Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 239807, 9 pages
doi:10.1155/2011/239807
Research Article
A Characterization of Planar Mixed
Automorphic Forms
Allal Ghanmi
Department of Mathematics, Faculty of Sciences, Mohammed V University, P.O. Box 1014,
Agdal, Rabat 10000, Morocco
Correspondence should be addressed to Allal Ghanmi, allalghanmi@gmail.com
Received 29 December 2010; Accepted 5 April 2011
Academic Editor: Charles E. Chidume
Copyright q 2011 Allal Ghanmi. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
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.
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
F γ · x j ν γ, x j μ ρ γ , τx Fx,
1.1
for every x ∈ X and g ∈ Γ. Here, j α γ, x; α ∈ 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.
2
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In this paper, we are interested in the space of mixed automorphic forms MΓ,τ defined on the complex plane X with respect to a given lattice Γ in and an equivariant
ν,μ
pair ρ, τ. We find that MΓ,τ is isomorphic to the space of Landau automorphic forms 6,
ν,μ
ν,μ F z γ χτ γ j Bτ γ, z Fz;
z ∈ , γ ∈ Γ,
1.2
of “weight”
ν,μ
Bτ z
2 2 ∂τ ∂τ ν,μ
Bτ ,
ν μ − ∂z
∂z
1.3
with respect to a special pseudocharacter χτ defined on Γ and given explicitly through 5.3
ν,μ
below. The crucial point in the proof is to observe that the quantity Bτ is in fact a real constant
independent of the complex variable z.
The exact statement of our main result Theorem 5.1 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
ν,μ
planar mixed automorphic forms MΓ,τ . We have to give necessary and sufficient condition
to ensure the nontriviality of such functional space. In Section 4, we introduce properly the
ν,μ
function ϕτ that serves to define the pseudocharacter χτ .
2. Group Action
Let
G T g : a, b ≡
a b
0 1
; a ∈ T, b ∈ 2.1
be the semidirect product group of the unitary group T {λ ∈ ; |λ| 1} and the additive
group , . G acts on the complex plane by the holomorphic mappings g · z az b;
g a, b, z ∈ , so that can be realized as Hermitian symmetric space G/T.
By a G-equivariant pair ρ, τ, we mean that ρ is a G-endomorphism and τ : → is
a compatible mapping, that is,
τ g · z ρ g · τz;
g ∈ G, z ∈ .
2.2
ν,μ
Now, for given real numbers ν, μ, and an equivariant pair ρ, τ, we define Jρ,τ to be the
complex-valued mapping
ν,μ Jρ,τ g, z : j ν g, z j μ ρ g , τz ;
where j α ; α ∈
g, z ∈ G × ,
2.3
is the “automorphic factor” given by
−1
j α g, z e2iαz,g ·0 .
2.4
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3
Here and elsewhere, z denotes the imaginary part of the complex number z and z, w zw
the usual Hermitian scalar product on . Thus, one can check the following.
ν,μ
Proposition 2.1. The mapping Jρ,τ satisfies the chain rule
ν,μ
ν,μ
ν,μ ν,μ Jρ,τ gg , z e2iφρ g,g Jρ,τ g, g · z Jρ,τ g , z ,
2.5
ν,μ
where φρ g, g is the real-valued function defined on G × G by
g, g : ν g −1 · 0, g · 0 μ ρ g −1 · 0, ρ g · 0 .
ν,μ φρ
2.6
Proof. For every g, g ∈ G and z ∈ , we have
ν,μ Jρ,τ gg , z j ν gg , z j μ ρ gg , τz
j ν gg , z j μ ρ g ρ g , τz .
2.7
Next, one can see that the automorphic factor j α ·, · satisfies
−1
j α hh , w e2iαh ·0,h ·0 j α h, h · w j α h , w ,
2.8
for every h, h ∈ G and w ∈ . This gives rise to
ν,μ
ν,μ
ν,μ Jρ,τ gg , z e2iφρ g,g j ν g, g · z j μ ρ g , ρ g · τz Jρ,τ g , z .
2.9
Finally, 2.5 follows by making use of the equivariant condition ρg · τz τg · z.
