International 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 x ) j ν,x ) j μ( ρ ), x ) F x, 1.1 for every x X and g. Here, j α γ,x ; α R is an automorphic factor associated to an appropriate action of a group on X,and ρ, is an equivariant pair for the data,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 International Mathematics and Mathematical Sciences In this paper, we are interested in the space of mixed automorphic forms M, C defined on the complex plane X C with respect to a given lattice in C and an equivariant pair ρ,.wefindthatm, C is isomorphic to the space of Landau automorphic forms 6, F ( z γ ) ( ) χ γ j B ( ) γ,z F z ; z C, γ, 1.2 of weight ( B 2 z ν μ ) 2 B, 1.3 with respect to a special pseudocharacter χ defined on and given explicitly through 5.3 below. The crucial point in the proof is to observe that the quantity B 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 planar mixed automorphic forms M, C.Wehavetogivenecessaryandsufficientcondition to ensure the nontriviality of such functional space. In Section 4, weintroduceproperlythe function ϕ that serves to define the pseudocharacter χ. 2. Group Action Let { ( ) } a b G T oc g : a, b ; a T, b C 0 1 2.1 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,sothatC can be realized as Hermitian symmetric space C G/T. By a G-equivariant pair ρ,,wemeanthatρ is a G-endomorphism and : C C is a compatible mapping, that is, ( g z ) ρ ( g ) z ; g G, z C. 2.2 Now, for given real numbers ν, μ, and an equivariant pair ρ,, wedefinej ρ, complex-valued mapping to be the J ρ, ( g,z ) : j ν ( g,z ) j μ( ρ ( g ), z ) ; ( g,z ) G C, 2.3 where j α ; α R is the automorphic factor given by j α( g,z ) e 2iαI z,g 0. 2.4
International Mathematics and Mathematical Sciences 3 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 mapping J ρ, satisfies the chain rule J ρ, ( gg,z ) e 2iφ ρ g,g J ( ρ, g,g z ) J ( ρ, g,z ), 2.5 where φ ρ g,g is the real-valued function defined on G G by φ ( ρ g,g ) ( : I( ν g 0,g 0 μ ρ g ) 0,ρ ( g ) ) 0. 2.6 Proof. For every g,g G and z C,wehave J ( ρ, gg,z ) j ν( gg,z ) j μ( ρ ( gg ), z ) j ν( gg,z ) j μ( ρ ( g ) ρ ( g ), z ). 2.7 Next, one can see that the automorphic factor j α, satisfies j α( hh,w ) e 2iαI h 0,h 0 j α( h, h w ) j α( h,w ), 2.8 for every h, h G and w C. This gives rise to J ρ, ( gg,z ) e 2iφ ρ g,g j ν( g,g z ) j μ( ρ ( g ),ρ ( g ) z ) J ρ, ( g,z ). 2.9 Finally, 2.5 follows by making use of the equivariant condition ρ g z g z. Remark 2.2. According to Proposition 2.1 above, the unitary transformations T g ] f z : J ρ, ( 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 C. 3. The Space of Planar Mixed Automorphic Forms Let be a uniform lattice of the additive group C, that can be seen as a discrete subgroup of G by the identification γ 1,γ ] ( ) 1 γ, 3.1 0 1
4 International Mathematics and Mathematical Sciences so that the action of on C is the one induced from this of G, thatis, γ z z γ. 3.2 Associated to such and given fixed data of ν, μ > 0and ρ, as above, we perform M, C the vector space of smooth complex-valued functions F on C satisfying the functional equation F z ) J ρ,,z ) F z j ν,z ) j μ( ρ ), z ) F z, 3.3 for every γ and z C. Definition 3.1. The space M, C is called the space of planar mixed automorphic forms of biweight ν, μ with respect to the equivariant pair ρ, and the lattice. We assert the following. Proposition 3.2. The functional space M, C is nontrivial if and only if the real-valued function 1/π φ ρ in 2.6 takes integral values on. 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 M, C is nontrivial, and let F be a nonzero function belonging to M, C. According to 2.5,weget F ( γ ) z ) 3.2 F ( γ ) z ) 3.3 J ρ, γ,z ) F z 2.5 e 2iφ ρ γ,γ J ρ,,γ z ) 3.4 J ( ) γ,z F z, ρ, for every γ,γ. On the other hand, we can write F ( γ ) z ) F ( z γ γ ) F z γ ]) F γ z ]) J ( ρ, γ,γ z ) F z ) 3.5 J,γ z ) J ( ) γ,z F z. 3.3 3.3 ρ, ρ, 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 e 2iφ ρ γ,γ 1, γ,γ. 3.6
International Mathematics and Mathematical Sciences 5 Conversely, by classical analysis, we pick an arbitrary nonzero C and compactly supported function ψ with support contained in a fundamental domain Λ of the lattice. Next, we consider the associated Poincaré seriesp ψ given by ] P ψ z J ( ) ( ) ρ, γ,z ψ γ z. 3.7 γ Then, it can be shown that the function P ψ is C and a nonzero function on C for being discrete and Supp ψ Λ. Indeed, for every z Supp ψ,wehave Furthermore, under the condition that 1/π φ ρ ψ belongs to M P P ψ ] (γ z ) P ψ ] z ψ z. 3.8 takes integral values on, weseethat, C. Infact,foreveryγ and z C n,wehave J ρ, k k J ρ, ( k, γ z ) ψ ( k z )) ( k, γ z ) ψ (( k γ ) z ) h γ k J ( ρ, h γ,γ z ) ψ h z h 3.9 2.5 h e 2iφ ρ h,γ J ( ρ, h, γ (γ z )) J ( ρ, γ,γ z ) ψ h z. Finally, since e 2iφ ρ γ,h 1 by hypothesis on φ ρ, it follows that P ψ ] (γ z ) h J ρ, h, z J ( ρ, γ,γ z ) ψ h z J ( ρ, γ,γ z ) h J ρ, h, z ψ h z 3.10 J ( ) ρ, γ,z P ] z. ψ The last equality, that is, J ρ,,γ z ) J ( ) γ,z ρ, 3.11 holds for every γ using the chain rule 2.5 and taking into account the assumption made on φ ρ. This completes the proof. Remark 3.3. The condition involved in Proposition 3.2 ensures that M, C can be realized as the space of cross-sections on a line bundle over the complex torus C /.
