Modular forms on the orthogonal group

Contents

Chapter I. Integral orthogonal groups 1

1. Bilinear forms 1 2. The orthogonal group 2 3. The Grassmannian 4 4. The orthogonal half plane 6 5. Two examples 11 6. Embedded half planes 14

Chapter I. Orthogonal modular groups 16

1. Discrete sub-groups of real vector spaces 16 2. Lattices 17 3. The discrimant group 18

Chapter II. The Baily Borel compactification 21

1. The simplest case 21 2. Boundary components 23 3. Rational boundary components 28

Chapter I. Basic facts about orthogonal modular forms 33

1. The notion of an orthogonal modular form 33 2. The automorphy factor 35 3. The Koecher principle 37 4. The Siegel operator 40

Chapter I. Constructions of modular forms 44

1. Poincarè series 44 2. Eisenstein series (scetch) 44

Contents II

3. Generators of some modular groups 44 4. Gritsenko’s singular modular form 47 5. Jacobi forms 48 6. Hecke operators, generalities 51 7. Hecke operators for Jacobi forms 52

Chapter I. Integral orthogonal groups

1. Bilinear forms

Let V be a finite dimensional vector space over a field K. A map

B : V × V ⟶ K

is called bilinear if it is linear in both variables. Let e_{1}, …, e_{n} be a basis of V. The matrix

G = ( \begin{matrix}B(e_{1}, e_{1}) & … & B(e_{1}, e_{n}) \\ ⋮ & & ⋮ \\ B(e_{n}, e_{1}) & … & B(e_{n}, e_{n}) \end{matrix} )

is called the Gram matrix of B with respect to the given basis. Let f_{1}, … f_{n} be another basis and if

A = ( \begin{matrix}a_{11} & … & a_{1n} \\ ⋮ & & ⋮ \\ a_{n1} & … & a_{nn} \end{matrix} ) , f_{i} = ∑_{j} a_{ij}e_{j},

then A′GA is the Gram matrix with respect to the new basis.

Bnd 1.1 Remark. For a bilinear form B the following statements are equivalent:

1. B(a, x) = 0 for all x ∈ L ⟹ a = 0.

2. B(x, a) = 0 for all x ∈ L ⟹ a = 0.

3. The Gram matrix with respect to some basis is invertible.

4. The Gram matrix with respect to an arbitrary basis is invertible.

A bilinear from with the properties 1-4 is called non-degenerated.

A bilinear form is called symmetric if B(x, y) = B(y, x). The Gram matrices of symmetric bilinear forms are symmetric matrices. We now assume that the characteristic of K is different from 2. The function

q(x) := \frac{1}{2}B(x, x)

is called the quadratic from associated to the symmetric bilinear form B. It determines B because of the polarization formula

B(x, y) = q(x + y) − q(x) − q(y).

Qur 1.2 Definition. A quadratic space (V, B) is a finite dimensional vector space over a field K of characteristic ≠ 2, equipped with a non-degenerated symmetric bilinear form B.

Two quadratic spaces (V_{i}, B_{i}) over the same field are called isomorphic, if there exists an ismorphism of vector spaces σ : V_{1} → V_{2}, such that B_{2}(σ(x), σ(y)) = B_{1}(x, y).

Gdia 1.3 Lemma. Every quadratic space (V, B) admits a basis such that the corresponding Gram matrix is diagonal.

This diagonal matrix is not unique. Replacing the basis elements e_{i} by α_{i}e_{i} means to replace the diagonal elements d_{i} by d_{i}α_{i}^{2}. This gives:

Cis 1.4 Remark. Two quadratic spaces over the field of complex numbers are isomorphic if their dimensions agree.

If K is the field of real numbers, we cannot change the sign of d_{i} using the transformation d_{i} ↦ d_{i}α_{i}^{2}. Actually one has:

Sig 1.5 Proposition. Let ℝ^{p,q} denote the vector space ℝ^{p+q} equipped with the bilinear form

B(x, y) = x_{1}y_{1} + ⋯ + x_{p}y_{p} − x_{p+1}y_{p+1} − ⋯ − x_{p+q}y_{p+q}.

