1 Lecture III

G = SL(2, ℝ) , Γ = SL(2, ℤ) , y = Γ\G .

1. Our goal is to construct some automorphic forms on y. Let us start with an automorphic form φ on y_{p} and consider a family of automorphic forms on y_{p} u_{s} = φ · y^{s}. It is easy to see that for Res ≫ 0 the Eisenstein operator E(u_{s}) is given by an absolutely convergent series. Hence it defines an automorphic form on y, which we denote E(φ; s). Clearly that this defines a function s → E(φ, s) from ℂ to F^{temp}(y) in the area Res ≫ 1 and this function is holomorphic (see a more detailed discussion of this notion in the appendix). Main theorem. The function E(φ, s) has a meromorphic continuation to the whole plane ℂ of parameter s. For every s this gives us an automorphic form E(φ, s). These forms are called Eisenstein series.

2. First we prove this theorem in the simplest case φ = y^{1/2} (this is the unique up to scalar k-invariant function such that δφ = 0; here δ is a generator of z(L) and δ(y^{1/2+s}) = 2sy^{1/2+s}. What we can tell about the Eisenstein series E(s) := E(φ, s) when Res ≫ 0.

(1) E(s) is a well defined element of F^{temp}(y).

(2)) (i) E(s) is k-invariant. (ii) δ^{2}E(s) = 4s^{2}E(s), where δ^{2} we consider as an element of z(G).

Let us note that if we have any function f on y which satisfies equations 2(i),(ii), then the constant term cf satisfies the same equation and hence

cf = y^{1/2}(α(s)y^{s} + β(s) · y^{−s})

and this is true for any s (for s = 0 we have to modify this a little bit). We will see that the function E(s) for Res ≫ 0 satisfies an additional condition

(3)  α(s) = 1 , i.e. cE(s) = y^{1/2}(y^{s} + β(s)y^{−s})

we will soon show the following

Proposition. Consider the following system of equations for a function f ∈ F^{tem}(y)Ξ(s):

(i) δ^{2}f = 4s^{2}f

(ii) f is K − invariant

(iii) cf = (y^{s}+?y^{−s})y^{1/2}.

Then in the domain U = {s ∈ ℂ | Res > 2} this system has a unique solution.

3. Now I would like to invoke the following very general

Continuation principle. Let V be a topological vector space, S – a connected complex manifold.

Suppose that for every s ∈ S we are given a system of linear (nonhomogeneous) equations Ξ(s) for a vector v ∈ V. We assume that this system of equations is holomorphic in s (see precise definition in the appendix).

Suppose we know that for s in some open non-empty domain U ⊂ S the system Ξ(s) has a unique solution v(s) ∈ V. Then this system has a unique solution v(s) for almost all s and the function s ↦ v(s) extends to a meromorphic function on S with values in V.

This principle seems to be very general. I do not claim that it always holds, but it is applicable in a very broad range of situations.

Clearly that together with the proposition above this principle implies the derived meromorphic continuation of the Eisenstein series E(s).

4. Let us discuss how to make a theorem from the continuation principle.

Definition. Let Ξ(s) be a system of linear equations for a vector in a vector space V which holomorphically depends on point s of a complex manifold S. For every s ∈ S we denote by Sol(Ξ(s)) the subset of vectors v ∈ V which satisfy the system of equations Ξ(s).

Definition. (i) We say that the system Ξ(s) is of finite type if there exists a finite dimensional vector space L and a holomorphic family of morphisms λ(s) : L → V such that for every s ∈ S λ(s)(L) ⊃ Sol(Ξ(s)).

(ii) We say that the system Ξ(s) is of locally finite type if every point s ∈ S has a neighborhood W ∃s such that Ξ(s) is of finite type over this neighborhood.

Theorem. Suppose the system (Ξ(s), V) is of locally finite type. Then the continuation principle holds for Ξ(s).

Proof. Step 1. Reduction to systems of finite type. Suppose we have already shown that the continuation principle holds for systems of finite type in a strong form where “for almost all s” means “for s in an open dense subset”.

