New models for the action of Hecke operators in spaces of Maass wave forms.

Ian Kiming

Dept. of Mathematics, Univ. of Copenhagen, Universitetsparken 5, 2100 Copenhagen Ø, Denmark

Abstract

Utilizing the theory of the Poisson transform, we develop some new concrete models for the Hecke theory in a space M_{λ}(N) of Maass forms with eigenvalue 1/4 − λ^{2} on a congruence subgroup Γ_{1}(N). The main result states that if Φ = (ν_{p} | p ∤ N) is a packet of Hecke eigenvalues occurring in M_{λ}(N), then either every ν_{p} is algebraic over a certain field F_{λ} which consists entirely of algebraic numbers if λ = 0, or else Φ will - for some m ∈ ℕ - occur in the first cohomology of a certain space W_{λ,m} which is a space of continuous functions on the unit circle with an action of SL_{2}(ℝ) well-known from the theory of (non-unitary) principal representations of SL_{2}(ℝ).

1 Introduction

We shall fix the following notation throughout the paper: N is a natural number, p always denotes a prime number, Γ is the congruence subgroup Γ_{1}(N) ≤ SL_{2}(ℤ), G := SL_{2}(ℝ), T := ℝ/πℤ, λ is a complex number, and M_{λ}(N) denotes the complex vector space of Maass forms g on Γ with eigenvalue \frac{1}{4} − λ^{2} for the Laplacian ([8]), i.e. real-analytic functions g(x, y) on the upper halfplane ℌ := {x + iy | y > 0} with the following 3 properties:

(i)  −y^{2} (\frac{∂^{2}}{∂x^{2}} + \frac{∂^{2}}{∂y^{2}} ) g = (\frac{1}{4} − λ^{2}) g ,

(ii)  g(γ.z) = g(z), z = x + iy, for all γ ∈ Γ,

where γ. denotes the usual action of γ ∈ Γ on ℌ, (iii) There is a positive number d such that

g(x + iy) = O(y^{d}) for y → +∞

and

g(x + iy) = O(y^{−d}) for y → 0+,

uniformly in x.

As usual, an element g ∈ M_{λ}(N) is said to be a cusp form, if it is bounded as a function on ℌ.

Let Δ_{1}(N) be the set of matrices ( \begin{matrix}a & b \\ c & d \end{matrix} ) with a, b, c, d ∈ ℤ, a ≡ 1 (N),

c ≡ 0 (N), and positive determinant. Then for α ∈ Δ_{1}(N) we have a Hecke operator T_{α} acting on M_{λ}(N): Suppose that ΓαΓ = ⋃_{i} Γα_{i} as a disjoint union, and that g ∈ M_{λ}(N). Then g | T_{α} is the form given by:

(g | T_{α})(z) := (det α)^{−1/2} ∑_{i} g(α_{i}.z) .

For primes p ∤ N we shall denote as usual by T_{p} the Hecke operator belonging

to α = ( \begin{matrix}1 & 0 \\ 0 & p \end{matrix} ). Also, 𝕋 = 𝕋_{N} will denote the Hecke algebra generated over

ℂ by the operators T_{p}, p ∤ N, acting on M_{λ}(N). Thus, 𝕋 is a commutative ℂ-algebra.

If W is a complex vector space which is also a 𝕋-module, and if Φ = (ν_{p} | p ∤ N) is a system of complex numbers, we say that Φ occurs in W, if 𝕋 has a (non-zero) eigenvector in W such that ν_{p} equals the eigenvalue corresponding to T_{p}, for all p ∤ N.

From an arithmetical point of view, the interest in the Hecke theory of the spaces M_{λ}(N) centers around the case λ = 0, because a part of ’Langlands’ philosophy’ predicts that a system Φ = (ν_{p} | p ∤ N) occurs as a system of Hecke eigenvalues in the subspace of cusp forms in M_{0}(N), precisely if there exists an irreducible Galois representation ρ : Gal(ℚ̄/ℚ) → GL_{2}(ℂ) with finite image, Artin conductor dividing N, (det ρ)(complex conjugation) = 1, and such that ν_{p} = Trρ(Frobenius at p), ∀p ∤ N. A consequence of this prediction is the conjecture that for every such system Φ, every ν_{p} is an algebraic integer. In [2], [3], an interesting attempt was made to prove this conjecture; however, as discovered by Henniart [7], the approach was unfortunately irreparably flawed. Thus, the conjecture is - in its full generality - still completely open.

It seems clear that any progress in connection with this problem must involve the development of new and more structured models for the Hecke theory in the spaces M_{λ}(N). The purpose of the present paper is to suggest and initiate the study of some new such models. Here is a rough description of the ideas involved. First we utilize the Poisson transform to transform the space M_{λ}(N) to a space of distributions on the unit circle T, - equipped with a certain on λ depending action of GL_{2}^{+}(ℝ). The basic idea is then - roughly speaking - to study ’point values’ - and the action of Hecke operators on them - of these distributions for points in a certain countable dense subset Ξ ⊆ T. The actual realization of this idea is a bit technical: By representing the space of distributions in question as derivatives of continuous functions, one ultimately ends up - for some k ∈ ℕ_{0} - with a very short exact sequence of (cohomological) Hecke modules

(∗)  H^{0}(Γ, U_{λ,k}) ⟶ M_{λ}(N) ⟶ H^{1}(Γ, V_{λ,k}) ;

here, U_{λ,k} is the space of (k + 1)-tuples of continuous functions on T with absolutely converging Fourier series and equipped with a certain GL_{2}^{+}(ℝ)- action, and V_{λ,k} is a subspace of U_{λ,k} that is shown to have a natural filtration whose successive quotients are isomorphic to spaces W_{λ,m}, 1 ≤ m ≤ k, introduced in definition 3 below: These are spaces consisting of continuous functions on T with a certain growth condition on their Fourier coefficients and equipped with an - on (λ, m) depending - action of GL_{2}^{+}(ℝ) that is easily recognizable from the theory of (non-unitary) principal series representations (of SL_{2}(ℝ)). Apart from some remarks at the end of the paper, we shall make no further references to representation theory of SL_{2}(ℝ) as we shall introduce all concepts in a completely explicit and self-contained manner.