Every quadratic space V is isomorphic to a ℝ^{(p,q)}. The pair (p, q) is uniquely determined. It is called the signature of V.

2. The orthogonal group

The orthogonal group O(V, q) of a quadratic space (V, q) is the subgroup of automorphims of V which preserve q. We will use the notation

(a, b) := B(a, b), (a, a) = 2q(a).

Orthogonal complements

§2. The orthogonal group 3

Let (V, q) a quadratic space over a field K of characteristic ≠ 2. Let A ⊂ V be a subspace. The orthogonal complement A^{⊥} is the set of all vectors v ∈ V with the property

(a, v) = 0 for all a ∈ A.

From linear algebra follows

dim A + dim A^{⊥} = dim V.

Assume that A is non degenerated (with respect to the restriced bilinear form). Then obviously A ∩ A^{⊥} = 0 and we obtain in this case

A ⊕ A^{⊥} = V.

Reflections

A vector a of a non-isotropic quadratic space is called isotropic if q(a) = 0. If a is non-isotropic then Ka is non-degenerated and we have

V = Ka ⊕ a^{⊥}.

The reflection along a is the automorphism of V, which maps a to −a and which is the identity on the orthogonal complement of a. A closed formula is given by

σ_{a}(x) = x − \frac{B(a, x)}{q(a)}a.

A reflection is an orthogonal transformation and its determinant is −1. An important result states

REf 2.1 Proposition. The orthogonal group O(V, q) is generated by reflections.

A fundamental result of Witt states:

WIT 2.2 Proposition. Let (V, q) be a quadratic space (over a field of characterstic ≠ 2). Let V_{1}, V_{2} be two subspaces and τ : V_{1} → V_{2} an isomorphism of vector spaces which preserves q. Then τ is the restriction of an orthogonal transformation σ ∈ O(V, q).

(We do not assume that (V_{i}, q|V_{i}) are non-degenerated.)

Hyperbolic planes

A hyperbolic plane is a two dimensional quadratic space (V, q) which admit a basis e, f with the property

(e, e) = 0, (f, f) = 0, (e, f) = 1.

The Gram matrix with respect to this basis is

( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

If the ground field is ℝ, the signature of a hyperbolic plane is (1, 1).

HypP 2.3 Lemma. Let (V, q) be a two dimensional quadratic space over the field of real numbers, which is not definit. Then (V, q) is a hyperbolic plane.

The proof is trivial.  ⊔ ⊓

A quadratic space over ℝ of signature (p, q) is p ≤ q is isomorphic to an orthogonal direct sum of hyperbolic planes H and a negative definit space W,

V ≅ H^{p} ⊥ W.

(The orthogonal direct sum of two two quadratic spaces (V_{1}, q_{1}), (V_{2}, q_{2}) is V_{1} ⊕ V_{2} equipped with

(x_{1} + y_{1}, x_{2} + y_{2}) := B_{1}(x_{1}, y_{1}) + B_{2}(x_{2}, y_{2}).

3. The Grassmannian

Let V be a quadratic space of signature (m, n) and G = O(V) the (real) orthogonal group of V. We dente by G the set of all subspaces W ⊂ V of dimension m such that the restriction of the quadratic form to W is positive definite. We simply call such subspaces “positive” and write W > 0. The group G acts on G

G × G → G ; (g, W) ↦ g(W).

This action is transitive. If we fix one W_{0} ∈ G we get a surjective map

G → G , g ↦ g(W_{0}).

For any W ∈ G we consider the orthogonal complement

W^{⊥} = {x ∈ V ; (x, y) = 0 ∀y ∈ W }.

It is clear that the restriction of the quadratic form to W^{⊥} is negative definit and that

V = W ⊕ W^{⊥} (direct sum).

The orthogonal groups O(W) and O(W^{⊥}) are compact. We have a natural embedding

\begin{matrix}O(W) × O(W^{⊥}) ⟶ O(V), \\ (g_{1}, g_{2})(a + b) = g_{1}(a) + g_{2}(b) (a ∈ W, b ∈ W^{⊥}). \end{matrix}

The group K = O(W) × O(W^{⊥}) is compact. One can show that the image of K in G = O(V) is a maximal compact subgroup. We consider the coset space

G/K = {gK ; g ∈ G}.

