FOURIER EXPANSIONS OF COMPLEX-VALUED EISENSTEIN SERIES ON FINITE UPPER HALF PLANES

We consider complex-valued modular forms on finite upper half planes Hq and obtain Fourier expansions of Eisenstein series invariant under the groups Γ=SL(2,Fp) and GL(2,Fp). The expansions are analogous to those of Maass wave forms on the ordinary Poincare upper half plane —the K-Bessel functions being replaced by Kloosterman sums.


Introduction
Before outlining our results, let us give a brief summary of the classical results for which we have found finite analogs.This work is a part of a continuing project to seek out finite analogs of Terras [17,Chapter 2].
An example of a Maass wave form for SL(2, Z) is provided by the Maass Eisenstein series on H defined, for z ∈ H and s ∈ C with Res > 1, by Here Γ ∞ is the subgroup of Γ consisting of matrices which fix ∞ (i.e., the lower left element is 0).Define the gamma function for Res > 0 by y s e −y dy y . (1.2) Next define the K-Bessel function for Rea > 0 by y s e −a/2(y+(1/ y)) dy y .
(1.3) Set where ζ(s) is Riemann's zeta function.Then the Fourier expansion of E s (z) is (1.5) Here the divisor function is See [17, pages 208-209] for a sketch of three proofs.
The upper half plane H has a finite analog called the finite "upper" half plane H q attached to a finite field F q with q elements.We assume q is an odd number.It is likely that most of our results go through in characteristic 2, but we leave this for a future paper.Then we take a fixed nonsquare δ ∈ F q and define H q = z = x + y δ | x, y ∈ F q , y = 0 . (1.7) The finite upper half plane is considered in detail in [18,Chapter 19].See also [13].An application to coding theory is to be found in [19].References which cover finite upper half planes in both even and odd characteristics include Angel [1], Angel and Evans [3], Evans [8], as well as Soto-Andrade and Vargas [15].
Recall that an element of the general linear group has entries in F q and nonzero determinant.Then for z ∈ H q , (1.9) Define the "distance" between z,w ∈ H q by It is easily checked that for g ∈ GL(2,F q ), d(gw,gw) = d(z,w).For a ∈ F q , define the finite upper half plane graphs X q (δ,a) to have vertices the elements of H q and an edge between z and w in H q if and only if d(z,w) = a.For a = 0, 4δ, the graph X q (δ,a) is (q + 1)-regular, connected, and Ramanujan.See [18] for more information.Ramanujan graphs are of interest to computer scientists because they provide efficient communications networks.Chai and Li [7] have proved some interesting connections of the spectra of the finite upper half plane graphs and modular forms of function fields.The GL(2,F q )-invariant operators on H q analogous to the non-Euclidean Laplacian on the Poincaré upper half plane H are the adjacency operators on the finite upper half plane graphs defined for a ∈ F q by (1.11) These operators generate a commutative algebra of operators.
One can view H q as a finite symmetric space G/K as in [18,Chapter 19].From that point of view, the adjacency operators A a correspond to G-invariant differential operators on a symmetric space.This subject may also be studied from the point of view of association schemes (Bannai and Ito [4]) or Hecke operators (Krieg [11]).
A (complex-valued) modular form on the finite upper half plane H q is a function f : H q → C having an invariance property for a subgroup Γ of GL(2, F q ).See Definition 2.4.Here we consider Γ = SL(2,F p ) and GL(2,F p ).Such modular forms which are eigenfunctions of the adjacency operators for the finite upper half plane graphs attached to H q are analogs of Maass wave forms on the Poincaré upper half plane.Now we consider one of our finite upper half plane analogs of the Maass Eisenstein series in formula (1.1).Let Γ be a subgroup of GL(2,F q ) and let χ be a multiplicative character on F * q .Define the Eisenstein "series" for z ∈ H q as E χ,Γ (z) = γ∈Γ χ Im(γz) . (1.12) We will see that E χ,Γ is an eigenfunction of the adjacency operators of finite upper half plane graphs if it does not vanish identically on H q .To see when this happens, recall that q is a cyclic group with generator g. (See [18] and the references given there for more information about finite fields.)Thus a multiplicative character χ of F * q has the following form for integers a, b: (1.13) We will find that for q = p n , with q > 2 and n > 2, E χ,GL (2,Fq) is nonzero if and only if χ = χ a when a is a multiple of (p − 1).This last condition is equivalent to saying that χ a | F * p = 1.