Using (∗) combined with a study of the evaluation of elements of H^{0}(Γ, U_{λ,k}) at points of the above Ξ ⊆ T, and by using the Ash-Stevens ’lifting lemma’ for packages of Hecke eigenvalues ([1]), we are then able to prove the following theorem.

Theorem 1 Let Φ = (ν_{p} | p ∤ N) be a system of Hecke eigenvalues occurring in M_{λ}(N). Then either

(1) Every ν_{p} is algebraic over the field ℚ(λ, \sqrt{n}, n^{λ/2} | n ∈ ℕ), or

(2) Φ occurs in H^{1}(Γ, W_{λ,m}) for some m ∈ ℕ.

2

In this section we prove Theorem 1. We proceed first by recalling some facts about the action of Hecke operators on cohomology of Hecke-modules, then with various preliminary constructions before introducing the spaces W_{λ,m} and proving Theorem 1.

2.1

In the terminology of [1], section 1.1, we will be concerned with the Hecke algebra H of the Hecke pair (Γ, Δ_{1}(N)), so that our 𝕋 introduced above is a subalgebra of H ⊗ ℂ. So, if M is a right ℂΔ_{1}(N)-module, we have a natural right action of 𝕋 on the cohomology groups H^{i}(Γ, M), i ≥ 0, cf. loc. cit.. In particular, we have a right action of 𝕋 on the Γ-fixed points of M that we consistently denote M^{Γ}, and also on H^{1}(Γ, M): Explicitly, if α ∈ Δ_{1}(N) with ΓαΓ = ∪_{i}Γα_{i} (disjoint), the action of the Hecke operator T_{α} on a homogeneous 1-cocycle c: Γ × Γ → M is given by

(c | T_{α})(γ_{0}, γ_{1}) := ∑_{i} c(t_{i}(γ_{0}), t_{i}(γ_{1})) | α_{i} ,

where t_{i}: Γ → Γ is the map determined by the requirements Γα_{i}γ = Γα_{j}, for some j depending on i and γ, and α_{i}γ = t_{i}(γ)α_{j}.

Let us also recall that the right action of 𝕋 commutes with the long exact cohomology sequence associated with a short exact sequence of ℂΔ_{1}(N)- modules.

2.2

In this subsection, we shall consider the Poisson-transformation associated with the upper halfplane ℌ interpreted as the symmetric space SL_{2}(ℝ)/SO_{2}(ℝ), use this to pull back the action of GL_{2}^{+}(ℝ) on Δ-eigenfunctions on ℌ to the space of distributions on T = ℝ/πℤ, and establish an isomorphism as Heckemodules between M_{λ}(N) and a certain space of distributions on T. Let us first introduce the appropriate actions:

For γ = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ GL_{2}^{+}(ℝ) and θ ∈ T, we put:

j(γ, θ) := ((a cos θ + b sin θ)^{2} + (c cos θ + d sin θ)^{2})^{1/2} ,

so that for fixed γ, j(γ, ·) is a C^{∞} function on T. Secondly, we define γ.θ ∈ T by the requirement

(cos γ.θ, sin γ.θ) = ± (\frac{a cos θ + b sin θ}{j(γ, θ)}, \frac{c cos θ + d sin θ}{j(γ, θ)}) ,

where the sign is chosen such that γ.θ ∈ [0, π[. One immediately verifies that (γ, θ) ↦ γ.θ actually defines an action of GL_{2}^{+}(ℝ) on T, and that we have

j(γ_{1}γ_{2}, θ) = j(γ_{1}, γ_{2}.θ)j(γ_{2}, θ).

With this, we see that we have a right action |_{λ} of GL_{2}^{+}(ℝ) on C^{∞}-functions on the torus T given by:

(φ |_{λ} γ)(θ) := (det γ)^{1+λ/2}j(γ, θ)^{−1−λ} · φ(γ.θ) , for φ ∈ C^{∞}(T) .

Definition 1 Define D′_{λ} to be the complex vectorspace of distributions on T with the (on λ depending) right action of GL_{2}^{+}(ℝ) given by:

(Λ |_{λ} γ)φ := Λ(φ |_{λ} γ^{−1})

for Λ ∈ D′_{λ}, φ ∈ C^{∞}(T), and γ ∈ GL_{2}^{+}(ℝ).

With this definition, the space D′_{λ} is also endowed with the structure of a 𝕋-module. Explicitly, if Λ ∈ D′_{λ} and α ∈ Δ_{1}(N) with ΓαΓ = ⋃_{i} Γα_{i} as a disjoint union, then

Λ | T_{α} := ∑_{i}(Λ |_{λ} α_{i}) .

Proposition 1 Suppose that Re(λ) ≥ 0. Then

D′_{λ}^{Γ} ≅ M_{λ}(N)

as 𝕋-modules.

Proof: Consider the maximal compact subgroup

K := SO_{2}(ℝ) = \{r(θ) := ( \begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix} ) | θ ∈] − π, π]\}

in G. We identify ℌ with the symmetric space G/K; explicitly, x + iy ∈ ℌ is identified with the coset g_{x,y}K where

g_{x,y} := ( \begin{matrix}y^{1/2} & xy^{−1/2} \\ 0 & y^{−1/2} \end{matrix} ) .

We shall now utilize the Poisson transform and the Helgason isomorphism associated with this situation. We shall use the particular version given in [6], Chap. IV, Theorem 5, and proceed now with introducing the necessary notation.