§3. The Grassmannian 5

We have a natural projection

G → G/K ; g ↦ gK.

Two elements g_{1}, g_{2} ∈ G have the same image if and only if g_{2}^{−1}g_{1} ∈ K. Let g ∈ G be an element which normalizes W, i.e. g(W) = W. Then g normalizes also the orthogonal complement of W and we obtain g ∈ K. (We identify now K = O(W) × O(W^{⊥}) with its image in G).

CoS 3.1 Proposition. Let W ∈ G be a fixed element. The assignment g ↦ g(W) defines a bijection

G/K ⟶^{∼} G ; gK ⟼ g(W).

We equip G/K with the quotient topology, i.e. a set U ⊂ G/K is called open if its inverse image in G is open. We equip G with a structure as topological space such that the map in 3.1 is topological. The map

G × G ⟶ G, (g, W) ⟼ g(W)

is continuous. As a consequence the map

G ⟶^{∼} G ; W ⟼ g(W),

is topological for each g ∈ G.

The group G acts also on the coset space G/K,

G × G/K ⟶ G/K, (g, hK) ⟼ (gh)K.

This action is compatible with the action on G, i.e. the diagram

G × G/K ⟶ G/K
↓  ↓
G × G ⟶ G

commutes. In this connection it is worth while to mention a general result (s. Helgason):

Let G be a topological group which acts continuousely and transitively on a topological space, G × X → X. Assume that G and X are locally compact and with countable basis of the topology. Denote by K the stabilizer of a distinguished point x ∈ X. The bijection

G/K ⟶^{∼} X, gK ⟼ g(x),

is topological.

The hyperbolic space

Let (V, q) now be a real quadratic space of signature (1, n). We consider the set

C = { y ∈ V; (y, y) > 0 }.

The set of one-dimensional sub-spaces of V is the projective space P (V). The Grassmanian G in this case is a subset of P (V), called the hyperbolic space associated to (V, q). It is nothing else but the image of C,

G := { [y] ∈ P (V); (y, y) > 0 }.

HyP 3.2 Lemma. Let (V, q) be a quadratic space of signature (1, n). The set C has two connected components,

C = C^{+} ∪ C^{−}

which are interchanged by y ↦ −y.

Proof. We can assume V = ℝ^{1,n}, q(x) = x_{0}^{2} − x_{1}^{2} − ⋯ − x_{n}^{2}. Then C is given by

x_{0}^{2} > x_{1}^{2} + ⋯ + x_{n}^{2}.
The two components are defined by x_{1} > 0 and x_{1} < 0.  ⊔ ⊓

The special case n = 0 is already interesting. Here C is the punctured line ℝ^{•} = ℝ − {0}.

A subset A ⊂ V is called a cone, if a ∈ A and t ∈ ℝ, t > 0 implies ta ∈ A and if ±a ∈ A implies a = 0.

CoN 3.3 Remark. The two connected components C^{±} are cones.

The orthogonal group O(V) acts on C. The group O′(V), which preserves the two components is a subgroup of index two. This subgroup is open and closed in O(V).

4. The orthogonal half plane

Let (V_{1}, q) be a quadratic space of signature (1, n − 1). Besides the hyperbolic space we also have to consider the tube domin

H := V_{1} + i P ⊂ V_{1}(ℂ) = V_{1} ⊗_{ℝ} ℂ.

Here P denotes one of the two connected components of C. The group O_{1}(V) acts on H as group of biholomorphic transformation. But H admits biholomorphic transformation of very different nature, for example translations

z ⟼ z + a, a ∈ V_{1}.

§4. The orthogonal half plane 7

The group of all biholomorphic transformation of H requires an orthogonal group O(2, n) of a quadratic space V which contains V_{1} as subspace.