Remark 2.2. According to Proposition 2.1 above, the unitary transformations
ν,μ
ν,μ Tg f z : Jρ,τ g, z f g · z
2.10
for varying g ∈ G define then a projective representation of the group G on the space of C∞
functions on .
3. The Space of Planar Mixed Automorphic Forms
Let Γ be a uniform lattice of the additive group , that can be seen as a discrete subgroup
of G by the identification
γ ∈ Γ −→ 1, γ 1 γ
0 1
,
3.1
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so that the action of Γ on is the one induced from this of G, that is,
γ · z z γ.
3.2
Associated to such Γ and given fixed data of ν, μ > 0 and ρ, τ as above, we perform MΓ,τ the vector space of smooth complex-valued functions F on satisfying the functional
equation
ν,μ
ν,μ F γ · z Jρ,τ γ, z Fz j ν γ, z j μ ρ γ , τz Fz,
3.3
for every γ ∈ Γ and z ∈ .
Definition 3.1. The space MΓ,τ 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.
ν,μ
We assert the following.
Proposition 3.2. The functional space MΓ,τ is nontrivial if and only if the real-valued function
ν,μ
1/πφρ in 2.6 takes integral values on Γ × Γ.
ν,μ
Proof. The proof can be handled in a similar way as in 6 making use of 2.5 combined
ν,μ
with the equivariant condition 2.2. Indeed, assume that MΓ,τ is nontrivial, and let F be
ν,μ
a nonzero function belonging to MΓ,τ . According to 2.5, we get
F
3.3 ν,μ 3.2 γ γ · z F γ · γ · z Jρ,τ γ · γ, z Fz
2.5
ν,μ ν,μ e2iφρ γ ,γ Jρ,τ γ , γ
ν,μ · z Jρ,τ γ, z Fz,
3.4
for every γ, γ ∈ Γ. On the other hand, we can write
F
γ γ · z F z γ γ F γ · z γ F γ · γ · z
ν,μ Jρ,τ γ , γ · z F γ · z
3.3
3.5
ν,μ ν,μ Jρ,τ γ , γ · z Jρ,τ γ, z Fz.
3.3
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
ν,μ
e2iφρ
γ ,γ 1,
∀γ, γ ∈ Γ.
3.6
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5
Conversely, by classical analysis, we pick an arbitrary nonzero C∞ and compactly
supported function ψ with support contained in a fundamental domain ΛΓ of the lattice
ν,μ
Γ. Next, we consider the associated Poincaré series PΓ ψ given by
ν,μ ν,μ PΓ ψ z Jρ,τ γ, z ψ γ · z .
3.7
γ ∈Γ
Then, it can be shown that the function PΓ ψ is C∞ and a nonzero function on for Γ being
discrete and Supp ψ ⊂ ΛΓ. Indeed, for every z ∈ Supp ψ, we have
ν,μ
ν,μ PΓ ψ z ψz.
3.8
ν,μ
Furthermore, under the condition that 1/πφρ takes integral values on Γ × Γ, we see that
ν,μ
ν,μ
PΓ ψ belongs to MΓ,τ . In fact, for every γ ∈ Γ and z ∈ n , we have
ν,μ ν,μ
PΓ ψ γ · z Jρ,τ k, γ · z ψ k · γ · z
k∈Γ
ν,μ Jρ,τ k, γ · z ψ k γ · z
k∈Γ
hγ k
3.9
ν,μ Jρ,τ h · γ −1 , γ · z ψh · z
h∈Γ
ν,μ
2.5 −2iφν,μ h,γ −1 ν,μ e ρ
Jρ,τ h, γ −1 · γ · z Jρ,τ γ −1 , γ · z ψh · z.
h∈Γ
ν,μ
Finally, since e2iφρ
γ,h
ν,μ
1 by hypothesis on φρ , it follows that
ν,μ ν,μ
ν,μ Jρ,τ h, z Jρ,τ γ −1 , γ · z ψh · z
PΓ ψ γ · z h∈Γ
ν,μ
ν,μ Jρ,τ γ −1 , γ · z
Jρ,τ h, zψh · z
3.10
h∈Γ
ν,μ ν,μ Jρ,τ γ, z PΓ ψ z.
