Eigenfunctions on the Finite Poincaré Plane by Jinghua Kuang ^{∗}

1. Introduction. Let F_{q} be a finite field with q elements (q odd.) Fix a non-square elemnet δ ∈ F.

H_{q} = F_{q}(\sqrt{δ}) − F_{q} = \{x + y\sqrt{δ} | x, y ≠ 0 ∈ F_{q}\}

is called the finite Poincaré plane. From [1], [2], [4], [7], we have the following facts:

1. G = GL_{2}(F_{q}) acts on H_{q} by linear transformation:

gz = \frac{az + b}{cz + d}, ∀g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ G, ∀z ∈ H_{q}.

Let K = \{( \begin{matrix}a & bδ \\ b & a \end{matrix} )\}. One may identify G/K with H_{q} and with P = \{( \begin{matrix}y & x \\ 0 & 1 \end{matrix} ) | y ≠ 0, x ∈ F_{q}\}.

2. There is a G-invariant pseudo-distance Δ on H_{q}:

Δ(z_{1}, z_{2}) = \frac{N(z_{1} − z_{2})}{Imz_{1} · Imz_{2}} ∀z_{1}, z_{2} ∈ H_{q},

where N is the norm from F_{q}(\sqrt{δ}) to F_{q} and Imz = y if z = x + y\sqrt{δ}.

3. For a ∈ F_{q}^{×} and a ≠ 4δ, a graph structure X_{q}(δ, a) is defined on H_{q}: (z_{1}, z_{2}) is an edge iff Δ(z_{1}, z_{2}) = a. Katz [3] and Li [5] proved that X_{q}(δ, a) are Ramanujan graphs using different methods. However, distributions of eigenvalues of the adjacency matrices have not been well understood. We will

calculate the first and second moments of the asymptotic distribution of the eigenvalues of the adjacency matrices and provide evidence for Terras’ conjecture in [7].

Let V be the space of all complex functions on H_{q}. Using character sums, Evans [2] constructed an orthogonal basis of V that diagonalizes all the adjacency matrices. Kuang [4], using group representation theory, also constructed such an orthogonal basis. This paper will also compare these two bases.

2. Hecke Algebra. Since K is the analogue of O(2) in the classical Poincaré upper half plane, it is natural to consider the Hecke algebra H(G, K), which is defined as the algebra of all bi-K-invariant complex functions on G under convolution. Since K is the isotropic subgroup of \sqrt{δ}, Δ(g, 1) = Δ(g^{−1}, 1) implies that KgK = Kg^{−1}K. Therefore, H(G, K) is commutative. Obviously, H(G, K) has dimension q. Hence there exist q idempotents in H(G, K).

3. Idempotents. For an irreducible representation π of G, let η′_{π}(g) = \frac{1}{|K|} ∑_{k∈K} tr π(kg). π is spherical (i.e. Res_{K}^{G}(π) ⊃ 1_{K}) iff η′_{π}(g) ≠ 0. From [4], we have

η′_{π_{1}} ∗ η′_{π_{2}} = \{ \begin{matrix}0, & ifπ_{1} ≄ π_{2}; \\ \frac{|G|}{dim(π)}η′_{π}, & ifπ_{1} ≃ π_{2} ≃ π. \end{matrix}

Let ω be a multiplicative character of F_{q}(\sqrt{δ}) of order q^{2} − 1, χ be a multiplicative character of order q − 1, and s = χ^{(q−1)/2}. There are q irreducible spherical representations of G (see [6]):

π = 1_{G};
or = ρ(s, s) = q dimensional irreducible component of Ind_{B}^{G}(s, s);
or = ρ(χ^{j}, χ̄^{j}) = Ind_{B}^{G}(χ^{j}, χ̄^{j}), j = 1, …, \frac{q − 3}{2};
or = ρ_{ν_{j}} = cuspidal representation corresponding to ν_{j} = ω^{j(q−1)}, j = 1, …, \frac{q − 1}{2}.

(B is the Borel subgroup of G.) Let η_{π} = \frac{dim(π)}{|G|}η′_{π}. Let η_{0} = η_{1_{G}}, η_{j} = η_{ρ(χ^{j},χ̄^{j})}, j = 1, …, (q − 3)/2, η_{\frac{q−1}{2}} = η_{ρ(s,s)} and η_{\frac{q−1}{2} +j} = η_{ρ_{ν_{j}}} , j = 1, …, (q − 1)/2. Then

∗ Research at MSRI supported in part by NSF grant #DMS 9022140 Theorem 1 (Kuang [4]). η_{0}, η_{1}, …, η_{q−1} are the q idempotents in H(G, K).