When n = 2, we will find that E λ,GL (2,Fq) is nonzero, for χ = 1 or χ = λ as in formula (1.20).We will also find experimentally that E χ,GL (2,Fq) is nonzero for p = 3,5 when a is a multiple of (p − 1) (see Tables 2.1 and 2.2 near the end of the paper).
As we saw in formula (1.5), Fourier expansions of Maass Eisenstein series on H involve K-Bessel functions.In the finite case under consideration, the Bessel functions will be replaced with Kloosterman sums.Thus we need to define these sums as well as Gauss sums in order to state the Fourier expansion of the Eisenstein series E χ,GL(2,Fp) (z).
Let Ψ be an additive character of F q .Then Ψ = Ψ b , where Ψ b (u) = e 2πiTr(bu)/p , for b,u ∈ F q . (1.14) Here the trace in the exponent is (for q = p n , p = prime) Let Ψ be an additive character and χ a multiplicative character.Define the Gauss sum by (1.16) The Gauss sum is analogous to the gamma function of formula (1.2).See [10] by Ireland and Rosen for more information on Gauss sums.In particular, they show that if χ and Ψ b are not trivial, then (1.17) Another basic result says that if χ and Ψ are not trivial, then The Kloosterman sum for a,b ∈ F q is defined by The Kloosterman sum is analogous to the K-Bessel function of formula (1.3).These sums have great importance in number theory.See [12] by Sarnak who discusses the connection with modular forms and "Kloostermania."Define the quadratic residue character λ on (1.20) One of our main results (see formula (2.34)) is a finite analog of formula (1.5) and says that the Fourier expansion of the Eisenstein series E χ,GL(2,Fq) (z) is given by (1.21) A. Shaheen and A. Terras 5 We will also prove the analogous result for Γ = SL(2,F p ) in formula (2.32).Both of these Fourier expansions result from that of Theorem 2.9 which gives the Fourier expansion of what we call a G-type Eisenstein series (see Definition 2.2).
The main idea of the proof of (1.21) is to imitate the second proof of formula (1.5) given by Terras [17].This proof goes back to Bateman and Grosswald [5] and is extended by Terras [16].The main fact needed is the Poisson summation formula for Z as a subgroup of R. For our finite upper half plane analog we will use Poisson summation for the additive subgroup F p of F q .See Proposition 2.5 or [18, page 199].
One can also consider finite field valued modular forms on finite upper half planes.These are perhaps more analogous to holomorphic modular forms.See [13] by Shaheen where an analog of the Selberg trace formula is developed which is more like that of Selberg (Hejhal [9]) than the version of Terras [18].

Lattices, modular forms on H q , and Fourier expansions
Classically Eisenstein series are sums over lattices such as Zw 1 ⊕ Zw 2 for elements w i in C which are linearly independent over R. Our finite analog of a lattice is defined as follows.
Here q = p n , with n > 1.
Note that if z ∈ H q , we have an example of an Proof.Imitate the classical proof found in [14].
We now give our most general definition of a complex-valued modular form.
Definition 2.4.Let χ : F q ( √ δ) * → C be a multiplicative character.Define f : H q → C to be a complex-valued modular form for χ and a subgroup Γ of GL(2, (2.3) We denote the complex vector space of such forms by M χ (Γ).