Consider the following standard subgroups of G:

A := \{( \begin{matrix}a & 0 \\ 0 & a^{−1} \end{matrix} ) | a ∈ ℝ_{+}\} ,

whose Lie algebra 𝔞 we identify with ℝ, and

N := \{( \begin{matrix}1 & t \\ 0 & 1 \end{matrix} ) | t ∈ ℝ\} , M := \{( \begin{matrix}±1 & 0 \\ 0 & ±1 \end{matrix} )\} ,

so that we have the Iwasawa decomposition G = KAN, and can identify K/M with T. Explicitly, r(θ)M is identified with

θ (mod π) ∈ T,

and the Iwasawa decomposition of an element g = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ G has the

shape

g = r(θ) ( \begin{matrix}u & 0 \\ 0 & u^{−1} \end{matrix} ) ( \begin{matrix}1 & v \\ 0 & 1 \end{matrix} ) ,

with u := (a^{2} +c^{2})^{1/2} and θ ∈]−π, π] determined by cos θ = a/u, sin θ = c/u. The map h: G ⟶ 𝔞 is the map uniquely determined by the requirement

g ∈ K exp(h(g))N, for g ∈ G.

Explicitly, one finds (with the identification of 𝔞 with ℝ)

h (( \begin{matrix}a & b \\ c & d \end{matrix} )) = log(a^{2} + c^{2})^{1/2}.

Finally, define for g ∈ G:

|g| := trace_{𝔤}(ad(gg^{t})) ,

where g^{t} is the transpose of g, and 𝔤 the Lie algebra of G. One shows that |k_{1}gk_{2}| = |g| for k_{1}, k_{2} ∈ K, so that it makes sense to define the notion of at the most exponential growth for functions f on G/K by the requirement

|f(g)| ≤ a|g|^{b}

with some constants a ∈ ℝ_{+}, b ∈ ℝ.

Now we specialize Chap. IV, 2 of [6], in particular Theorem 5 (ii), to our present situation: We identify elements λ of the space of linear forms on the complexified Lie algebra 𝔞_{ℂ} with complex numbers. The Poisson transform P_{λ} associated with λ is then a linear map from the space D′(K/M) of distributions on K/M to the space E_{λ} of C^{∞} functions on G/K with at the most exponential growth and are eigenfunctions for the Laplace-Beltrami operator on G/K with an eigenvalue that we specify below; P_{λ} is given explicitly by

(∗)  (P_{λ}Λ)(gK) := Λ (kM ↦ e^{(−λ−1)·h(g^{−1}k)} , k ∈ K) ,

for Λ ∈ D′(K/M). Furthermore, Theorem 5 (ii) of loc. cit. implies that P_{λ} is an isomorphism, provided that Re(λ) ≥ 0 which is insured in the present case by assumption. Using the above identification of K/M with T and of G/K with ℌ via x + iy ↔ g_{x,y}K, it is straightforward to verify that we can rewrite (∗) as

(∗∗)  (P_{λ}Λ)(x + iy) = Λ (θ ↦ b(x, y, θ)^{\frac{−1−λ}{2}} , θ ∈ T) ,

where

b(x, y, θ) := y^{−1} (cos^{2} θ + x^{2} sin^{2} θ − 2x cos θ sin θ) + y sin^{2} θ ,

which arises because of

h(g_{x,y}^{−1}r(θ)) = \frac{1}{2} log b(x, y, θ) .

From this, one readily checks that elements of E_{λ} viewed as C^{∞} functions on ℌ have Δ-eigenvalue 1/4−λ^{2}. Furthermore, it is tedious but straightforward to check that we have defined the action of GL_{2}^{+}(ℝ) on D′ precisely so as to make P_{λ} equivariant w.r.t. to GL_{2}^{+}(ℝ)-action. Thus, we may conclude that

D′^{Γ} ≅ E_{λ}^{Γ},

as 𝕋-modules. Hence the proof is concluded by showing that

M_{λ}(N) = E_{λ}^{Γ}.

Now, the growth condition on elements of f ∈ E_{λ} viewed as functions on ℌ requires the existence of constants a ∈ ℝ_{+} and b ∈ ℝ (depending on f) such that

|f(x + iy)| ≤ a|g_{x,y}|^{b},

and as a simple computation shows that

|g_{x,y}| = 1 + 2x^{2}y^{−2} + (y + x^{2}y^{−1})^{2} + y^{−2} ≥ max{y^{2}, y^{−2}},

it is immediately clear that M_{λ}(N) ≤ E_{λ}^{Γ}. On the other hand, one sees that |g_{x,y}| ≤ (const.) · (y^{2} + y^{−2}) for x ∈ [0, 1], so that if f ∈ E_{λ}^{Γ} then certainly for some positive d we have

f(x + iy) = O(y^{d}) as y → +∞

and

f(x + iy) = O(y^{−d}) as y → 0+

uniformly in x ∈ [0, 1]. But since f is invariant under the substitution x ↦ x + 1, this holds uniformly in x ∈ ℝ. Thus, f ∈ M_{λ}(N).

If f is a continuous complex-valued function on T, we shall write f ∼ ∑_{n∈ℤ} a_{n}e^{2inθ} (only) to signal that the Fourier coefficients of f are being denoted by a_{n}, n ∈ ℤ.

If Λ ∈ D′(T), the Fourier coefficients of Λ are

a_{n} := Λ(e^{−2inθ}),

and as is well-known, a given sequence (a_{n}) of complex numbers is the sequence of Fourier coefficients of some distribution Λ ∈ D′(T) if and only if there exists an integer m ≥ 0 such that

∑_{n}(1 + n^{2})^{−m}|a_{n}|^{2} < ∞ ;

if this is the case, there is then an integer s ≥ 0 such that the series

∑_{n}(1 + 2in)^{−s} · a_{n}

converges absolutely. Then

f(θ) := ∑_{n}(1 + 2in)^{−s}a_{n}e^{2inθ}

defines a continuous function on T, and with the notation

∂ := 1 + \frac{d}{dθ} ,

we have Λ = ∂^{s}f in the usual sense, i.e.

Λφ = \frac{1}{π} ∫_{T} f(θ)(∂^{s}φ)(θ)dθ

for φ ∈ C^{∞}(T). It will be convenient for us to consider distributions representable in slightly more general form

(†)  Λ = ∑_{j=0}^{s} ∂^{j}f_{j} ,

where the f_{j} are continuous functions on T with absolutely converging Fourier series. We now define some auxiliary objects, and then the spaces occurring in Theorem 1.