4. Representation of H(G, K). H(G, K) has a natural representation on L^{2}(H_{q}) = L^{2}(P), the space of all complex functions on H_{q} = P:

T_{φ}(f)(p) = \frac{1}{|K|}f ∗ φ(p) = \frac{1}{|K|} ∑_{h∈G} f(h)φ(h^{−1}p) = ∑_{p_{1}∈P} f(p_{1})φ(p_{1}^{−1}p),

for φ ∈ H(G, K), f ∈ L^{2}(P) and p ∈ P.

For the usual inner product ⟨, ⟩ on L^{2}(P), we have ⟨T_{φ}(f_{1}), f_{2}⟩ = ⟨f_{1}, Tφ̂(f_{2})⟩, where φ̂(h) = \overline{φ(h^{−1})}. That is, H(G, K) acts on L^{2}(P) self-adjointly. Hence, there exists a basis of simultaneous eigenfunctions of H(G, K).

5. Construction of the Bases. Let ψ be a fixed non-trivial additive character of F_{q}. Denote ψ_{a}(x) = ψ(ax). Define

χ_{i}(g) = χ^{i}(y) if gK = ( \begin{matrix}y & x \\ 0 & 1 \end{matrix} ) K, i = 1, …, q − 1.

Ψ_{a}(g) = δ(y)ψ_{a}(x) if gK = ( \begin{matrix}y & x \\ 0 & 1 \end{matrix} ) K, a ∈ F_{q}^{×},

where δ(y) = 1 if y = 1; = 0 otherwise. Then

Theorem 2 (Kuang [4]). χ_{i}, i = 1, …, q − 1 and Ψ_{a} ∗ η_{i}, i = 1, …, q − 1, a ∈ F_{q}^{×} are simultaneous eigenfunctions of H(G, K) and make up an orthogonal basis of L^{2}(P). Moreover, T_{η_{j}}(χ_{i}) = χ_{i} if η_{j} = η_{ρ(χ^{±i},χ̄^{±i})}; = 0 otherwise. T_{η_{j}}(Ψ_{a} ∗ η_{i}) = Ψ_{a} ∗ η_{i} if i = j; = 0 otherwise.

Terras [8] constructed the following functions:

K_{i,ψ_{a}}(( \begin{matrix}y & x \\ 0 & 1 \end{matrix} )) = ψ_{a}(−x)χ^{i}(y) ∑_{u∈F_{q}} ψ_{a}(u)χ̄^{i}(u^{2} − δy^{2}).

Evans [2], based on Velasquez’s work according to [9], constructed the following functions:

H_{t,i,ψ_{a}}(( \begin{matrix}y & x \\ 0 & 1 \end{matrix} )) = \frac{ψ_{a}(−x)}{q + 1} ∑_{u∈F_{q}} ^{∗}ψ_{a}(u) ∑_{α∈U_{1}} ω^{i}(α)s (α + \frac{1}{α} + \frac{δy}{t} + \frac{t}{δy} − \frac{u^{2}}{ty} ) ,

where U_{1} = {α ∈ F_{q}^{×}| N(α) = 1} and the asterisk indicates that when y = ±t, the u = 0 term is multiplied by q + 1.

Theorem 3 (Terras, Velasquez, Evans [2]). K_{i,ψ_{a}}, i = 1, …, q − 1, a ∈ F_{q} and H_{t,i,ψ_{a}}, t ∈ F_{q}^{×}, i = 1, …, q − 1, a ∈ F_{q} are simultaneous eigenfunctions of H(G, K). There exist a t_{0} for any fixed i and a such that K_{i,ψ_{0}}, i = 1, …, q − 1, K_{i,ψ_{a}}, i = 1, …, (q − 1)/2, a ∈ F_{q} and H_{t_{0},i,ψ_{a}} i = 1, …, (q − 1)/2, a ∈ F_{q}^{×} make up an orthogonal basis of L^{2}(P).

6. Comparison of the Bases. We now prove that the two bases constructed above are essentially same. Let us denote by π_{i} the spherical representation that give rise to η_{i}.

Theorem 4. The following identities hold.

\overline{C_{i}(0)} · χ_{i} = K_{i,1}, i = 1, …, q − 1;
Ψ_{a} ∗ η_{i} = \frac{dim(π_{i})}{|G|(q + 1)}C_{i}(a)K_{i,ψ_{−a}}, i = 1, …, (q − 1)/2;
Ψ_{a} ∗ η_{i+(q−1)/2} = \frac{dim(π_{i+(q−1)/2})}{|G|}H_{−δ,i,ψ_{−a}}, i = 1, …, (q − 1)/2,

where C_{i}(a) = ∑_{x∈F_{q}} ψ_{a}(x)χ^{i}(x^{2} − δ).