Proposition 2.3 can then be restated to say that, as a function of z, G χ (z Note that M χ (GL(2,F p )) = {0} implies that χ(a) = 1 for all a ∈ F * p .Similarly, implies that χ(−1) = 1.This is analogous to the fact that classical holomorphic modular forms for SL(2, Z) must have even weight.Fundamental domains for Γ\H q are discussed by Shaheen [13].It turns out that the classical perpendicular bisector method for constructing such fundamental domains (discussed by Terras [17], e.g.) has a finite analog.
Tessellations of H q given by letting Γ act on a fundamental domain are as beautiful as those for the modular group acting on the Poincaré upper half plane.One such tessellation is given in Figure 2.1.
Let χ be a multiplicative character of F * q and let N = N Fq( √ δ)/Fq be the norm map.The character χ • N −1 seems like a reasonable character to investigate.In fact, it comes up when we consider the Eisenstein series E χ,Γ from formula (1.12) of the introduction when Γ = GL(2,F p ) as we will soon see.
Here we will need the Poisson summation formula for a subgroup B of a finite abelian group A proved by Terras [18, page 199].The formula relates Fourier analysis on A/B and that on A itself.Proposition 2.5 (Poisson summation formula for finite abelian groups).Suppose B is a subgroup of the finite additive abelian group A and f :

.5)
Write A to denote the group of additive characters of A.Here the dual group to B in A is A. Shaheen and A. Terras 7 and the Fourier transform of f at Note that the left-hand side of formula (2.5) is a function of the coset a + B in A/B while the right-hand side involves Fourier transforms on A.
We want to apply the Poisson summation formula to B = F p as an additive subgroup of A = F q .Therefore, we will need to know the characters in F # p .As in formula (1.14), Ψ a (u) = e 2πiTr(au)/p is an arbitrary element of F q .Here the trace is Tr = Tr Fq/Fp as in formula (1.15).Then, Ψ a (u) = e 2πiTr(au)/p = 1 for all u ∈ F p if and only if p divides Tr(au) = uTr(a) for all u ∈ F p .Therefore, Ψ a ∈ F # p if and only if Tr(a) = 0 in F p .We have proved the following lemma.
Lemma 2.6.Using the notation of the preceding proposition, if F p is considered as a subgroup of the additive group F q , the dual group to F p in F q is (2.8) We will also find that the Kloosterman sum appears in a different guise in our computations.So let us define the finite k-Bessel function in analogy to that considered on the Poincaré upper half plane in [17, page 136].
Definition 2.7.For a multiplicative character χ of F * q and an additive character Ψ of F q , if z ∈ H q , define the k-Bessel function by (2.9) ) is an eigenfunction of all the adjacency operators A a for the finite upper half plane graphs from formula (1.11).
The following proposition is proved by Celniker et al. [6] (see also [2]).It is a finite analog of a classical integral formula for K-Bessel functions.
Proposition 2.8 (Relation between k-Bessel function and Kloosterman sum).For a nontrivial multiplicative character χ of F * q and a nontrivial additive character where g a = u∈Fq Ψ a (u 2 ) is a Gauss sum, the Kloosterman sum K is defined by formula (1.19), and the multiplicative character λ is given in formula (1.20).
Why do we say that g a is a Gauss sum?It is easily seen that for a = 0, Theorem 2.9.Let N : F q ( √ δ) * → F * q be the norm map, χ = χ a the multiplicative character of F * q given by (1.13), and where m/2 = (q − 1)/(p − 1) − 1, then for z = x + y √ δ ∈ H q , the G-Eisenstein series has the following Fourier expansion: (2.13) Here ψ b is the additive character of F q given by (1.14), Γ(χ,Ψ) is the Gauss sum from formula (1.16), K ψ is the Kloosterman sum from formula (1.19), and λ is the multiplicative character from formula (1.20).If then G χ•N −1 = 0.
Furthermore, if q = p 2 , then one can show that G χ•N −1 (z) is not the zero function by showing that the constant term of its Fourier expansion is nonzero.When q = p 2 , our later computations show that certain Eisenstein series must be nonzero also.But we leave the general q = p 2 question open for the moment.
