VALUE DISTRIBUTION FOR EIGENFUNCTIONS OF DESYMMETRIZED QUANTUM MAPS

PÄR KURLBERG AND ZEÉV RUDNICK

Abstract. We study the value distribution and extreme values of eigenfunctions for the “quantized cat map”. This is the quantization of a hyperbolic linear map of the torus. In a previous paper it was observed that there are quantum symmetries of the quantum map - a commutative group of unitary operators which commute with the map, which we called “Hecke operators”. The eigenspaces of the quantum map thus admit an orthonormal basis consisting of eigenfunctions of all the Hecke operators, which we call “Hecke eigenfunctions”.

In this note we investigate suprema and value distribution of the Hecke eigenfunctions. For prime values of the inverse Planck constant N for which the map is diagonalizable modulo N (the “split primes” for the map), we show that the Hecke eigenfunctions are uniformly bounded and their absolute values (amplitudes) are either constant or have a semi-circle value distribution as N tends to infinity. Moreover in the latter case different eigenfunctions become statistically independent. We obtain these results via the Riemann hypothesis for curves over a finite field (Weil’s theorem) and recent results of N. Katz on exponential sums. For general N we obtain a nontrivial bound on the supremum norm of these Hecke eigenfunctions.

1. Introduction

In the past few years, much attention has been devoted to the behavior of eigenfunctions of classically chaotic quantum systems. One aspect of this topic concerns their value distribution and specifically their extreme values [Be, AS, AQC, IS, HR, ABST]. Our aim is to explore this for one of the best-studied models in quantum chaotic dynamics, the “quantized cat map” [HB]. This is the quantization of a hyperbolic linear map A of the torus. For a brief background about this model, see section 2.

Date: February 21, 2001. Partially supported by a grant from THE ISRAEL SCIENCE FOUNDATION

founded by the Israel Academy of Sciences and Humanities, and by the National Science Foundation grant no. DMS-0071503.

2 PÄR KURLBERG AND ZEÉV RUDNICK

For each integer N ≥ 1 (the inverse Planck constant) let U_{N}(A) denote the quantization of A as a unitary operator on H_{N} = L^{2}(Z/NZ). In a previous paper [KR] it was observed that there are quantum symmetries of U_{N}(A) - a commutative group of unitary operators which commute with U_{N}(A). We called these Hecke operators, in analogy with the classical theory of modular forms. The eigenspaces of U_{N}(A) thus admit an orthonormal basis consisting of eigenfunctions of all the Hecke operators, which we called Hecke eigenfunctions. These can be thought of as the eigenfunctions for the desymmetrized quantum map.

In [KR] we showed that the Hecke eigenfunctions become uniformly distributed as N → ∞. In this note we investigate suprema and value distribution of the Hecke eigenfunctions. For general N we obtain a nontrivial bound on the supremum norm of these Hecke eigenfunctions. For prime values of N for which A is diagonalizable modulo N (the “split primes” for A), we obtain much more refined, optimal results via the modern theory of exponential sums. We show that for these values of N, the Hecke eigenfunctions are uniformly bounded and their absolute values (amplitudes) are either constant or have a semi-circle value distribution as N → ∞. Moreover in the latter case different eigenfunctions become statistically independent.

Below is a detailed description of our results.

1.1. Suprema - general N. The trivial bound

||ψ||_{∞} ≤ N^{1/2} (1)

is a consequence of the L^{2}-normalization

||ψ||_{2}^{2} = \frac{1}{N} ∑_{Q∈Z/NZ} |ψ(Q)|^{2} = 1 .

Equidistribution of the eigenfunctions [KR] implies that one can do better: ||ψ||_{∞} = o(N^{1/2}). Our first result gives a quantitative improvement on this:

Theorem 1. Let ψ be a Hecke eigenfunction, normalized by ||ψ||_{2} = 1. Then the supremum of ψ satisfies

||ψ||_{∞} ≪_{ϵ} N^{3/8+ϵ}

for all ϵ > 0, the implied constant depending only on ϵ and not on ψ.

1.2. The split primes. We next consider the case when N is a prime for which A is diagonalizable modulo N (these constitute half the primes). In this case, the group of Hecke operators is isomorphic to the multiplicative group (Z/NZ)^{∗} and the Hecke eigenfunctions correspond to eigencharacters of the group of Hecke operators, namely to Dirichlet characters χ modulo N. For nontrivial χ, the corresponding eigenspace is one-dimensional, while for the trivial character χ_{0} the eigenspace is two-dimensional. For nontrivial χ denote by ψ_{χ,N} a Hecke eigenfunction of norm 1. For simplicity, we also assume that A is not triangular modulo N (which holds for all but finitely many N). The suprema of the Hecke eigenfunctions in this case are given by

Theorem 2. If N is a split prime for A such that A is not triangular modulo N then

1. The Hecke eigenspace corresponding to the trivial character has an orthonormal basis ψ_{0,N}, ψ_{∞,N} for which the amplitude is constant: |ψ_{∞,N}| = 1 and |ψ_{0,N}| = \sqrt{1 − 1/N}.