The last equality, that is,
ν,μ ν,μ Jρ,τ γ −1 , γ · z Jρ,τ γ, z
3.11
holds for every γ ∈ Γ using the chain rule 2.5 and taking into account the assumption made
ν,μ
on φρ . This completes the proof.
Remark 3.3. The condition involved in Proposition 3.2 ensures that MΓ,τ can be realized as
the space of cross-sections on a line bundle over the complex torus /Γ.
ν,μ
6
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ν,μ
4. On the Function ϕτ
ν,μ
In order to prove the main result of this paper, we need to introduce the function ϕτ .
Proposition 4.1. The first-order differential equation
2 2 ν,μ
∂τ ∂τ ∂ϕ
∂τ
∂τ
τ
τ
−iμ
−τ
− − z
∂z
∂z
∂z
∂z
∂z
ν,μ
admits a solution ϕ
τ : →
4.1
ν,μ
such that ϕ
τ is constant.
Proof. By writing the G-endomorphism ρ : G → G T
differentiating the equivariant condition
as ρg χg, ψg, and
τ g · z ρ g · τz χ g τz ψ g ,
4.2
it follows that
∂τ
∂g · z ∂τ g·z χ g
z,
∂z
∂z
∂z
∂τ
∂g · z ∂τ g·z χ g
z.
∂z
∂z
∂z
4.3
Hence, for ∂g · z/∂z and χg being in T, we deduce that
ν,μ Bτ g
∂τ 2 ∂τ 2
· z ν μ g · z − g · z ∂z
∂z
2 2 ∂τ
∂τ
ν,μ
Bτ z.
ν μ z − z
∂z
∂z
4.4
ν,μ
Therefore, z → Bτ z is a real-valued constant function since the only G-invariant functions
on are the constants. Now, by considering the differential 1-form
ν,μ
θτ z : i νzdz − zdz μτdτ − τdτ ,
ν,μ
ν,μ
ν,μ
4.5
ν,μ
one checks that dθτ dθBτ , where θBτ : iBτ zdz−zdz. Therefore, there exists a function
ν,μ
ν,μ
: → such that ϕ
Constant and satisfying the first-order partial differential
ϕ
τ
τ
equation
ν,μ
∂ϕ
∂τ
∂τ
ν,μ
τ
−i ν − Bτ z μ τ
−τ
∂z
∂z
∂z
2 2 ∂τ ∂τ ∂τ
∂τ
−iμ
τ
−τ
− − z .
∂z
∂z
∂z
∂z
This completes the proof.
4.6
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7
ν,μ
Remark 4.2. The partial differential equation 4.6 satisfied by ϕ
τ can be reduced further to
the following:
ν,μ
∂ψτ
∂τ
τ ,
∂z
∂z
∂ψτ
1
∂τ
ν,μ
ν − Bτ z τ ,
∂z
μ
∂z
ν,μ
4.7
ν,μ
ν,μ
ν,μ
2
2
with ϕ
τ z iBτ − ν|z| − μ|τz| 2iμψτ z.
5. Main Result
ν,μ
ν,μ
ν,μ
ν,μ
Let ϕτ be the real part of ϕ
τ − ϕτ 0, where ϕτ is a complex-valued function on as in
ν,μ
Proposition 4.1. Define Wτ to be the special transformation given by
ν,μ ν,μ
Wτ f z : eiϕτ z fz.
5.1
We have the following.
Theorem 5.1. The image of MΓ,τ by the transform 5.1 is the space of Landau Γ, χτ -automorphic functions. More exactly, one has
ν,μ
ν,μ
Wτ
ν,μ
with Bτ
ν,μ ν,μ
MΓ,τ F; C∞ , F z γ χτ γ j Bτ γ, z Fz ,
ν μ|∂τ/∂z|2 − |∂τ/∂z|2 ∈
5.2
and χτ is the pseudocharacter defined on Γ by
ν,μ −1 ·0 .
χτ γ exp 2iϕτ γ − 2iμ τ0, ρ γ
5.3
For the proof, we begin with the following.