Definition 2 For s ∈ ℕ_{0}, denote by D′_{λ,s} the subspace of D′(T) consisting of distributions Λ representable in the form (†) and with the right GL_{2}^{+}(ℝ)- action given in Definition 1. Define also U_{λ,s} to be the complex vector space consisting of (s + 1)-tuples (f_{0}, …, f_{s}) of continuous functions on T with absolutely converging Fourier series, and define V_{λ,s} to be the subspace of U_{λ,s} consisting of tuples (f_{0}, …, f_{s}) with

∑_{j} ∂^{j}f_{j} = 0

in the distribution sense. Because of Proposition 1 and [8], Satz 5, the space D′_{λ}^{Γ} is finite-dimensional. Consequently, there exists a non-negative integer k such that

D′_{λ}^{Γ} ≤ D′_{λ,k} ,

so that obviously,

D′_{λ,k}^{Γ} = D′_{λ}^{Γ}.

Definition 3 Let m ∈ ℕ. We define W_{λ,m} as the complex vector space consisting of continuous functions f ∼ ∑_{n∈ℤ} a_{n}e^{2inθ} on T with

∑_{n∈ℕ}|n||a_{n}| < ∞ ,

and with the following on (λ, m) depending right action of GL_{2}^{+}(ℝ):

(f |_{λ,m} γ)(θ) := f(γ.θ)(det γ)^{−m−λ/2}j(γ, θ)^{(2m−1)+λ}.

Using the fact that j(γ, ·) ∈ C^{∞}(T), we have j(γ, ·) ∈ W_{λ,m} for every m ∈ ℕ, and it is easily verified that · |_{λ,m} γ actually maps W_{λ,m} into itself so that the definition makes sense.

The definitions of U_{λ,s} and V_{λ,s} are such that we have a natural exact sequence of complex vector spaces

0 ⟶ V_{λ,s} ⟶ U_{λ,s} ⟶ D′_{λ,s} ⟶ 0 ,

where the map U_{λ,s} ⟶ D′_{λ,s} is given by (f_{0}, …, f_{s}) ↦ ∑_{j} ∂^{j}f_{j}. In the next subsection we lift the action of GL_{2}^{+}(ℝ) on D′_{λ,s} to the space U_{λ,s} so that this sequence becomes an exact sequence of 𝕋-modules, and we show that the W_{λ,m} appear as subquotients of the modules V_{λ,s}. In subsection 2.4, we prove a theorem concerning eigenvalues of Hecke operators acting on the space U_{λ,s}^{Γ} of Γ-fixed points of U_{λ,s}. With this preparation, we then proceed in subsection 2.5 to finish the proof of Theorem 1.

2.3

Lemma 1 For each s ∈ ℕ_{0} and to each γ ∈ GL_{2}^{+}(ℝ) there exist on λ depending, uniquely determined C^{∞} functions u_{γ}^{s,j}(θ), j ∈ ℤ, on T with u_{γ}^{s,j} = 0 for j < 0 and for j > s, and with the following properties.

(1) For all φ ∈ C^{∞}(T) we have

∂^{s}(φ |_{λ} γ)(θ) = (det γ)^{1+λ/2} ∑_{j=0}^{s} u_{γ}^{s,j}(θ) · (∂^{j}φ)(γ.θ) .

(2) For all s,

u_{γ}^{s,s}(θ) = j(γ, θ)^{−1−λ−2s} · (det γ)^{s}.

(3) For all s and j = 0, … s, the function j(γ, θ)^{λ}u_{γ}^{s,j}(θ) is a polynomial in the functions

\frac{d^{ℓ}}{dθ^{ℓ}} j(γ, θ) , ℓ = 0, …, s − j,

with coefficients in ℤ[λ, (det γ)].

(4) For γ_{1}, γ_{2} ∈ GL_{2}^{+}(ℝ),

u_{γ_{1}γ_{2}}^{s,j}(θ) = ∑_{ℓ=j}^{s} u_{γ_{2}}^{s,ℓ}(θ) · u_{γ_{1}}^{ℓ,j}(γ_{2}.θ).

Proof: This is a completely trivial exercise in differential calculus so we shall be very brief. Define first u_{γ}^{s,j}(θ) := 0 for j < 0 and for j > s. Put u_{γ}^{0,0}(θ) := j(γ, θ)^{−1−λ}, and recursively (w.r.t. s)

u_{γ}^{s+1,j}(θ) := ∂u_{γ}^{s,j}(θ) + (det γ) · j(γ, θ)^{−2} (u_{γ}^{s,j−1}(θ) − u_{γ}^{s,j}(θ)) ,

for j = 0, … s + 1. Then (2) and (3) are immediately clear, and (1) is easily proved by induction on s using the relation ∂(fg) = −fg + f(∂g) + (∂f)g and that

\frac{d(γ.θ)}{dθ} = (det γ) · j(γ, θ)^{−2}.

Notice that the case s = 0 in (1) is merely the definition of the action |_{λ} γ on C^{∞} functions. Uniqueness for a given γ of functions u_{γ}^{s,j}(θ) with property (1), is shown by assuming functions w_{γ}^{s,j}(θ) given with

0 = ∑_{j=0}^{s} w_{γ}^{s,j}(θ) · (∂^{j}φ)(γ.θ)

for all φ ∈ C^{∞}(T). Using this on test functions φ(θ) = e^{2imθ}, m ∈ ℕ, changing variables θ ↦ γ^{−1}.θ, and letting m → ∞, one obtains successively w_{γ}^{s,s} = 0, …, w_{γ}^{s,0} = 0.

Finally, uniqueness for each fixed γ of functions u_{γ}^{s,j} with (1) proves (4): Applying ∂^{s} to the a function φ |_{λ} γ_{1}γ_{2} = (φ |_{λ} γ_{1}) |_{λ} γ_{2}, we see that (1) holds for γ = γ_{1}γ_{2} if u_{γ_{1}γ_{2}}^{s,j} is replaced by the function

∑_{ℓ=0}^{s} u_{γ_{2}}^{s,ℓ}(θ) · u_{γ_{1}}^{ℓ,j}(γ_{2}.θ) = ∑_{ℓ=j}^{s} u_{γ_{2}}^{s,ℓ}(θ) · u_{γ_{1}}^{ℓ,j}(γ_{2}.θ).