2. For nontrivial χ mod N, ||ψ_{χ,N}||_{∞} ≤ 2\sqrt{1 − 1/N}.

We finally turn to the question of value distribution. The issue here is only for nontrivial characters χ. We will show that the value distribution of \frac{1}{2}|ψ_{χ,N}| tends to the semi-circle measure, which is the measure μ_{sc} on [0, 1] such that

μ_{sc}(I) = ∫_{I} \frac{4}{π} \sqrt{1 − u^{2}}du

for any interval I ⊂ [0, 1]. This measure is the image of Haar measure on SU(2) under the map g ↦ | tr(g)|/2.

The precise result is:

Theorem 3. If N is restricted to vary only over split primes for A then

1. For any nontrivial character χ modulo N, the amplitude |ψ_{χ,N}(t)| has a semi-circle limit distribution as N → ∞, i.e. for any subinterval I ⊂ [0, 1] we have as N → ∞ through split primes for A

\frac{1}{N} #\{t mod N : |ψ_{χ,N}(t)|/2 ∈ I\} → μ_{sc}(I) .

2. For r ≥ 2, and any choice of distinct nontrivial characters χ_{1}, …, χ_{r}, the amplitudes |ψ_{χ_{1},N}|, …, |ψ_{χ_{r},N}| are statistically independent, that is for any choice of subintervals I_{1}, …, I_{r} ⊂ [0, 1] we have

\frac{1}{N} #\{t mod N : |ψ_{χ_{i},N}(t)|/2 ∈ I_{i}, ∀i = 1, …, r\} → ∏_{i=1}^{r} μ_{sc}(I_{I})

as N → ∞ through split primes for A.

Theorem 3 is a consequence of a corresponding theorem by N. Katz [Ka] on the value distribution of certain exponential sums, see Section 5.

4 PÄR KURLBERG AND ZEÉV RUDNICK

1.3. Comparison with Maass forms. To put our results in perspective, we briefly survey the situation in another well studied example in quantum chaology - the eigenfunctions of the hyperbolic Laplacian Δ on modular surfaces. These are the quotient H/Γ of the hyperbolic plane H by a congruence subgroup Γ of the units of a quaternion algebra (in the compact case) or of the modular group SL(2, Z), see Sarnak’s survey [AQC]. In these cases, there is a commuting family of self-adjoint operators which commute with the Laplacian, and thus there is an orthonormal basis ψ_{j}, j = 0, 1, … of L^{2}(H/Γ) consisting of (real-valued) eigenfunctions of the Laplacian (Δψ_{j} + λ_{j}ψ_{j} = 0) and of all the Hecke operators. These are called Maass-Hecke forms. To compare with our result, note that the Laplace eigenvalue λ scales with Planck’s constant h like 1/h^{2}. In the case of the cat map, the inverse Planck constant N equals 1/h.

A general bound for the eigenfunctions of the Laplacian on any compact Riemannian surface gives [Ho]:

||ψ_{j}||_{∞} ≪ λ_{j}^{1/4} ∼ 1/h^{1/2}

which is analogous to the trivial bound (1). Iwaniec and Sarnak [IS] studied the supremum of the Maass-Hecke forms, and showed that for L^{2}-normalized forms one has

||ψ_{j}||_{∞} ≪_{ϵ} \sqrt{λ_{j}}^{5/12+ϵ} ∼ 1/h^{5/12+ϵ}

for all ϵ > 0. This is analogous to our result in Theorem 1.

Unlike in the cat map case, for modular surfaces equidistribution of eigenfunctions is still open, though recent work of Sarnak [S] establishes this for a subsequence of eigenfunctions of “CM type” for congruence subgroups of SL(2, Z), and a recent formula of Watson [W] confirms that it is implied by the Generalized Riemann Hypothesis for certain automorphic L-functions.

Concerning the question of value distribution for modular surfaces, numerical experiments indicate that the Hecke eigenfunctions have a locally Gaussian value distribution [HR, AS]. In the compact case, this means that if ψ is an L^{2}-normalized cusp form with eigenvalue λ > 0, which is an eigenfunction of all Hecke operators, then the measure of the set of z ∈ H/Γ where the amplitude |ψ(z)| < r is asymptotic to \sqrt{2/π} ∫_{0}^{r} e^{−t^{2}/2}dt as λ → ∞. Here “measure” means the hyperbolic measure dxdy/y^{2} on H/Γ normalized to have total area unity. Moreover these experiments indicate that eigenfunctions corresponding to different eigenvalues are statistically independent[HR, AS].

At present, we seem to be far from being able to prove such statements for modular surfaces (not to mention doing it for a generic system). Recently some progress has been made toward Gaussian value distribution: Watson [W] showed that the third moment ∫_{H/Γ} ψ^{3} of the eigenfunctions vanishes as λ → ∞ (all odd moments vanish for the Gaussian distribution), and Sarnak showed that for “CM forms”, the fourth moment agrees with the Gaussian moment, i.e. ∫_{H/Γ} |ψ|^{4} → 3 as λ → ∞ (recall that we normalized the total area of H/Γ to be unity).