We now consider a quadratic space (V, q) of signature (2, n). We want to equip the Grassmanian G with a complex structure in this special case. For this purpose we consider

V (ℂ) = V ⊗_{ℝ} ℂ

and extend (·, ·) to a ℂ-bilinear form

V (ℂ) × V (ℂ) ⟶ ℂ.

We denote this extension also with (·, ·) and the corrsponding quadratic form with q(z) = (z, z)/2. Let P (V (ℂ)) be the projective space of V (ℂ). this is the set of all complex lines in V (ℂ). We have a natural projection

V (ℂ) − {0} ⟶ P (V (ℂ)); z ⟼ [z] := ℂz.

It is well known that P (V (ℂ)) is a compact topological space if we equip it with the quotient topology. Moreover it carries a structure as complex analytic manifold.

We consider the so-called zero quadric.

N := { [z] ∈ P (V (ℂ)), (z, z) = 0 }.

The orthogonal group (O(V (ℂ), q) acts transitively on N by Witts result.

defK 4.1 Definition. The set K ⊂ N is the open part of the zero quadric defined by (z, z̄) > 0

An easy calculation gives:

KGras 4.2 Lemma. For a point z = x + iy ∈ V (ℂ), x, y ∈ V, the following two conditions are equivalent:

1. (z, z) = 0, (z, z̄) > 0.

2. (x, x) = (y, y) > 0 and (x, y) = 0.

The vectors x, y span a plane in V, which can be considered as an element of the Grassmannian G. If a ∈ ℂ is a complex number different from zero then the planes associated to z and az are the same. This gives us a map

K ⟶ G, [z] ⟼ ℝx + ℝy.

We have to investigate how far the point [z] ∈ P (V (ℂ)) is determined by the plane. So let x̃, ỹ be another basis of then plane. We may assume

(x, x) = (y, y) = (x̃, x̃) = (ỹ, ỹ) = 1 (and (x, y) = (x̃, ỹ) = 0).

We have

x̃ = αx + βy, ỹ = γx + δy.

The matrix

M = ( \begin{matrix}α & β \\ γ & δ \end{matrix} )

is an orthogonal matrix (M′M = 1). We assume for a moment that its determinant is 1. Then we can use a basic fact about the group SO(2, ℝ) of all orthogonal 2 × 2-matrices of determinant 1. This group is abelian. The assignment

M ⟼ ξ := α + iβ

defines an isomorphism

SO(2, ℝ) ⟶^{∼} { ξ ∈ ℂ, |ξ| = 1 }.

The complex number ξ in our case has the property

ζz = z̃ (= x̃ + iỹ)

and we obtain

[z] = [z̃].

If the determinant of M is −1 we can write M as a product of a matrix with determinant 1 and the diagonal matrix with entries 1, −1. In this case we obtain

[z̄] = [z̃].

The points [z] and [z̄] are two different points of K. hence we have proved:

embC 4.3 Proposition. The assignment

[z] ↦ W = ℝx + ℝy

defines a surjective map of K into G. The inverse image of an element W ∈ G consists of a pair of conjugate points [z], [z̄].

The tube domain model

Every quadratic space (V, q) of signature (2, n), n > 0 containes a hyperbolic plane H. We can consider the orthogonal decomposition

V = H ⊕ V_{1}.

The signature of V_{1} is (1, n). As in the last section we consider the set

C = { x ∈ V_{1}; q(x) > 0 }

and we have to consider the complex domain

K′ := V_{1} + i C ⊂ V_{1}(ℂ) = V_{1} ⊗_{ℝ} ℂ.

This is an open domain of the complex vector space K′. We will construct a biholomorphic map K′ → K. This map depends on the choice of a basis e_{1}, e_{2} of H, where e_{2} is isotropic, (e_{2}, e_{2}) ≠ 0. Of course (e_{1}, e_{2}) ≠ 0. We use the notation

(z_{1}, z_{2}, Z) := z_{1}e_{1} + z_{2}e_{2} + Z (z_{1}, z_{2} ∈ ℂ, Z ∈ V_{1}(cz)).