By uniqueness, (4) follows.

Definition 4 Define for s ∈ ℕ_{0} and γ ∈ GL_{2}^{+}(ℝ) the (on λ depending) (s + 1) × (s + 1) matrix function

A_{γ}^{(s)}(θ) := ((det γ)^{−λ/2}j(γ, θ)^{−2}u_{γ^{−1}}^{μ,ν}(γ.θ))_{0≤μ,ν≤s} ,

θ ∈ T, where the u_{γ^{−1}}^{μ,ν} are the uniquely determined functions from Lemma 1. Define also for (f_{0}, …, f_{s}) ∈ U_{λ,s}

(f_{0}, …, f_{s}) |_{λ} γ = (f̃_{0}, …, f̃_{s}),

where

(f̃_{0}(θ), …, f̃_{s}(θ) := (f_{0}(γ.θ), …, f_{s}(γ.θ)) · A_{γ}^{(s)}(θ),

and the linear map h_{s}: U_{λ,s} ⟶ D′_{λ,s} by

h_{s}(f_{0}, …, f_{s}) := ∑_{j} ∂^{j}f_{j}.

Proposition 2 (1) The map |_{λ} γ defines a right GL_{2}^{+}(ℝ)-action on U_{λ,s} such that h_{s} is a homomorphism of GL_{2}^{+}(ℝ)-modules.

(2) If s > 0, the natural injection ι_{s−1}: U_{λ,s−1} ⟶ U_{λ,s} given by

(f_{0}, …, f_{s−1}) ↦ (f_{0}, …, f_{s−1}, 0)

is a GL_{2}^{+}(ℝ)-homomorphism that injects V_{λ,s−1} into V_{λ,s}.

(3) For s > 0, if V_{λ,s−1} is viewed as a GL_{2}^{+}(ℝ)-submodule of V_{λ,s}, we have

V_{λ,s}/V_{λ,s−1} ≅ W_{λ,s}

as GL_{2}^{+}(ℝ)-modules, and hence also as 𝕋-modules.

Proof: (1) Notice first that A_{γ}^{(s)}(θ) has C^{∞} coefficients so that |_{λ} γ actually maps U_{λ,s} into itself. That we have thus defined a right GL_{2}^{+}(ℝ)-action follows from the formula

A_{γ_{1}γ_{2}}^{(s)}(θ) = A_{γ_{1}}^{(s)}(γ_{2}.θ)A_{γ_{2}}^{(s)}(θ)

which is easily verified on the basis of Lemma 1, (4).

To see that h_{s} commutes with the action of GL_{2}^{+}(ℝ), suppose that F := (f_{0}, …, f_{s}) ∈ U_{λ,s}, put Λ := h_{s}(F), and suppose that φ ∈ C^{∞}(T). Then by definition of |_{λ} γ and A_{γ}^{(s)}(θ), the function h_{s}(F |_{λ} γ)φ equals

\frac{1}{π} ∫_{T} ∑_{ℓ,j} (f_{j}(γ.θ) · (det γ)^{−λ/2}j(γ, θ)^{−2}u_{γ^{−1}}^{j,ℓ}(γ.θ)) · (∂^{ℓ}φ)(θ)dθ,

where the summation is over ℓ, j = 0, …, s. Making the change of variables θ ↦ γ^{−1}.θ, using \frac{d(γ.θ)}{dθ} = (det γ) · j(γ, θ)^{−2}, and remembering the properties of the u_{γ^{−1}}^{j,ℓ}, we find that this integral equals

Λ(φ |_{λ} γ^{−1}) = (Λ |_{λ} γ)φ.

(2) That ι_{s−1} is a GL_{2}^{+}(ℝ)-homomorphism is a consequence of the definition of the action of GL_{2}^{+}(ℝ) combined with the observation that the matrix A_{γ}^{(s)}(θ) is an upper triangular (s + 1) × (s + 1) matrix whose upper left s × s minor coincides with A_{γ}^{(s−1)}(θ). That ι_{s−1} maps V_{λ,s−1} into V_{λ,s} is trivial.

(3) Suppose that s > 0 and that (f_{0}, …, f_{s}) ∈ V_{λ,s}. Denote the Fourier coefficients of f_{j} by a_{n}(f_{j}), n ∈ ℤ. By definition of V_{λ,s}, the distribution Λ = ∑_{j} ∂^{j}f_{j} ∈ D′(T) is 0, and hence every Fourier coefficient a_{n}(Λ) of Λ vanishes. But we have

a_{n}(Λ) = ∑_{j=0}^{s}(1 + 2in)^{j}a_{n}(f_{j}),

hence

(1 + 2in)a_{n}(f_{s}) = − ∑_{j=0}^{s−1}(1 + 2in)^{j−s+1}a_{n}(f_{j})

for each n ∈ ℤ. As each of the series ∑_{n} a_{n}(f_{j}) is absolutely convergent, the same holds for the series ∑_{n}(1 + 2in)a_{n}(f_{s}). It follows that f_{s} lies in the space W_{λ,s}. Consequently, we can define a linear map ψ_{s}: V_{λ,s} ⟶ W_{λ,s} by

ψ_{s}(f_{0}, …, f_{s}) := f_{s}.

Then clearly the kernel of ψ_{s} is V_{λ,s−1} viewed as a subspace of V_{λ,s} via ι_{s−1}. We claim that ψ_{s} is surjective. Suppose that f ∈ W_{λ,s} with Fourier coefficients a_{n}(f). Then if we define the numbers

a_{n} := −(1 + 2in)a_{n}(f)

for n ∈ ℤ, the series ∑_{n} a_{n} converges absolutely. There is thus a continuous function g on T with Fourier coefficients a_{n}(g) = a_{n}, and F := (0, …, 0, g, f) is an element of U_{λ,s}. The Fourier coefficients of the distribution

∂^{s−1}g + ∂^{s}f

are all 0, so F is in fact an element of V_{λ,s}. But ψ_{s}(F) = f.