Proof. K_{i,1} = \overline{C_{i}(0)} · χ_{i} is obvious. Since for any pair of a, b ∈ F_{q}, K_{i,ψ_{b}} and Ψ_{a} ∗ η_{i} are eigenfunctions of H(G, K) of the same eigenvalues (i = 1, …, (q − 1)/2), Ψ_{a} ∗ η_{i} is a complex linear combination of K_{i,ψ_{b}}, (b ∈ F_{q}). Now,

⟨K_{i,ψ_{b}}, Ψ_{a} ∗ η_{i}⟩ = ∑_{p∈P} K_{i,ψ_{b}}(p)\overline{Ψ_{a} ∗ η_{i}}(p)

= ∑_{p∈P} ∑_{u∈F_{q}} ψ_{b}(u − x)χ^{i}(y)\overline{χ}^{i}(u^{2} − δy^{2})\overline{η_{i}}(p) ∑_{v∈F_{q}} ψ(−(a + b)v)

= 0 if a + b ≠ 0

Hence Ψ_{a} ∗ η_{i} is a multiple of K_{i,ψ_{−a}}. Using the equation (2.17) in [2], we calculate

\begin{aligned}Ψ_{a} ∗ η_{i}(1) & = ∑_{v∈F_{q}} ψ_{a}(u)η_{i}(( \begin{matrix}1 & −u \\ 0 & 1 \end{matrix} )) \\ & = \frac{dim(π_{i})}{|G|(q + 1)}C_{i}(a)\overline{C_{i}(a)}. \end{aligned}

And K_{i,ψ_{−a}}(1) = \overline{C_{i}(a)}. Therefore, Ψ_{a} ∗ η_{i} = \frac{dim(π_{i})}{|G|(q+1)}C_{i}(a)K_{i,ψ_{−a}}.

For the third identity, we use the equation (2.16) in [2]. Let |S(z)| be the cardinality of the K-orbit of z in H_{q}. Let î = i + (q − 1)/2. Then,

Ψ_{a} ∗ ηî(p) = ∑_{u∈F_{q}} ψ_{a}(u)ηî(( \begin{matrix}y & x − u \\ 0 & 1 \end{matrix} ))

= ∑_{u∈F_{q}} ψ_{a}(x − u)ηî(( \begin{matrix}y & u \\ 0 & 1 \end{matrix} ))

= ψ_{a}(x) ∑_{u∈F_{q}} ψ_{a}(−u)\frac{dim(πî)}{|G|} \frac{1}{|S(u + y\sqrt{δ})|} ∑_{α∈U_{1}} ω^{i}(α)s(α + \frac{1}{α} − 2 + \frac{Δ(u + y\sqrt{δ}, \sqrt{δ})}{δ})

= ψ_{a}(x) ∑_{u∈F_{q}} ψ_{a}(−u)\frac{dim(πî)}{|G|} \frac{1}{|S(u + y\sqrt{δ})|} ∑_{α∈U_{1}} ω^{i}(α)s(α + \frac{1}{α} + \frac{u^{2}}{yδ} − y − \frac{1}{y} )

= \frac{dim(πî)}{|G|}H_{−δ,i,ψ_{−a}}(p)

7. Comments and Questions. For any function f on G, define the Fourier coefficient

F (g; a, f) = ∑_{x∈F_{q}} f(( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) g)ψ_{a}(x).

From [4], we see ⟨Ψ_{a} ∗ η_{i}, Ψ_{a} ∗ η_{i}⟩ = |K| · q · F (1; a, η_{i}). So F (1; a, η_{i}) ≠ 0. But can one evaluate F (1; a, η_{i})? We have F (1; a, η_{i}) = Ψ_{−a} ∗ η_{i}(1). Hence, for i = 1, …, (q − 1)/2,

F (1; a, η_{i}) = \frac{dim(π_{i})}{|G|(q + 1)}|C_{i}(−a)|^{2} = \frac{dim(π_{i})}{|G|(q + 1)}|C_{i}(a)|^{2}
F (1; a, ηî) = \frac{dim(πî)}{|G|} ∑_{u∈F_{q}} ψ_{a}(u)\frac{1}{|S(u + \sqrt{δ})|} ∑_{α∈U_{1}} ω^{i}(α)s(α + \frac{1}{α} + \frac{u^{2}}{δ} − 2)

However, we don’t know much about C_{i}(a) and about the last set of sums. We note that C_{i}(0) is essentially a Jacobi sum, in fact, C_{i}(0) = −χ^{i}(−δ)J(χ^{i}, s).