Proof.If a = 0, then the result is clear.If a = 0, we show that the constant term of the Fourier expansion of G χa•N −1 (z) is nonzero.We have two cases to consider.If λχ a is the identity character, then the constant term of (1 Suppose that λχ a is not the identity character.We take the absolute value of the second term in the constant term and note that it is not 1 using formula (1.18).Therefore, the constant term of G χa•N −1 (z) is nonzero.
Next we consider the Fourier expansion of some Eisenstein series which are analogous to the Maass Eisenstein series (1.1).First, we define the finite analog of the power function y s on the Poincaré upper half plane.
Let χ : F * q → C be a multiplicative character.Define the "power function" p χ : H q → C to be p χ (z) = χ Im(z) . (2.26) A. Shaheen and A. Terras 11 Lemma 2.11 (The power function is an eigenfunction of all adjacency operators).If A a denotes the adjacency operator defined by formula (1.11), then Proof.See [18, page 324] by Terras.
Now we want to relate the Eisenstein series of E-type from formula (1.12)-an Eisenstein series analogous to the Maass Eisenstein series of formula (1.1)-and the G-type Eisenstein series from Definition 2.2.If Γ is a subgroup of GL(2,F q ), χ is a multiplicative character on F * q , and p χ (z) is the power function from formula (2.26), define the finite upper half plane E-type Eisenstein series E χ,Γ (z) by formula (1.12).Then we see that (2.28) Note that E χ,Γ ∈ M χ1 (Γ), where χ 1 denotes the trivial character.
Lemma 2.12.Given c,d ∈ F p not both zero and u ∈ F * p , there exist p pairs (a,b) ∈ F 2 p such that det( a b c d ) = u.Proof.We need to solve the equation xd − yc = u = 0 for x, y ∈ F p .If c = 0, then we can pick any x ∈ F p and solve for y = c −1 (xd − u).This gives us p different choices for x.If c = 0, then y can be any element of F p and x = ud −1 .Again, we have p choices for y.
We see from the preceding results that for our two favorite subgroups of GL(2,F q ), we have the following relations between the Eand G-type Eisenstein series: Here the G-type Eisenstein series G χ is defined by formula (2.1).In both cases of formula (2.29), when χ 2 | F * p = 1, the Eisenstein series is 0.
It follows from Lemma 2.11 that if E χ,Γ ≡ 0 on H q , then E χ,Γ (z) is an eigenfunction of the adjacency operator A a of formula (1.11)  (2.30) Thus the Eisenstein series, E χ,Γ , are the analogs of the classical Maass Eisenstein series E s from formula (1.1) at least when they are not identically zero.Angel et al. [2] considered Eisenstein series for Γ = K, the subgroup of GL(2,F q ) fixing √ δ.In this case, the Eisenstein series are usually called spherical functions.The general case was introduced in [13] in an attempt to find elements of L 2 (Γ\H q ) that simultaneously diagonalize the matrices A Γ a = A a | L 2 (Γ\Hq) .
We can now easily find the Fourier expansions of E χ,SL (2,Fp) and E χ,GL (2,Fp) using that of G χ•N −1 (z) in Theorem 2.9.
In order to compute some values of these Eisenstein series, we need to think about fundamental domains.Here we will only consider q = p 2 for p = 3,5.The fundamental domains in Examples 2.13 and 2.14 and the Eisenstein series in Tables 2.1 and 2.2 were calculated using Mathematica.
A. Shaheen and A. Terras 13  ( More general results on fundamental domains can be found in [13]. Finally, we compute values of E χ,SL(2,Fp) (z) for the preceding examples.One can compute E χ,GL(2,Fp) (z) for the same values of p and q using formula (2.29).

Figure 2 . 2 .
Figure 2.2.The fundamental domain for SL(2,F 3 \H 9 ) is given by the 6 points which are either circled or boxed.