It remains to be seen that ψ_{s} commutes with the action of the group GL_{2}^{+}(ℝ). Let f := (f_{0}, …, f_{s}) ∈ V_{λ,s} and let γ ∈ GL_{2}^{+}(ℝ). By the definition of f |_{λ} γ we find

f |_{λ} γ = (…, f_{s}(γ.θ)(det γ)^{−λ/2}j(γ, θ)^{−2}u_{γ^{−1}}^{s,s}(γ.θ)).

Together with Lemma 1 (2) and j(γ^{−1}, γ.θ) = j(γ, θ)^{−1}, this gives

(ψ_{s}(f |_{λ} γ))(θ) = f_{s}(γ.θ)(det γ)^{−s−λ/2}j(γ, θ)^{2s−1+λ},

which is precisely the definition of f_{s} |_{λ,s} γ for f_{s} ∈ W_{λ,s}.

2.4

The purpose of this subsection is to prove a statement concerning possible eigenvalues of Hecke operators acting on the spaces U_{λ,s}^{Γ}. We proceed with some preparations.

Let us denote by F_{λ} the field occurring in Theorem 1 (1), i.e.,

F_{λ} := ℚ(λ, \sqrt{n}, n^{λ/2} | n ∈ ℕ).

Denote also by Ξ the subset of T = ℝ/πℤ consisting of those θ ∈ T for which cot θ ∈ ℙ^{1}(ℚ), i.e.,

Ξ := {0} ∪ {0 ≠ θ ∈ T | cot θ ∈ ℚ} .

Lemma 2 (1) Ξ is dense in T and stable under the action on T of the group GL_{2}^{+}(ℚ).

(2) Suppose that θ_{0} ∈ Ξ, ℓ ∈ ℕ_{0}, and γ ∈ GL_{2}^{+}(ℚ). Then

\frac{d^{ℓ}}{dθ^{ℓ}} j(γ, θ)_{|θ=θ_{0}} ∈ F_{0}.

(3) Suppose that θ_{0} ∈ Ξ, s ∈ ℕ_{0}, and γ ∈ GL_{2}^{+}(ℚ). Then the matrix

A_{γ}^{(s)}(θ_{0})

(definition 4) is a (s + 1) × (s + 1) matrix with coefficients in F_{λ}.

Proof: (1) The first statement is clear, and the second follows from the formula

cot γ.θ = \frac{a cot θ + b}{c cot θ + d}

for γ ∈ GL_{2}^{+}(ℝ).

(2) Notice first that cos θ, sin θ ∈ F_{0} if θ ∈ Ξ. Now, if ℓ = 0, the statement is clear if sin θ_{0} = 0, and also if sin θ_{0} ≠ 0 since then

j(γ, θ_{0}) = ± sin θ_{0} ((a cot θ_{0} + b)^{2} + (c cot θ_{0} + d)^{2})^{1/2} ∈ F_{0}.

For ℓ ≥ 1, the statement follows from this by induction on ℓ when one shows by induction on ℓ ≥ 1 that

\frac{d^{ℓ}}{dθ^{ℓ}} j(γ, θ) = j(γ, θ)^{−ℓ} · p_{γ,ℓ}(θ),

where p_{γ,ℓ}(θ) is a polynomial with rational coefficients in cos θ, sin θ and the \frac{d^{μ}}{dθ^{μ}} j(γ, θ), μ = 0, …, ℓ − 1.

(3) This follows from (2), the definition of A_{λ}^{(s)}(θ), and from Lemma 1 (3). Now let s ∈ ℕ_{0} and α ∈ Δ_{1}(N). We shall consider the action of the

Hecke operator T_{α} on a space U_{λ,s}^{Γ} = H^{0}(Γ, U_{λ,s}). The action of T_{α} on this space is as a cohomological Hecke operator which we can describe explicitly as follows. Let

ΓαΓ = ∪_{μ=1}^{r}Γα_{μ} , α_{μ} ∈ Δ_{1}(N),

as a disjoint union. Then if f = (f_{0}, …, f_{s}) ∈ U_{λ,s}^{Γ}, we have

(f |_{λ} T_{α})(θ) :=  ∑_{μ=1}^{r}((f_{0}, …, f_{s}) |_{λ} α_{μ})(θ)
= ∑_{μ=1}^{r}(f_{0}(α_{μ}.θ), …, f_{s}(α_{μ}.θ))A_{α_{μ}}^{(s)}(θ).

Let us also as usual denote by Γ_{∞} := ⟨± ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )⟩ the stabilizer in Γ of

the cusp ∞ ∈ ℙ^{1}(ℚ). Notice that if Γ is viewed as acting on T then Γ_{∞} is precisely the stabilizer of 0 ∈ T. Choose a decomposition of SL_{2}(ℤ) in double cosets w.r.t. (Γ, Γ_{∞}):

SL_{2}(ℤ) = ∪_{ν=1}^{m}Γγ_{ν}Γ_{∞},

so that the set {γ_{ν}} is in 1-1 correspondence with the cusps w.r.t. Γ. We define a linear map ϕ_{s} of U_{λ,s}^{Γ} into Mat_{m,s+1}(ℂ) ≅ ℂ^{m(s+1)} by

ϕ_{s}(f_{0}, …, f_{s}) := ( \begin{matrix}f_{0}(γ_{1}.0) & … & f_{s}(γ_{1}.0) \\ ⋮ & & ⋮ \\ f_{0}(γ_{m}.0) & … & f_{s}(γ_{m}.0) \end{matrix} ) .

Theorem 2 The space U_{λ,s}^{Γ} is finite-dimensional and for any α ∈ Δ_{1}(N), the eigenvalues of the Hecke operator T_{α} acting on U_{λ,s}^{Γ} are algebraic over the field F_{λ}.