Let S_{0} ⊂ S be the set of points t in S such that the system Ξ(s) has a unique solution in some neighborhood of t. Then for any point s ∈ S there is a neighborhood W such that Ξ(s) is of finite type on W, and hence by assumption above S_{0} ∩ W is either empty or dense in W.

But this immediately implies that the closure \overline{S}_{0} is open in S. Since S is connected and S_{0} is not empty this implies that \overline{S}_{0} = S, i.e. S_{0} is dense in S.

Remark. In fact we will show that S\S_{0} is a proper analytic subset of S.

Step 2. Suppose that Ξ(S) is a system of finite type. Fix a finite dimensional space L and a holomorphic family of morphisms λ(s) : L → U such that for all s λ(s)L ⊃ SolΞ(s). Let us show that it is enough to establish continuation principle for the case when λ(S) : L → V is monomorphic for all s. Indeed, let us above a point s_{0} ∈ s such k(s) = dim(λ(s)(L)) is maximal at point s_{0}. Choose a subspace L_{0} ⊂ L of dimension k = k(s_{0}) such that λ(S)L_{0} = λ(S)L. Then it is easy to see that for some open dense subset S_{0} ⊂ S λ(S)L_{0} = λ(S)L and λ(S) is a monomorphism on L_{0}. Replacing L by L_{0} and S by S_{0} we can assume that λ(S) : L → V is mono for all s.

Step 3. Since for all S ∈ S λ(s) : L → U is mono and SolΞ(s) ⊂ L we can replace our space V by L and the system of equations Ξ by an induced system Ξ′(s) on L. Hence we can assume that L = V is finite dimensional.

Choose coordinates x_{1}, …, x_{k} on L. Then our system of equations can be written as some infinite system

∑_{i} a_{iα}(s)x_{i} = b_{α}(s) ,

Pick a point s ∈ U. Since at this point our system has a unique solution, we can find indices α_{1}, …, α_{k} such that the determinant of k × k matrix (a_{i}, α_{i}(s)) is not 0. But then this determinant is not 0 for some open dense subset S_{0} ⊂ S.

Using Cramer’s rule we can solve our k × k subsystem of equations Ξ̃(s) and get a meromorphic vector function s ↦ v(s) which is holomorphic for s ∈ S_{0}.

Note that for s ∈ S_{0} ∩ U our function is the only solution of the system Ξ̃ and hence it coincides with the only solution of the larger system of equations Ξ. Thus our function v(s) is a meromorphic continuation of the solution of the system Ξ on U. This implies that it is a solution of Ξ everywhere on S_{0}.

This shows that on the dense subset S_{0} ⊂ S our function v(s) represents unique solution for Ξ(s). Moreover, we know that v(s) extends to a meromorphic function on the whole set S. Q.E.D.

5. How to check that some system of equations Ξ(s) in a vector space V is of locally finite type. For example, how to see that the equations (i), (ii), (iii) in Proposition 2 above for Eisenstein functions E(s) are of locally finite type.

First we can forget equation (iii). We will show that the system (i), (ii) of linear homogeneous equations is already of locally finite type.

We will present the detailed proof later. But here I would like to make the following metamathematical remark.

Suppose we have a system of linear homogeneous equations Ξ(s) in a space V. If it is of locally finite type then we have (∗) for every s ∈ S the subspace SolΞ(s) ⊂ V is finite dimensional.

So this is a necessary condition. Now suppose we have a proof that for a given holomorphic system of linear equations Ξ(s) the condition (∗) holds. Then I claim that always we can tighten up this proof so that it will show that the system Ξ(s) is of locally finite type. It will require somewhat more refined analysis, but it will be essentially the same proof.

In the case of the Eisenstein series we have already shown that for every s ∈ S the solutions of the system of equations (i), (ii) in Proposition (2) from a finite dimensional subspace of automorphic forms A_{σ_{0}}^{I(s)}. So I would like to emphasize that the proof of the fact that the system Ξ(s) is of locally finite type is a not very interesting technical extension of things we have already done.