Acknowledgments: We thank P. Sarnak for stimulating discussions on the subject matter of this paper and N. Katz for discussions on exponential sums and for providing us with [Ka].

2. Background

The full details on the cat map and its quantization can be found in [KR]. For the reader’s convenience we briefly recall the setup:

2.1. Classical dynamics. The classical dynamics are given by a hyperbolic linear map A ∈ SL(2, Z) so that x = ( \begin{matrix}p \\ q \end{matrix}) ∈ T^{2} ↦ Ax is a symplectic map of the torus. Given an observable f ∈ C^{∞}(T^{2}), the classical evolution defined by A is f ↦ f ∘A, where (f ∘A)(x) = f(Ax).

2.2. Kinematics: The space of states. As the Hilbert space of states, we take distributions ψ(q) on the line R which are periodic in both the position and the momentum representation. This restricts h, Planck’s constant, to take only inverse integer values. With h = 1/N, the space of states, denoted H_{N}, is of dimension N and consists of periodic point-masses at the coordinates q = Q/N, Q ∈ Z. We identify H_{N} with L^{2}(Z/NZ), where the inner product ⟨ · , · ⟩ is given by

⟨ϕ, ψ⟩ = \frac{1}{N} ∑_{QmodN} ϕ(Q) \overline{ψ}(Q).

2.3. Observables: The basic observables are given by the operators T_{N}(n_{1}, n_{2}) acting on ψ ∈ L^{2}(Z/NZ) via:

(T_{N}(n_{1}, n_{2})ψ) (Q) = e^{\frac{iπn_{1}n_{2}}{N}} e(\frac{n_{2}Q}{N})ψ(Q + n_{1}). (2)

For any smooth classical observable f ∈ C^{∞}(T^{2}) with Fourier expansion

f(x) = ∑_{n_{1},n_{2}∈Z} f̂(n_{1}, n_{2})e(n_{1}p + n_{2}q), x = ( \begin{matrix}p \\ q \end{matrix}) ∈ T^{2},

its quantization, Op_{N}(f), is given by

Op_{N}(f) := ∑_{n_{1},n_{2}∈Z} f̂(n_{1}, n_{2})T_{N}(n_{1}, n_{2})

2.4. Dynamics: We let Γ(4, 2N) ⊂ SL(2, Z) be the subgroup of matrices that are congruent to the identity matrix modulo 4 (resp., 2) if N is even (resp., odd). For A ∈ Γ(4, 2N) we can assign unitary operators U_{N}(A), acting on L^{2}(Z/NZ), having the following important properties:

• “Exact Egorov”: For all observables f ∈ C^{∞}(T^{2})

U_{N}(A)^{−1} Op_{N}(f)U_{N}(A) = Op_{N}(f ∘ A).

• The quantization depends only on A modulo 2N: if A, B ∈ Γ(4, 2N) and A ≡ B mod 2N then

U_{N}(A) = U_{N}(B)

• The quantization is multiplicative: if A, B ∈ Γ(4, 2N), then

U_{N}(AB) = U_{N}(A)U_{N}(B) (3)

2.5. Hecke eigenfunctions. If α is an eigenvalue of A, form 𝔒 = Z[α] which is an order in the real quadratic field K = Q(α). (Note that 𝔒 is not necessarily equal to 𝔒_{K}, the full ring of integers in K.) Let v = (v_{1}, v_{2}) ∈ 𝔒^{2} be a vector such that vA = αv. Let I := Z[v_{1}, v_{2}] ⊂ 𝔒. Then I is an 𝔒-ideal, and the matrix of α acting on I by multiplication in the basis v_{1}, v_{2} is precisely A. The choice of basis of I gives an identification

ι : I → Z^{2} . (4)

The action of 𝔒 on the ideal I by multiplication gives a ring homomorphism ι : 𝔒 → Mat_{2}(Z) with the property that the determinant of ι(β), β ∈ 𝔒, is given by N (β), where N : Q(α) → Q is the norm map.

Reducing the norm map modulo 2N gives a well defined map

N_{2N} : 𝔒/2N𝔒 → Z/2NZ,

and we let C_{A}^{θ}(2N) be the elements in the kernel of this map that are congruent to 1 modulo 4𝔒 (resp., 2𝔒) if N is even (resp., odd).

Now, reducing ι modulo 2N gives a map

ι_{2N} : C_{A}^{θ}(2N) → SL_{2}(Z/2NZ).

Since C_{A}^{θ}(2N) is commutative, the properties in section 2.4 imply that

{U_{N}(ι_{2N}(β)) : β ∈ C_{A}^{θ}}

forms a family of commuting operators. Analogously with modular forms, we call these Hecke operators, and functions ψ ∈ H_{N} that are simultaneous eigenfunctions of all the Hecke operators are denoted Hecke eigenfunctions. Note that a Hecke eigenfunction is an eigenfunction of U_{N}(ι_{2N}(α)) = U_{N}(A).