Proof: Retain the above notation. We first show that the linear map ϕ_{s} is injective which will prove the first part of the theorem. So, suppose that (f_{0}, …, f_{s}) ∈ U_{λ,s}^{Γ} with

ϕ_{s}(f_{0}, …, f_{s}) = 0.

We must show f_{0} = … = f_{s} = 0. View SL_{2}(ℤ) as acting on T. One finds that SL_{2}(ℤ).0 = Ξ which is dense in T (Lemma 2 (1)). As the f_{j} are continuous it thus suffices to show that f_{0}(g.0) = … = f_{s}(g.0) = 0 for all g ∈ SL_{2}(ℤ). Let then g ∈ SL_{2}(ℤ) and write

g = γ · γ_{ν} · γ_{∞},

with 1 ≤ ν ≤ m, γ ∈ Γ, and γ_{∞} ∈ Γ_{∞}. Then,

(f_{0}(g.0), …, f_{s}(g.0)) =  (f_{0}(γ.(γ_{ν}.0)), …, f_{s}(γ.(γ_{ν}.0)))
= ((f_{0}, …, f_{s}) |_{λ} γ)(γ_{ν}.0) · A_{γ}^{(s)}(γ_{ν}.0)^{−1}
= (f_{0}(γ_{ν}.0), …, f_{s}(γ_{ν}.0))A_{γ}^{(s)}(γ_{ν}.0)^{−1}
=  (0, …, 0).

Secondly we show the existence of an endomorphism t_{α} of ℂ^{m(s+1)} with the following 2 properties:

(i) The diagram

\xymatrix{U_{λ,s}^{Γ} \ar[d]_{T_{α}} \ar[r]^{ϕ_{s}} & ℂ^{m(s+1)} \ar[d]^{t_{α}} \\ U_{λ,s}^{Γ} \ar[r]^{ϕ_{s}} & ℂ^{m(s+1)}}

is commutative. (ii) t_{α} is defined over the field F_{λ} (i.e. given by a matrix with coefficients in F_{λ}).

Together with the injectivity of ϕ_{s}, this will then prove the rest of the theorem.

Let μ ∈ {1, …, r} and ν ∈ {1, …, m}. Now, α_{μ}γ_{ν}.0 ∈ Ξ ⊆ T, and as Ξ = SL_{2}(ℤ).0 there is g_{μ,ν} ∈ SL_{2}(ℤ) such that g_{μ,ν}.0 = α_{μ}γ_{ν}.0. We can write

g_{μ,ν} = β_{μ,ν}γ_{ξ_{μ}(ν)}γ_{∞} ,

where β_{μ,ν} ∈ Γ, ξ_{μ} is some map of {1, …, m} into itself, and γ_{∞} ∈ Γ_{∞}. Then

α_{μ}γ_{ν}.0 = β_{μ,ν}γ_{ξ_{μ}(ν)}.0 .

Define then the endomorphism t_{α} of Mat_{m,s+1}(ℂ) by

t_{α} ( \begin{matrix}x_{1,0} & … & x_{1,s} \\ ⋮ & & ⋮ \\ x_{m,0} & … & x_{m,s} \end{matrix} ) :=

∑_{μ=1}^{r} ( (x_{ξ_{μ}(1),0}, …, x_{ξ_{μ}(1),s})A_{β_{μ,1}}^{(s)}(γ_{ξ_{μ}(1)}.0)^{−1}A_{α_{μ}}^{(s)}(γ_{1}.0) ; ⋮ ; (x_{ξ_{μ}(m),0}, …, x_{ξ_{μ}(m),s})A_{β_{μ,m}}^{(s)}(γ_{ξ_{μ}(m)}.0)^{−1}A_{α_{μ}}^{(s)}(γ_{m}.0) ) .

Then claim (ii) above is clear because of Lemma 2 (3). We proceed to show (i). So, let f = (f_{0}, …, f_{s}) ∈ U_{λ,s}^{Γ}. Then,

ϕ_{s}(f |_{λ} T_{α}) = ∑_{μ=1}^{r} ϕ_{s}(f |_{λ} α_{μ})

=  ∑_{μ=1}^{r} ( ⋮ ; (f_{0}(α_{μ}γ_{ν}.0), …, f_{s}(α_{μ}γ_{ν}.0))A_{α_{μ}}^{(s)}(γ_{ν}.0) ; ⋮ )

= ∑_{μ=1}^{r} ( ⋮ ; (f_{0}(β_{μ,ν}γ_{ξ_{μ}(ν)}.0), …, f_{s}(β_{μ,ν}γ_{ξ_{μ}(ν)}.0))A_{α_{μ}}^{(s)}(γ_{ν}.0) ; ⋮ ) .

As f ∈ U_{λ,s}^{Γ} we have

(f_{0}(β_{μ,ν}.θ), …, f_{s}(β_{μ,ν}.θ)) = (f_{0}(θ), …, f_{s}(θ))A_{β_{μ,ν}}^{(s)}(θ)^{−1},

for all θ ∈ T. Consequently,

ϕ_{s}(f |_{λ} T_{α}) =

∑_{μ=1}^{r} ( ⋮ ; (f_{0}(γ_{ξ_{μ}(ν)}.0), …, f_{s}(γ_{ξ_{μ}(ν)}.0))A_{β_{μ,ν}}^{(s)}(γ_{ξ_{μ}(ν)}.0)^{−1}A_{α_{μ}}^{(s)}(γ_{ν}.0) ; ⋮ )

= t_{α} ( ⋮ ; f_{0}(γ_{ν}.0), …, f_{s}(γ_{ν}.0) ; ⋮ ) = t_{α}ϕ_{s}(f),

as desired.