6 International Mathematics and Mathematical Sciences 4. On the Function ϕ In order to prove the main result of this paper, we need to introduce the function ϕ. Proposition 4.1. The first-order differential equation ( ( ϕ iμ ) ( 2 ) ) 2 z 4.1 admits a solution ϕ : C C such that I ϕ is constant. Proof. By writing the G-endomorphism ρ : G G T oc differentiating the equivariant condition as ρ g χ g,ψ g, and ( g z ) ρ ( g ) z χ ( g ) z ψ ( g ), 4.2 it follows that ( ) g z ( ) ( ) g z χ g z, ( g z ) ( ) ( ) g z χ g z. 4.3 Hence, for g z / and χ g being in T,wededucethat B ( ( ) ( ) 2 g z ν μ g z ( ) ) 2 g z ( ν μ z 2 z ) 2 B z. 4.4 Therefore, z B 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 one checks that dθ dθ B ϕ : C C such that I equation θ z : i { ν zdz zdz μ d d }, 4.5,whereθ B ϕ : ib zdz zdz. Therefore, there exists a function Constant and satisfying the first-order partial differential ϕ ( ] ( i ν B z μ )) ( ( iμ ) ( 2 ) ) 2 z. 4.6 This completes the proof.
International Mathematics and Mathematical Sciences 7 Remark 4.2. The partial differential equation 4.6 satisfied by the following: ϕ can be reduced further to ψ, ψ 1 μ ( ν B )z, 4.7 with ϕ z i B ν z 2 μ z 2 2iμψ z. 5. Main Result Let ϕ be the real part of Proposition 4.1. DefineW ϕ ϕ ϕ 0, where is a complex-valued function on C as in to be the special transformation given by W ( f ) ] z : e iϕ z f z. 5.1 We have the following. Theorem 5.1. The image of M, C by the transform 5.1 is the space of Landau,χ -automorphic functions. More exactly, one has ( ) W M, C {F; C,F ( z γ ) ( ) χ γ j B ( ) } γ,z F z, 5.2 with B ν μ / 2 / 2 R and χ is the pseudocharacter defined on by χ ) exp ( 2iϕ ( ) γ 2iμI 0,ρ ) ) 0. 5.3 For the proof, we begin with the following. Lemma 5.2. The function χ defined on C by ( ) χ z; γ : e i ϕ z γ ϕ z e 2i B ν I z,γ μi z,ρ γ 0 5.4 is independent of the variable z. Proof. Differentiation of χ z; γ with respect to the variable z gives χ i ( ϕ ( ) ϕ ) z γ z χ ( ] B (ρ ( ν γ μ ) γ 0 ) ( ( z ρ γ ) ) ]) 0 z χ. 5.5
8 International Mathematics and Mathematical Sciences On the other hand, using the equivariant condition z γ ρ γ z and 4.6, onegets ( ϕ i ( ) ϕ ) z γ z B γ S z S ( ) z γ ] B ν γ μ a γ z a γ ] z, 5.6 wherewehaveseta γ ρ γ 0. Thus, from 5.5 and 5.6, weconcludethat χ / 0. Similarly, one gets also χ / 0. This ends the proof of Lemma 5.2. Proof of Theorem 5.1. We have to prove that W F belongs to F B,χ : {F : C C C ; F ( z γ ) ( ) χ γ j B,z ) F z }, 5.7 whenever F M, C, where χ ) : exp ( 2iϕ ( ) γ 2iμI 0,ρ ) ) 0. 5.8 Indeed, we have W ] (z ) F γ : e iϕ z γ F ( z γ ) e iϕ z γ j ν,z ) j μ( ρ ), z ) F z e i ϕ z γ ϕ ( ) χ z; γ j B z j ν,z ) j μ( ρ ), z ) ] W F z ( ) ] γ,z W F z. 5.9 Whence by Lemma 5.2, weseethat χ z; γ χ 0; γ : χ γ, and therefore W F ] (z γ ) χ ) j B ( ) ] γ,z W F z. 5.10 The proof is complete. Corollary 5.3. The function χ γ exp 2iϕ γ 2iμI 0,ρ γ 0 satisfies the following pseudocharacter property: χ γ ) e 2iB I γ,γ χ ) χ ) 5.11 if and only if φ ρ in 2.6 takes its values in πz on.
International 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. iv 116, 1984. 3 Y. Choie, Construction of mixed automorphic forms, The 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 C n of magnitude ν, Mathematical Physics, vol. 49, no. 8, 2008.
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