ[aγ¯∂τ∂z(z)-aγ∂τ¯∂z(z)],
where we have set aγ=ρ(γ-1)·0. Thus, from (5.5) and (5.6), we conclude that ∂χτ̂/∂z=0. Similarly, one gets also ∂χτ̂/∂z¯=0. This ends the proof of Lemma 5.2.Proof of Theorem5.1.
We have to prove that𝒲τν,μF belongs to
(5.7)FΓ,χτBτν,μ:={F:C⟶C∞C;F(z+γ)=χτ(γ)jBτν,μ(γ,z)F(z)},
whenever F∈ℳΓ,τν,μ(ℂ), where
(5.8)χτ(γ):=exp⁡(2iφτν,μ(γ)-2iμI〈τ(0),ρ(γ)-1⋅0〉).
Indeed, we have
(5.9)[Wτν,μF](z+γ)≔eiφτν,μ(z+γ)F(z+γ)=eiφτν,μ(z+γ)jν(γ,z)jμ(ρ(γ),τ(z))F(z)=ei(φτν,μ(z+γ)-φτν,μ(z))jν(γ,z)jμ(ρ(γ),τ(z))[Wτν,μF](z)=χτ̂(z;γ)j-Bτν,μ(γ,z)[Wτν,μF](z).
Whence by Lemma 5.2, we see that χτ̂(z;γ)=χτ̂(0;γ)=:χτ(γ), and therefore
(5.10)[Wτν,μF](z+γ)=χτ(γ)j-Bτν,μ(γ,z)[Wτν,μF](z).
The proof is complete.Corollary 5.3.
The functionχτ(γ)=exp(2iφτν,μ(γ)-2iμℑ〈τ(0),ρ(γ)-1·0〉) satisfies the following pseudocharacter property:
(5.11)χτ(γ+γ′)=e2iBτν,μI〈γ,γ′〉χτ(γ)χτ(γ′)
if and only if ϕρν,μ in (2.6) takes its values in πℤ on Γ×Γ.



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