3. Proof of Theorem 1

3.1. Spectral expansions. We first display the intensity |ψ(Q)|^{2} for ψ ∈ H_{N} as an expectation value of an N-dependent observable. Choose f ∈ C_{c}^{∞}(R) so that f(0) = 1, ∫_{−∞}^{∞} f(x) dx = 0, and f is supported in (−1/2, 1/2). The function

G_{N}(x) = N ∑_{k∈Z} f(N(x − k))

is periodic, and its Fourier coefficients are given by

Ĝ_{N}(m) = f̂(\frac{m}{N}),

where f̂(y) = ∫_{−∞}^{∞} f(x)e^{−2πixy}dx is the Fourier transform of f on the line. For Q ∈ Z/NZ, set

g_{N,Q}(p, q) = G_{N}(q − \frac{Q}{N}), (p, q) ∈ T^{2} .

We obtain a function on the torus which is independent of the momentum variable p, and is strongly localized in the position variable q around Q/N.

Lemma 4. Let Q ∈ Z/NZ. Then for all ψ ∈ H_{N}

|ψ(Q)|^{2} = ⟨Op_{N}(g_{N,Q})ψ, ψ⟩

Proof. Recall that

Op_{N}(g_{N,Q}) = ∑_{m,n∈Z} ĝ_{N,Q}(m, n)T_{N}(m, n) .

Since g_{N,Q} is independent of p, we have ĝ_{N,Q}(m, n) = 0 unless m = 0, in which case we have

ĝ_{N,Q}(0, n) = e(−\frac{nQ}{N})Ĝ_{N}(n) = e(−\frac{nQ}{N})f̂(\frac{n}{N}) . (5)

8 PÄR KURLBERG AND ZEÉV RUDNICK

Since T_{N}(0, n)ψ(Q′) = e(nQ′/N)ψ(Q′), we get

Op_{N}(g_{N,Q})ψ(Q′) = ∑_{m,n∈Z} ĝ_{N,Q}(m, n)T_{N}(m, n)ψ(Q′)

= ∑_{n∈Z} ĝ_{N,Q}(0, n)T_{N}(0, n)ψ(Q′)

= ∑_{n∈Z} ĝ_{N,Q}(0, n)e(nQ′/N)ψ(Q′)

= g_{N,Q}(0, Q′)ψ(Q′) = G_{N}(\frac{Q′ − Q}{N})ψ(Q′)

Since the support of G_{N}(x) is contained in (−1/2N, 1/2N) modulo 1, and G_{N}(0) = N, we get

Op_{N}(g_{N,Q})ψ(Q′) = \begin{cases}Nψ(Q) & if Q′ = Q mod N \\ 0 & otherwise. \end{cases}

Hence

⟨Op_{N}(g_{N,Q})ψ, ψ⟩ = \frac{1}{N} ∑_{Q′ mod N} G_{N}(\frac{Q′ − Q}{N})|ψ(Q′)|^{2} = |ψ(Q)|^{2}

Lemma 5. Let ψ_{1}, …, ψ_{N} ∈ H_{N}. Then for all Q ∈ Z/NZ

∑_{j=1}^{N} |ψ_{j}(Q)|^{8} ≤ (∑_{n∈Z} |f̂(\frac{n}{N})| (∑_{j=1}^{N} |⟨T_{N}(0, n)ψ_{j}, ψ_{j}⟩|^{4})^{1/4})^{4}

Proof. For ease of notation we put

t_{j}(n) = |⟨T_{N}(0, n)ψ_{j}, ψ_{j}⟩|.

By the previous lemma,

|ψ_{j}(Q)|^{2} = ⟨Op_{N}(g_{N,Q})ψ_{j}, ψ_{j}⟩

≤ ∑_{n∈Z} |ĝ_{N,Q}(0, n)|t_{j}(n)

= ∑_{n∈Z} |f̂(\frac{n}{N})|t_{j}(n)

on using (5). Thus

\begin{aligned}∑_{j=1}^{N} |ψ_{j}(Q)|^{8} & ≤ ∑_{j=1}^{N} (∑_{n∈Z} |f̂(\frac{n}{N})|t_{j}(n))^{4} \\ & = ∑_{j=1}^{N} ∑_{n_{1},n_{2},n_{3},n_{4}∈Z} ∏_{k=1}^{4} |f̂(\frac{n_{k}}{N})|t_{j}(n_{k}) . \end{aligned}