Assume Z ∈ V_{1}(ℂ). Then we can determine z_{2} such that

q(1, z_{2}, Z) = 0 namely z_{2} = \frac{−q(e_{1}) − q(Z)}{(e_{1}, e_{2})}.

TUB 4.4 Lemma. Let (V, q) be a real quadratic space of signature (2, n), n > 0, and H ⊂ V a hyperbolic plane,

V = H ⊕ V_{1} orthogonal decomposition.

The assignment

Z ⟼ [1, z_{2}, Z], z_{2} = \frac{−q(e_{1}) − q(Z)}{(e_{1}, e_{2})},

defines a biholomorphic map

K′ = V_{1} + i C ⊂ V_{1}(ℂ) ⟶^{∼} K .

Corollary. The set K has two connected components,

K = K^{+} ∪ K^{−}.

The restricion of the map K → G defines bijections

K^{±} ⟶ G.

The induced bijection K^{+} ↦ K^{−} is given by z ↦ z̄.

The orthogonal group O(V) = O(2, n) of course acts on the domain K as group of biholomorphic mappings There is an open an closed subgroup of index 2

O′(V) ⊂ O(V) (O(V) = O(2, n))

which preserves the two components. and O′(V) acts on the two connected components as group of biholomorphic mappings.

BiT 4.5 Remark. The bijections

K^{±} ⟶ G

are topological.

It is not difficult to give a direct proof.

Usually we will distinguish one connected component of K and denote it by

H := K^{+}.

We want to derive a refined form of .4.4in the case n ≥ 2 in which the two connected components are better visible.

Let (V, q) be a real quadratic space of signature (2, n). We assume now n ≥ 2. Then we can choose a negative definit subspace W ⊂ V of dimension n − 2. The orthogonal complement of W has signature (2, 2). The space W is isomorphic to the orthogonal direct sum of two hyperbolic planes because this sum also has signature (2, 2). Hence we can choose a basis of this complement e_{1}, …, e_{4} ∈ V with the properties

\begin{aligned}(e_{1}, e_{2}) & = 1, \\ (e_{3}, e_{4}) & = 1, \\ (e_{i}, e_{i}) & = 0 (j = 1, …, 4), \\ (e_{i}, e_{j}) & = 0 (i = 1, 2, j = 3, 4). \end{aligned}

We use the following notation for an arbitrary element of V,

(z_{1}, z_{2}, z_{3}, z_{4}, Z) := z_{1}e_{1} + z_{2}e_{2} + z_{3}e_{3} + z_{4}e_{4} + Z

and denote as usual by [z_{1}, z_{2}, z_{3}, z_{4}, Z] the image in the projective space if this element is not zero. We use the embedding

\begin{matrix}ℂ × ℂ × W (ℂ) & ⟶ N , \\ (z_{1}, z_{2}, Z) & ⟼ [z_{1}, z_{2}, 1, ∗, Z], \end{matrix}

Here z_{1}, …, z_{4} are complex numbers and Z is contained in W (ℂ). The entry ∗ is determined such that the whole element is in the zero quadric (−z_{1}z_{2}−q(Z)). Now we apply Lemma .4.4 for the hyperbolic plane H = ℝe_{3} + ℝe_{4} and obtain:

TUBn 4.6 Proposition. Consider the set

K′ := { (z_{1}, z_{2}, Z) ∈ ℂ × ℂ × W (ℂ); y_{1}y_{2} + q(𝔶) > 0 }.

The assignment

(z_{1}, z_{2}, Z) ⟼ [z_{1}, z_{2}, 1, ∗, Z]

defines a biholomorphic map of K′ onto K.

§5. Two examples 11

The condition y_{1}y_{2}+q(𝔶) > 0 implies that y_{1}y_{2} > 0 because q is negative definit on W. Hence the connected components of K′ must be given by by y_{1} > 0 and y_{1} < 0. We define

H′ := { (z_{1}, z_{2}, Z) ∈ ℂ × ℂ × W (ℂ); y_{1}y_{2} + q(𝔶) > 0, y_{1} > 0 }.

aTUBn 4.7 Additional Remark to .4.6. If one replaces e_{1} ↦ −e_{1}, e_{3} ↦ −e_{3}, then the H′ is replaced by the other component of K′. Hence it is no loss of generality to assume that H′ corresponds to H, i.e. the map in .4.6 defines a biholomorphic map

H′ ⟶^{∼} H.