8. Connection to X_{q}(δ, a). There are q double cosets: D_{a} = KgK, where Δ(g, 1) = a ∈ F_{q}. Let φ_{a} be the characteristic function of the set D_{a}. Then A_{a} = \frac{1}{|K|}T_{φ_{a}} acting on L^{2}(H_{q}) is the adjacency matrix of X_{q}(δ, a). Therefore each basis constructed in Section 5 diagonalizes all the adjacency matrices. Let S_{a} = D_{a} ∩ P. Then

λ_{i}(a) = \frac{|G|}{dim(π_{i})}|S_{a}|η_{i}(D_{a}), i = 0, 1, …, q − 1

are all the eigenvalues of A_{a}. Katz [3], Li [5] proved that |λ_{i}(a)| ≤ 2\sqrt{q} for i = 1, …, q − 1, which confirms that X_{a}(δ, a) is Ramanujan.

Fix a ≠ 4δ ∈ F^{×}, Terras [7] conjectured that \{λ_{i}(a)/\sqrt{q} | i = 1, …, q − 1\} asymptotically has Sato-Tate distribution, i.e. for E ⊂ [−2, 2],

lim_{q→∞} \frac{1}{q − 1}|\{λ_{i}(a) | λ_{i}(a)/\sqrt{q} ∈ E\}| = \frac{1}{2π} ∫_{E} \sqrt{4 − x^{2}}dx.

9. Moments. Define a q × q matrix

M = (\sqrt{\frac{|S_{a}|}{dim(π_{i})}}η_{i}(S_{a}))_{i=0,1,...,q−1, a∈F_{q}} .

The idempotent property of η_{i} (i = 0, …, q − 1) implies that MM′ = \frac{1}{|K||G|}I_{q}. (M′ is the transpose of M.) Hence, M′M = \frac{1}{|K||G|}I_{q}, that gives

\begin{aligned}\frac{1}{q − 1} ∑_{i=0}^{q−1} \frac{dim(π_{i})}{q} · \frac{λ_{i}(a)}{\sqrt{q}} & = 0 \\ \frac{1}{q − 1} ∑_{i=0}^{q−1} \frac{dim(π_{i})}{q} · (\frac{λ_{i}(a)}{\sqrt{q}})^{2} & = \frac{|G||S_{a}|}{|K|q^{2}(q − 1)} \end{aligned}

That, in turn, implies

\begin{aligned}lim_{q→∞} \frac{1}{q − 1} ∑_{i=1}^{q−1} \frac{λ_{i}(a)}{\sqrt{q}} & = 0 \\ lim_{q→∞} \frac{1}{q − 1} ∑_{i=1}^{q−1} (\frac{λ_{i}(a)}{\sqrt{q}})^{2} & = 1. \end{aligned}

Therefore, the first and second moments of \{λ_{i}(a)/\sqrt{q} | i = 1, …, q − 1\} asymptotically match with those of the Sato-Tate distribution.

Acknowledgement. This is a written-up of a talk that the author delivered at the Workshop on Spectral Theory of Automorphic Forms and Number Theory, in MSRI, Berkeley, California, October 24-28, 1994. Some calculation was completed while the author was visiting MSRI. He thanks the MSRI for its hospitality.

References. [1]. J. Angel, N. Celniker, S. Poulos, A. Terras, C. Trimble, E. Velasquez Special Functions on Finite Upper Half Planes, in AMS Contemporary Math. Vol. 138, 1992. [2]. R. Evans Character sums as orthogonal eigenfunctions of adjacency operators for Cayley graphs, in Proc. of a conference on finite fields, AMS contemporary Math. Vol. 168, 1994. [3]. N. Katz Estimates for Soto-Andrade Sums, J. für die Reine und Angew. Math. 438 (1993), 143-161. [4]. J. Kuang A Natural Orthogonal Basis of Eigenfunctions of the Hecke Algebra Acting on Cayley Graphs, to appear in the Proceeding of American Mathematical Society. [5]. W. Li A survey of Ramanujan graphs, Proc. of a Conference on Arithmetic Geometry and Coding Theory at Luminy, 1993 (to appear).

[6]. I. Piatetski-Shapiro Complex Representation of GL_{2}(F_{q}), AMS Contemporary Math. Vol. 16, 1983. [7]. A. Terras Are Finite Upper Half Plane Graphs Ramanujan? in DIMACS Series in Discrete Mathematics and Theoretic Computer Science, Vol. 10 (1993), 125-142 [8]. A. Terras Fourier Analysis on Finite Groups and Applications, U. C. S. D. Lecture Notes, 1991-2. [9]. E. Velasquez, Private Conversation, October 27, 1994.

Department of Mathematics Pennsylvania State University, at Fayette Uniontown, PA 15401 E-mail address: kuang@math.psu.edu