Applying Cauchy-Schwartz twice we get that

\begin{aligned}∑_{j=1}^{N} t_{j}(n_{1})t_{j}(n_{2})t_{j}(n_{3})t_{j}(n_{4}) & ≤ \sqrt{∑_{j=1}^{N} t_{j}(n_{1})^{2}t_{j}(n_{2})^{2}}\sqrt{∑_{j=1}^{N} t_{j}(n_{3})t_{j}(n_{4})} \\ & ≤ ∏_{k=1}^{4} (∑_{j=1}^{N} t_{j}(n_{k})^{4})^{1/4} \end{aligned}

and hence

\begin{aligned}∑_{j=1}^{N} |ψ_{j}(Q)|^{8} & ≤ ∑_{n_{1},n_{2},n_{3},n_{4}∈Z} ∏_{k=1}^{4} (|f̂(\frac{n_{k}}{N})| (∑_{j=1}^{N} t_{j}(n_{k})^{4})^{1/4}) \\ & = (∑_{n∈Z} |f̂(\frac{n}{N})| (∑_{j=1}^{N} t_{j}(n)^{4})^{1/4})^{4} \end{aligned}

3.2. A counting problem. Recall that we identified the action of the matrix A on Z^{2} with multiplication by its eigenvalue α on the ideal I ⊆ Z[α]. Let ι : I → Z^{2} be the identification given in (4). We will need the following Proposition ([KR, Proposition 11]):

Proposition 6. Fix ν ∈ I. Then for any orthonormal basis of Hecke eigenfunctions ψ_{j},

∑_{j=1}^{N} |⟨T_{N}(ι(ν))ψ_{j}, ψ_{j}⟩|^{4} ≤ \frac{N}{|C_{A}^{θ}(2N)|^{4}} Sol(N, ν)

where Sol(N, ν) is the number of solutions of the equation

ν(β_{1} − β_{2} + β_{3} − β_{4}) ≡ 0 mod N (6)

with β_{1}, …, β_{4} in C_{A}^{θ}(N).

10 PÄR KURLBERG AND ZEÉV RUDNICK

It was also shown that equation (6) has less than N^{2+ϵ} solutions for ν fixed, as N tends to infinity. However, in the current setup, we need to make the dependence on ν more explicit^{1}. Recall that α is an eigenvalue of A, and that N : Q(α) → Q is the norm map.

Lemma 7. We have

∑_{j=1}^{N} |⟨T_{N}(0, n)ψ_{j}, ψ_{j}⟩|^{4} ≤ gcd(n, N)^{2}N^{−1+ϵ}

Proof. It is sufficient to show that equation 6 has less than

gcd(n, N)^{2}N^{2+ϵ}

solutions. We argue as in [KR], section 7, except for a small twist: If (0, 1) ∈ Z^{2} corresponds to ω ∈ I, then (0, n) corresponds to ν = nω since the action is Z-linear. We may thus make ν Galois stable by multiplying by \overline{ω}. Since ω is in Z[α], the number of solutions to

ν(β_{1} − β_{2} + β_{3} − β_{4}) ≡ 0 mod N, β_{i} ∈ C_{A}^{θ}(N)

is bounded by the number of solutions of

nN (ω)(β_{1} − β_{2} + β_{3} − β_{4}) ≡ 0 mod N, β_{i} ∈ C_{A}^{θ}(N)

which in turn is given by the number of solutions to

β_{1} − β_{2} + β_{3} − β_{4} ≡ 0 mod (N/ gcd(N, nN (ω))) , β_{i} ∈ C_{A}^{θ}(N). (7)

Let N′ = N/ gcd(N, nN (ω)). The number of solutions of equation 7 then equals the product of:

• The number of solutions to

β′_{1} − β′_{2} + β′_{3} − β′_{4} ≡ 0 mod N′ β′_{i} ∈ C_{A}^{θ}(N′).

where β′_{1}, …, β′_{4} ranges over all elements in C_{A}^{θ}(N′).

• The number of elements (β_{1}, β_{2}, β_{3}, β_{4}) ∈ C_{A}^{θ}(N)^{4} that reduce to

the same element (β′_{1}, β′_{2}, β′_{3}, β′_{4}) ∈ C_{A}^{θ}(N′)^{4}. From proposition 14 in [KR], applied with ν = 1, it follows that the first term is ≪ (N′)^{2+ϵ}. For the second term, lemma 20 in [KR] gives that the cardinality of the cokernel of the reduction map C_{A}^{θ}(N) → C_{A}^{θ}(N′) is uniformly bounded in N. Hence there are ≪ (C_{A}^{θ}(N)/C_{A}^{θ}(N′))^{4} elements (β_{1}, β_{2}, β_{3}, β_{4}) that reduce to (β′_{1}, β′_{2}, β′_{3}, β′_{4}) modulo N′.

Finally, from lemma 8 in [KR] we have

N^{1−ϵ} ≪ |C_{A}^{θ}(N)| ≪ N^{1+ϵ}

1If n ≡ 0 mod N, then there are about N^{4} solutions; it is essential to control the contribution from such terms.

VALUE DISTRIBUTION FOR QUANTUM MAPS 11

and thus

C_{A}^{θ}(N)/C_{A}^{θ}(N′) ≪ (N/N′)^{1+ϵ}.

The number of solutions is therefore bounded by (N′)^{2+ϵ} × (N/N′)^{4+ϵ} ≪ N^{2+ϵ} gcd(nN (ω), N)^{2+ϵ} ≪ N^{2+2ϵ} gcd(n, N)^{2} .