5. Two examples

We consider a quadratic space (V, q) of signature (2,1). We can assume V = ℝ^{3} and q(x_{0}, x_{1}, x_{2}) = (x_{0}x_{2} − x_{1}^{2}). We write

X = ( \begin{matrix}x_{0} & x_{1} \\ ∗ & x_{2} \end{matrix} )

and obtain

q(X) = − det X.

Let A ∈ GL(2, ℝ) be a matrix with det(A) = ±1. Then

X ⟼ AXA′

preserves the determinant and is hence an orthogonal transformation. Hence we obtain a homomorphism

{ A ∈ GL(2, ℝ); det A = ±1 } ⟶^{∼} O(V).

It can be proved that this homomorphism is an isogeny*). Its kernel is {±E} (E denotes the unit-matrix). It is also easy to see that the restriction to SL(2, ℝ) defines an isogeny

SL(2, ℝ) ⟶ O′(V)

again with kernel {±E}. The image is a subgroup of index 2. The full group O′(V) is generated by the image and the transformation x ↦ −x.

*) A homomorphism of topoplogical groups G → H is called am isogeny if its kernel is finite and if its image is an open subgroup of finite index

We define a bijection

P^{1}(ℂ) = ℂ ∪ {∞} ⟶^{∼} N .

topE 5.1 Remark. The assignment

τ ⟼ [ \begin{matrix}τ^{2} & τ \\ τ & 1 \end{matrix} ] , ∞ ⟼ [ \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ] .

defines a biholomorphic map

P^{1}(ℂ) = ℂ ∪ {∞} ⟶^{∼} N .

We describe the action of a matrix

M = ( \begin{matrix}a & b \\ c & d \end{matrix} ) .

uMob 5.2 Remark. The action of SL(2, ℝ) on P^{1}(ℂ) which is compatible with the action of O′(2, 1) on N is given by the usual Moebius transformations

τ ⟼ \frac{aτ + b}{cτ + d}.

Proof. A straightforward computation gives

( \begin{matrix}a & b \\ c & d \end{matrix} ) ( \begin{matrix}τ^{2} & τ \\ τ & 1 \end{matrix} ) ( \begin{matrix}a & c \\ b & d \end{matrix} ) = ( \begin{matrix}∗ & (aτ + b)(cτ + d) \\ ∗ & (cτ + d)^{2} \end{matrix} ) .  ⊔ ⊓

We identify the tube domain H in this model:

haP 5.3 Remark. The set K in the model P^{1}(ℂ) is the complement of the real axis including ∞, i.e. the union of the upper and the lower half-plane.

Usually we choose the connected component H of K such that it corresponds to the usual upper half-plane which we denote by

H = { τ ∈ ℂ; Im τ > 0 } ⟶^{∼} H.

We treat another special case:

Let V be the vector space of all real 2 × 2 matrices equipped with the quadratic form q(X) = − det X. The signature of V is (2, 2). There is a natural homomorphism

SL(2, ℝ) × SL(2, ℝ) ⟶ O(V), ((A, X), X) ⟼ AXB′.

This is an isogeny. The image is the connected component of O(V) which is contained in O′(V). The kernel is an obvious subgroup of order 4. We want to describe the zero quadric N. From classical geometry it is known that a non-singular quadric in P^{3}(ℂ) is biholomorphic equivalent to P^{1}(ℂ) × P^{1}(ℂ). Here is a concrete realization: topZ 5.4 Remark. The map

ℂ × ℂ ⟶ N; (z_{1}, z_{2}) ⟼ ( \begin{matrix}z_{1}z_{2} & z_{1} \\ z_{2} & 1 \end{matrix} )

extends to a biholomorphic map

P^{1}(ℂ) × P^{1}(ℂ) ⟶^{∼} N .