6. As we have seen the proof of meromorphic continuation of the function E(s) (and the same will be true in a general case) consists of 2 rather independent parts:

(I) Show that the system Ξ(s) is of locally finite type.

(II) Show that in some non-empty domain U ⊂ S the system Ξ(s) has a unique solution.

We will later describe how to modify the proof of finite dimensionality of spaces of automorphic forms to prove (I). Now I prefer to discuss more interesting uniqueness Part (II).

First of all notice that the equation (iii) in Proposition 2 is not really a linear equation. But we can safely replace it by a linear equation

(iii)′  (δ + 2s)cv = (δ + 2s)y^{s+1/2}

since clearly for s ≠ 0 this equation has exactly the same solutions (in the class of k-invariant functions).

Now how to prove that for Res > 5 our system has a unique solution. One solution is given by the Eisenstein series which converges in this region. Suppose we have two solutions f_{1}, f_{2} and denote by f̃ their difference, f̃ = f_{1} − f_{2}. Then the function f̃ is an automorphic form which satisfies

Df = 4s^{2}f for D = δ^{2}
cf = β · y^{−s}

We have shown before that any automorphic form is closely approximated by its constant term. This implies that the function f̃ lies in L^{2}(X, μ).

Now this function is an eigenfunction of the operator D = δ^{2} which is Casimir operator +1. We know that the Casimir operator on L^{2}(x, μ) is selfadjoint and negative. Hence any L^{2}-eigenfunction of the operator D showed have eigenvalue which is real and ≤ 1. This shows that in the domain Res > 1/2 the system of equations Ξ(s) has unique solution. Remark. Technically it is a little difficult to analyse the unbounded operator Casimir and show that it is selfadjoint and negative. But as usual we can replace this consideration with the consideration of some bounded condition operators Π(α) and get the same result. We will do this in an appendix.

7. In order to finish the proof of meromorphic continuation of the function E(s) we need to prove that in the area of convergence functions E(s) satisfy equation (3) from Section 2. Namely, that

CE(s) = y^{1/2}(y^{s} + β(s)y^{−s}) for some β(s) .

This follows from the following important proposition which gives some information about the composition of the operators C and E. Proposition. In the domain of convergence we can write C ∘ E = Id + I, where I is some operator I : F (y_{s}) → F (y_{s}) which satisfies

(∗)  Iδ = −δI and I is G − invariant

Clearly this proposition implies the formula above.

Proof of Proposition. CE(φ) = ∫_{u∈U/Γ_{U}} u(∑_{γ} γ(φ)du, where the sum is take over Γ/Γ_{p}.

Let us recall that we have a Bruhat decomposition for the group G,

namely G = G_{e} ∪ G_{ω}, where G_{e} = P, G_{ω} = U_{p}ωp with ω = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) –

nontrivial element of the Weyl group. Let us break our sum in the expression for CE into two terms I_{e} + I_{w} corresponding to decomposition Γ = Γ_{e} ∪ Γ_{ω} of elements γ ∈ Γ. It is clear that I_{e}(φ) ≡ φ.

In order to investigate the operator I_{ω} let us choose a system of representatives R of left cosets Γ_{U_{p}}\Γ_{ω}/Γ_{p}. Then clearly we can write I_{ω} = ∑_{v∈R} I_{v}, where I_{v}(φ) = ∫_{U∈U_{p}} Ur(φ)du.

Since the function φ is left U_{p}-invariant and r ∈ U_{ω}P it is easy to see that I_{v}(δφ) ≡ −δI_{v}(φ). Namely, we claim that it is true for arbitrary v ∈ U_{ω}P, not necessarily the element of Γ. In order to prove this, we can multiply the element v by elements of U on the left and on the right. Then we can reduce the claim to the situation when v ∈ ωL_{p}. In this case the formula follows from the fact that I_{r}Π(ℓ) = Π(ℓ^{−1})I_{r} for any element ℓ of the center C(L_{p}), where Π is the action of L_{p} on the space Aff_{p}. This formula is a simple computation.