Lemma 8. For all ϵ > 0,

∑_{n∈Z} |f̂(\frac{n}{N})| gcd(n, N)^{1/2} ≪_{ϵ} N^{1+ϵ}.

Proof. Let R = 1/ϵ. Since f is smooth and compactly supported, |f̂(\frac{n}{N})| ≪ 1 for |n| ≤ N^{1+1/R}, and |f̂(\frac{n}{N})| ≪ \frac{1}{(n/N)^{R}} for |n| ≥ N^{1+1/R}.

Trivially, gcd(n, N)^{1/2} ≤ N, so

∑_{|n|≥N^{1+1/R}} |f̂(\frac{n}{N})| gcd(n, N)^{1/2} ≪ ∑_{|n|≥N^{1+1/R}} \frac{N}{(n/N)^{R}}

= N^{R+1} ∑_{|n|≥N^{1+1/R}} n^{−R}

≪ \frac{N^{R+1}}{(N^{1+1/R})^{R−1}} = N^{1+1/R}

For the sum over small n, we have

∑_{|n|≤N^{1+1/R}} |f̂(\frac{n}{N})| gcd(n, N)^{1/2} ≪ ∑_{|n|≤N^{1+1/R}} gcd(n, N)^{1/2}

≪ N^{1/R} ∑_{n=0}^{N−1} gcd(n, N)^{1/2} .

(8)

We note that

∑_{n=0}^{N−1} gcd(n, N)^{1/2} ≪ N^{1+ϵ} . (9)

Indeed,

∑_{n=0}^{N−1} gcd(n, N)^{1/2} = ∑_{d|N} d^{1/2} ∑_{n≤N,gcd(n,N)=d} 1 ≤ ∑_{d|N} d^{1/2} \frac{N}{d} = N ∑_{d|N} d^{−1/2}.

Now ∑_{d|N} d^{−1/2} is bounded by the number of divisors of N, and hence is ≪ N^{ϵ} for all ϵ > 0.

Therefore from (8) and (9) we get

∑_{|n|≤N^{1+1/R}} |f̂(\frac{n}{N})| gcd(n, N)^{1/2} ≪ ∑_{|n|≤N^{1+1/R}} gcd(n, N)^{1/2} ≪ N^{1/R+1+ϵ} .

3.3. Conclusion of the proof. Let {ψ_{j}}_{i=1}^{N} be an orthonormal basis of H_{N} such that ψ_{1} = ψ, and each ψ_{j} is a Hecke eigenfunction. We will then bound |ψ(Q)|^{8} trivially by the sum ∑_{j=1}^{N} |ψ_{j}(Q)|^{8}.

By lemma 5,

∑_{j=1}^{N} |ψ_{j}(Q)|^{8} ≤ (∑_{n∈Z} |f̂(\frac{n}{N})| (∑_{j=1}^{N} |⟨T_{N}(0, n)ψ_{j}, ψ_{j}⟩|^{4})^{1/4})^{4}

and from Lemma 7 we have

∑_{j=1}^{N} |⟨T_{N}(0, n)ψ_{j}, ψ_{j}⟩|^{4} ≤ gcd(n, N)^{2}N^{−1+ϵ}

Lemma 8 then gives that

\begin{aligned}∑_{j=1}^{N} |ψ_{j}(Q)|^{8} & ≪ N^{−1+ϵ} (∑_{n∈Z} |f̂(\frac{n}{N})| gcd(n, N)^{1/2})^{4} \\ & ≪ N^{−1+ϵ} (N^{1+ϵ})^{4} ≪ N^{3+5ϵ} \end{aligned}

and thus

|ψ(Q)| ≪ N^{3/8+ϵ}

as required.

4. Uniform boundedness for split primes

4.1. Explicit construction of Hecke eigenfunctions. Let N = p be an odd “split” prime, i.e. a prime p which does not divide (tr A)^{2}−4 such that A is diagonalizable modulo p. We also assume that A is not triangular mod p, that is p does not divide any of the off-diagonal entries of A. For such p, we give an explicit construction of the Hecke eigenfunctions (see [DEGI] for an alternative approach to constructing these). This construction will enable us to prove Theorems 2 and 3.

The condition above implies that N = p is an odd prime that splits in K and does not divide the conductor^{2} of 𝔒. Since p does not divide the conductor of 𝔒, we have 𝔒/p𝔒 ≃ 𝔒_{K}/p𝔒_{K}. Moreover, since p splits in K, (𝔒_{K}/p𝔒_{K})^{×} ≃ (Z/pZ)^{×} × (Z/pZ)^{×}, and from lemma 19 in [KR] we get that C_{A}^{θ}(p) ≃ (Z/pZ)^{×}. On the other hand, since p is odd, we have C_{A}^{θ}(2p) ≃ C_{A}^{θ}(p), hence C_{A}^{θ}(2p) is a cyclic group of order p − 1. Let β be the generator of this group, and let B ∈ Γ(4, 2p) be congruent to ι(β) modulo 2p. The Hecke operators are then given by U_{N}(B^{k}). Since the order of B modulo p is p − 1, B can be diagonalized modulo p (all elements in (Z/pZ)^{×} are p−1 roots of unity.) Let (B_{2}, B_{p}) denote the mod 2 and mod p reductions of B. Since B ∈ Γ(4, 2p) we find that B_{2} is the identity matrix, and as B_{p} is diagonalizable there