2.5

Proof of Theorem 1: Fix λ ∈ ℂ with Re(λ) ≥ 0. Recall (Definition 2) that we have fixed a non-negative integer k such that

D′_{λ,k}^{Γ} = D′_{λ}^{Γ} ≅ M_{λ}(N),

where the isomorphism is as 𝕋-modules, cf. Proposition 1. Let Φ = (ν_{p} | p ∤ N) be a system of Hecke eigenvalues occurring in M_{λ}(N) and hence also in D′_{λ,k}^{Γ}. Let 0 ≠ v ∈ D′_{λ,k}^{Γ} be a corresponding 𝕋-eigenvector, i.e. T_{p}v = ν_{p}v for p ∤ N. Recall (Definition 2 and Proposition 2 (1)) that we have a short exact sequence of 𝕋-modules

0 ⟶ V_{λ,k} ⟶ U_{λ,k} ⟶ D′_{λ,k} ⟶ 0 ,

which gives rise to the very short exact sequence of 𝕋-modules

U_{λ,k}^{Γ} ⟶^{α} D′_{λ}^{Γ} ⟶^{β} H^{1}(Γ, V_{λ,k}) ,

with 𝕋 acting as an algebra of cohomological Hecke operators. Suppose that βv = 0 so that v ∈ Im(α). Then Φ occurs in Im(α), and because of Theorem 2 the tautological homomorphism

α: U_{λ,k}^{Γ} ⟶ Im(α)

is a surjective homomorphism of 𝕋-modules that are finite-dimensional as complex vector spaces. Using Proposition 1.2.2 of [1], we may then conclude that Φ occurs in U_{λ,k}^{Γ}. According to Theorem 2 all eigenvalues ν_{p} are then algebraic over the field

F_{λ} := ℚ(λ, \sqrt{n}, n^{λ/2} | n ∈ ℕ).

We then suppose that βv ≠ 0 and will show that Φ occurs in H^{1}(Γ, W_{λ,m}) for some 0 ≤ m ≤ k. This then finishes the proof of Theorem 1.

Recall that according to Proposition 2 (3) we may consider the spaces V_{λ,m}, 0 ≤ m ≤ k, as a filtration

0 = V_{λ,0} ≤ … V_{λ,m−1} ≤ V_{λ,m} … ≤ V_{λ,k}

of 𝕋-submodules of V_{λ,k} where the successive quotients are isomorphic to the W_{λ,m} as 𝕋-modules. Now, our assumption βv ≠ 0 implies that Φ occurs in a finite-dimensional sub-𝕋-module of H^{1}(Γ, V_{λ,k}) namely β(D′_{λ}^{Γ}); notice, that we must have k ≥ 1. We shall assume that Φ occurs in some finitedimensional, sub-𝕋-module X of H^{1}(Γ, V_{λ,m}) for some m with 1 ≤ m ≤ k and will show that then either Φ occurs in H^{1}(Γ, W_{λ,m}) or else in some finitedimensional, sub-𝕋-module of H^{1}(Γ, V_{λ,m−1}). By induction on k, this gives the desired conclusion as V_{λ,0} = 0.

Consider the short exact sequence of 𝕋-modules

0 ⟶ V_{λ,m−1} ⟶ V_{λ,m} ⟶ W_{λ,m} ⟶ 0

coming from Proposition 2 (3). This gives rise to a long exact sequence of 𝕋-modules:

W_{λ,m}^{Γ} ⟶ H^{1}(Γ, V_{λ,m−1}) ⟶^{ϵ} H^{1}(Γ, V_{λ,m}) ⟶^{η} H^{1}(Γ, W_{λ,m}) .

Now, it is easy to see that the space W_{λ,m}^{Γ} is 0: Suppose that f ∈ W_{λ,m}^{Γ}. As f is continuous, f is bounded. However, the definition of j(γ, θ) shows that j(γ_{n}, θ), n ∈ ℕ, is unbounded if 0 ≠ θ ∈ T where γ_{n} denotes the matrix

γ_{n} := ( \begin{matrix}1 & n \\ 0 & 1 \end{matrix} ) ∈ Γ .

So, the definition of the Γ-structure on W_{λ,m} implies that f(θ) vanishes for every θ ≠ 0 and hence for all θ.

Thus, ϵ is an injection and the very short exact sequence

ϵ^{−1}(X) ⟶ X ⟶ η(X)

is an exact sequence of finite-dimensional 𝕋-modules. Applying Proposition 1.2.2 of [1] as above, we conclude that either Φ occurs in ϵ^{−1}(X), and thus a fortiori in H^{1}(Γ, V_{λ,m−1}), or else in η(X) and so also in H^{1}(Γ, W_{λ,m}).

Remarks: The reader will notice that our use of continuous coefficients in the above – as opposed to L^{2}-coefficients – is necessitated by the use we made of the evaluation maps ϕ_{s} in the proof of Theorem 2 above. Thus, the use of continuous coefficients is indispensable for our approach. However, we wish to remark here that the there is a certain price to be paid for this, notably the following.

The author does not have concrete examples of triples (Γ, λ, s) where he can prove that the space U_{λ,s}^{Γ} is actually non-zero. However, given the injectivity of the evaluation map ϕ_{s}, the proof of Theorem 2 shows that any eigenvalue of a Hecke operator T_{α} acting on the space U_{λ,s}^{Γ} is also an eigenvalue of the linear operator t_{α} acting on ℂ^{m(s+1)}; moreover, the eigenvalues of t_{α} can – for any concretely given triple (Γ, λ, s) – be computed numerically. Such numerical experiments seem to indicate that Hecke eigenvalues on the spaces U_{λ,s}^{Γ} are probably not very interesting, and that one could at the most retrieve packages of Hecke eigenvalues which are readily recognizable as belonging to certain standard Eisenstein series. Thus, it would appear that the interesting packages of Hecke eigenvalues should be the ones occurring in the spaces H^{1}(Γ, W_{λ,m}).

There are certain reasons that make it not wholly unreasonable to venture the conjecture that the spaces H^{1}(Γ, W_{λ,m}) are in fact finite-dimensional. For instance, the methods of the papers [5] and more specifically [4], might show the way towards analyzing this question. It will be seen however, that because the W_{λ,m} are spaces of continuous functions, an attempt to use the methods of these papers to approach the question of finite-dimensionality of the H^{1}(Γ, W_{λ,m}) will quickly lead to some serious analytical difficulties. The author hopes to return to these problems in a future paper.

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