Lemma 5.2. The function χ
τ defined on × Γ by
ν,μ
ν,μ
ν,μ
−1
χ
τ z; γ : eiϕτ zγ −ϕτ z e−2iBτ −νz,γ μτz,ργ ·0
5.4
is independent of the variable z.
Proof. Differentiation of χ
τ z; γ with respect to the variable z gives
ν,μ
ν,μ
∂ϕτ
∂ϕτ ∂
χτ
i
zγ −
τ
z χ
∂z
∂z
∂z
∂τ
−1 ∂τ
ν,μ
−1
− Bτ − ν γ μ ρ γ
·0
·0
τ .
z − ρ γ
z χ
∂z
∂z
5.5
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International Journal of Mathematics and Mathematical Sciences
On the other hand, using the equivariant condition τz γ ργ · τz and 4.6, one gets
i
ν,μ
ν,μ
∂ϕτ
∂ϕτ zγ −
z
∂z
∂z
ν,μ
ν,μ
ν,μ Bτ γ Sτ z − Sτ
zγ
∂τ
∂τ
ν,μ
Bτ − ν γ μ aγ z − aγ z ,
∂z
∂z
5.6
χτ /∂z 0.
where we have set aγ ργ −1 · 0. Thus, from 5.5 and 5.6, we conclude that ∂
Similarly, one gets also ∂
χτ /∂z 0. This ends the proof of Lemma 5.2.
ν,μ
Proof of Theorem 5.1. We have to prove that Wτ F belongs to
ν,μ
Bτν,μ C∞
Bτ
,
γ
j
γ,
z
Fz
:
F
:
−→
;
F
z
γ
χ
FΓ,χ
τ
τ
5.7
whenever F ∈ MΓ,τ , where
ν,μ
ν,μ −1 ·0 .
χτ γ : exp 2iϕτ γ − 2iμ τ0, ρ γ
5.8
Indeed, we have
ν,μ ν,μ
Wτ F z γ : eiϕτ zγ F z γ
ν,μ
eiϕτ
zγ ν ν,μ
eiϕτ
j γ, z j μ ρ γ , τz Fz
ν,μ j γ, z j μ ρ γ , τz Wτ F z
zγ −ϕτ z ν ν,μ
5.9
ν,μ ν,μ χ
τ z; γ j −Bτ γ, z Wτ F z.
τ 0; γ : χτ γ , and therefore
Whence by Lemma 5.2, we see that χ
τ z; γ χ
ν,μ ν,μ ν,μ
Wτ F z γ χτ γ j −Bτ γ, z Wτ F z.
5.10
The proof is complete.
Corollary 5.3. The function χτ γ exp2iϕτ γ − 2iμτ0, ργ −1 · 0 satisfies the following
pseudocharacter property:
ν,μ
ν,μ
χτ γ γ e2iBτ γ,γ χτ γ χτ γ if and only if φρ in 2.6 takes its values in π on Γ × Γ.
ν,μ
5.11
International Journal of Mathematics and Mathematical Sciences
9
Acknowledgment
The author is indebted to Professor A. Intissar for valuable discussions and encouragement.
References
1 B. Hunt and W. Meyer, “Mixed automorphic forms and invariants of elliptic surfaces,” Mathematische
Annalen, vol. 271, no. 1, pp. 53–80, 1985.
2 P. Stiller, “Special values of Dirichlet series, monodromy, and the periods of automorphic forms,” Memoirs of the American Mathematical Society, vol. 49, no. 299, p. iv116, 1984.
3 Y. Choie, “Construction of mixed automorphic forms,” The Journal of the Australian Mathematical Society.
Series A, vol. 63, no. 3, pp. 390–395, 1997.
4 M. H. Lee, “Eisenstein series and Poincaré series for mixed automorphic forms,” Collectanea Mathematica, vol. 51, no. 3, pp. 225–236, 2000.
5 M. H. Lee, Mixed Automorphic Forms, Torus Bundles, and Jacobi Forms, vol. 1845 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2004.
6 A. Ghanmi and A. Intissar, “Landau automorphic forms of n of magnitude ν,” Journal of Mathematical
Physics, vol. 49, no. 8, 2008.
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