2Recall that 𝔒 can be written as Z + f𝔒_{K}; the integer f is called the conductor of 𝔒.

exists D, M ∈ Γ(4, 2p) such that D ≡ (\begin{matrix}t & 0 \\ 0 & t^{−1} \end{matrix}) mod 2p is diagonal

and

B ≡ MDM^{−1} mod 2p

Because all matrices lie in Γ(4, 2p), the multiplicativity property (3) implies that U_{N}(B) = U_{N}(M)U_{N}(D)U_{N}(M)^{−1}.

Now, U_{N} is constructed as a tensor product U_{N} = ⊗_{p^{k}||N}U_{p^{k}} and since B_{2} is trivial, we find the action is determined by B_{p}. Recall from section 4 of [KR] that the action of diagonal matrices on ψ ∈ L^{2}(Z/pZ) is given by

(U_{p} (D) ψ) (x) = Λ_{p}(t)ψ(tx)

where Λ_{p}(t) is the quadratic character of (Z/pZ)^{×}. Thus, if χ is a character on (Z/pZ)^{×}, extended to Z/pZ by letting χ(0) = 0, we find that χ is an eigenfunction of U_{p} (D). We also find that δ(x), where δ(0) = 1 and δ(x) = 0 for x ≠ 0, is an eigenfunction.

If f is an eigenfunction of U_{p}(D), then U_{p}(M)f is an eigenfunction of U_{p}(B) since

U_{p}(B)U_{p}(M)f = U_{p}(M)U_{p}(D)U_{p}(M^{−1})U_{p}(M)f = U_{p}(M)U_{p}(D)f

But U_{p}(B) generates the group of Hecke operators, hence any Hecke eigenfunction will either be of the form U_{p}(M)χ, for χ nontrivial, or a linear combination of U_{p}(M)χ_{0} and U_{p}(M)δ for χ_{0} trivial.

4.2. A reduction to exponential sums. We first note that M is not upper triangular modulo p, otherwise the same would hold for B. Since A is a power of B (modulo p), this would imply that A is upper triangular modulo p, which is contrary to our assumption on p. Thus we may use the Bruhat decomposition of SL(2, Z/pZ) to write

M = [\begin{matrix}1 & b_{1} \\ 0 & 1 \end{matrix}] [\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}] [\begin{matrix}1 & b_{2} \\ 0 & 1 \end{matrix}] [\begin{matrix}t & 0 \\ 0 & 1/t \end{matrix}] (10)

for some b_{1}, b_{2}, t (depending on p). From section 4 of [KR], we obtain that for ψ ∈ L^{2}(Z/pZ),

(U_{p}(M)ψ)(x) = Λ_{p}(t)\frac{e_{p}(r_{p}b_{1}x^{2})}{\sqrt{p}} ∑_{y=1}^{p} e_{p}(r_{p}(b_{2}y^{2} + 2xy))ψ(ty) (11)

where r_{p} is the inverse of 2 mod p, and e_{p}(x) := e^{2πix/p}.

4.2.1. The case ψ = U_{p}(M)χ. We begin with the following lemma on exponential sums:

Lemma 9. If r_{p} ≢ 0 mod p then

1. If χ is nontrivial and b_{2} ≠ 0 mod p then

|∑_{y=1}^{p} e_{p}(r_{p}(b_{2}y^{2} + 2xy))χ(y)| ≤ 2\sqrt{p}

2. If χ = χ_{0} is trivial or b_{2} = 0 mod p then

|∑_{y=1}^{p−1} e_{p}(r_{p}(b_{2}y^{2} + 2xy))χ(y)| = \sqrt{p}

Proof. If χ is trivial or b_{2} ≡ 0 mod p then we can express the sum as a classical Gauss sum, and in this case the result is well known. For χ nontrivial, we note that the degree of r_{p}(b_{2}y^{2} + 2xy) is coprime to p, hence we may apply Weil’s bound [Weil] on exponential sums (see [Sch, page 45, Theorem 2G ]) to obtain

|∑_{y=1}^{p} e_{p}(r_{p}(b_{2}y^{2} + 2xy))χ(y)| ≤ 2\sqrt{p}

Note that the bound is independent of the order of χ!

Corollary 10. Let ψ = \sqrt{p/(p − 1)}U_{p}(M)χ. Then ∥ψ∥_{2} = 1, |ψ| ≡ \sqrt{p/(p − 1)} if χ is the trivial character, and for nontrivial characters χ

∥ψ∥_{∞} ≤ 2\sqrt{p/(p − 1)} .

4.2.2. The case f = U_{p}(M)δ.

Lemma 11. Let ψ_{∞,p} = \sqrt{p}U_{p}(M)δ. Then ∥ψ_{∞,p}∥_{2} = 1 = ∥ψ_{∞,p}∥_{∞}.

Proof. ∥ψ_{∞,p}∥_{2} = 1 since ∥δ∥_{2}^{2} = 1/p and U_{p}(M) is unitary. From equation 11 we get

|ψ_{∞,p}(x)| = |\sqrt{p}(U_{p}(M)δ)(x)|

= |\sqrt{p}Λ_{p}(t)\frac{e_{p}(r_{p}b_{1}x^{2})}{\sqrt{p}} ∑_{y=1}^{p} e_{p}(r_{p}(b_{2}y^{2} + 2xy))δ(ty)| = 1

since δ(ty) = 0 unless y = 0. Hence ∥ψ_{∞,p}∥_{∞} = 1.

Theorem 2 follows immediately from Corollary 10 and Lemma 11.

5. Value distribution for split primes

Let p be a split prime for our map A. We assume that A is not triangular mod p. To prove Theorem 3, we again use (11), which says that we can write the normalized Hecke eigenfunctions ψ_{χ,p} for nontrivial χ as

ψ_{χ,p}(x) = \frac{Λ_{p}(t_{p})e_{p}(r_{p}b_{1,p}x^{2})}{\sqrt{p − 1}} ∑_{y mod p} e_{p}(r_{p}(b_{2,p}y^{2} + 2xy))χ(y)

where r_{p} is the inverse of 2 modulo p, Λ_{p} is the unique quadratic character mod p, and t_{p}, b_{1,p} and b_{2,p} come from the Bruhat decomposition (10) of the diagonalizing matrix M_{p} for A.

Note that b_{2,p} ≠ 0 mod p for p as in our assumptions, since otherwise from (10), we find that

M_{p} = [\begin{matrix}−b_{1,p}t_{p} & 1/t_{p} \\ −t_{p} & 0 \end{matrix}]

and consequently the matrix B is upper triangular:

B = MDM^{−1} = [\begin{matrix}1/t_{p} & b_{1,p}(t_{p} − 1/t_{p}) \\ 0 & t_{p} \end{matrix}] .

Since A is a power of B mod p, this implies that A is also upper triangular, contradicting our assumption on p .

Thus we may express the absolute value of ψ_{χ,p} in terms of the exponential sums

H_{p}(χ, R)(t) = ∑_{y mod p} e_{p}(R(y^{2} + ty))χ(y) .

First define (following N. Katz) a normalization F_{p}(χ, R)(t) of these sums as (R ≠ 0 mod p):

F_{p}(χ, R)(t) = −\frac{e_{p}(Rt^{2}/8)}{\sqrt{χ(−1/2)G(R, χ)G(R, Λ_{p})}}H_{p}(χ, R)(t)

where G(R, χ) = ∑_{x mod p} χ(x)e_{p}(Rx) are Gauss sums, and \sqrt{∗} denotes any choice of the square root. Then we have:

|ψ_{χ,p}(x)| = \frac{1}{\sqrt{1 − 1/p}} |F_{p}(χ, r_{p}b_{2,p})(\frac{2x}{b_{2,p}})| . (12)

Concerning the the normalized sums F_{p}(χ, R)(t), Katz proved the following value distribution and statistical independence theorem:

Theorem 12 ([Ka] ). Let p be an odd prime, χ be a nontrivial character mod p and R ≠ 0 mod p. Then

1. The normalized sums F_{p}(χ, R)(t) are real and take values in the interval [−2, 2].

2. As p → ∞, the p numbers {F_{p}(χ, R)(t)/2 : t mod p} become equidistributed in [−1, 1] with respect to the semi-circle measure \frac{2}{π} \sqrt{1 − u^{2}}du.

3. For any r ≥ 2, and a choice of r distinct nontrivial characters χ_{1}, …, χ_{r}, the p vectors

{(F_{p}(χ_{1}, R)(t)/2, …, F_{p}(χ_{r}, R)(t)/2) : t mod p}

become equidistributed in [−1, 1]^{r} with respect to the product of the semi-circle measures.

By virtue of the relation (12) between the normalized eigenfunctions f_{χ} and the normalized sums F_{p}(χ, R), Theorem 3 is an immediate consequence of Katz’s theorem and the following general lemma:

Lemma 13. Let {f_{p}(t), t = 1, …, p} be a sequence of p points in the interval [0, 1] which become equidistributed as p → ∞ with respect to a probability measure ρ(x)dx having a continuous density ρ. Suppose {g_{p}(t) : t = 1, …, p} is another sequence of points so that g_{p}(t) = θ_{p}f_{p}(t), with θ_{p} = 1 + o(1). Then {g_{p}(t) : t = 1, …, p} is also equidistributed in [0, 1] with respect to ρ(x)dx.

We leave the proof of this as a simple exercise for the reader.

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Department of Mathematics, University of Georgia, Athens, GA

30602, U.S.A. (kurlberg@math.uga.edu)

Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel (rudnick@post.tau.ac.il)