The action of SL(2, ℝ) × SL(2, ℝ) can be computed. A similar calculation as in the proof of .5.2 shows:

vMob 5.5 Remark. The action of SL(2, ℝ) × SL(2, ℝ) on P^{1}(ℂ) × P^{1}(ℂ) which is compatible with the action of O′(2, 2) on N is given by

(τ_{1}, τ_{2}) ⟼ (\frac{a_{1}τ_{1} + b_{1}}{c_{1}τ_{1} + d_{1}} , \frac{a_{2}τ_{2} + b_{2}}{c_{2}τ_{1} + d_{2}} ).

Finally we have to exhibit the halfplane H in this model.

haQ 5.6 Remark. In the model .5.5 the set K is given by the inequality

Im z_{1} · Im z_{2} > 0.

The connected component H can be chosen such that it corresponds to the product of two upper half planes,

H × H ⟶^{∼} H.

Proof. This can be considered as a special case of .4.6. On just has to identify

z_{1}e_{1} + z_{2}e_{2} + z_{3}e_{3} + z_{4}e_{4} ⟷ ( \begin{matrix}−z_{4} & z_{1} \\ z_{2} & z_{3} \end{matrix} )

Then the quadratic form z_{1}z_{2}+z_{3}z_{4} (sum of two hyperbolic planes) corresponds to the negative of the determinant and

z_{1}e_{1} + z_{2}e_{2} + e_{3} − z_{1}z_{2}e_{4} ⟷ ( \begin{matrix}z_{1}z_{2} & z_{1} \\ z_{2} & 1 \end{matrix} ) .

The spinor norm

We explain how the sub-group O′(V) ⊂ O(V) can be understood algebraically:

ReN 5.7 Remark. Let V be a quadratic space of signature (p, q), p ≤ 2, q > 0. The reflection σ_{a} with respect to a vector a is contained in O′(V) if and only if q(a) is negative.

This can be proved by means of the concrete models above. As a corollary we obtain that there is a unique homomorphism O(V) → {±1} which is one on vectors of positive norm (q(a) > 1) and −1 on vectors of negative norm.

This is a special case of the so-called spinor-norm, which we introduce without proof:

SpiN 5.8 Proposition. Let (V, q) be a quadratic space (over a field K of characteristic ≠ 2). We denote by K^{•} the multiplicative group of K and here by (K^{•})^{2} the subgroup of squares. There exists a unique homomorphism

O(V) ⟶ K^{•}/(K^{•})^{2}

which maps the reflection σ_{a} to the square class of a.

This homomorphism is the “spinor norm”. In the case K = ℝ we can identify

ℝ^{•}/(ℝ^{•})^{2} ≅ ℤ/2ℤ,

i.e. the spinor norm is only a sign.

TusK 5.9 Proposition. The connected component O^{∘}(2, n) of the group O(2, n), n > 0, is a normal subgroup of index 4. The factor group is isomorphic to ℤ/2ℤ × ℤ/2ℤ. As a consequence the group O(V) admits 4 characters which are trivial on the connected component, the trivial character, the determinant, the spinor norm and the product of both. One has

O^{∘}(2, n) = O′(2, n) ∩ SO(2, n).

The kernel of this homomorphism is the so-called “spinor kernel”

6. Embedded half planes

Let (V, q) a real quadratic space of signature (2, n) and V_{1} ⊂ V a subspace of signature (2, n_{1}). Then the orthogonal complement of V_{1} in V is a negative definite subspace of W and V_{1} is the orthogonal complement of W. If conversely W is a negative definit subspace of V then its orthogonal complement is of signature (2, n_{1}). We assume 1 < n_{1} < n. We consider the complexification, zero quadric, etc. for V_{1} insted of V and get a chain

H_{1} ⊂ K_{1} ⊂ N_{1} ⊂ P (V_{1}(ℂ)))
∩ ∩ ∩ ∩
H ⊂ K ⊂ N ⊂ P (V (ℂ)))

§6. Embedded half planes 15

The inclusion H_{1} ⊂ H of course demands that we have made compatible choices of the connected components.

We consider the orthogonal group O(V_{1}) as a subgroup of O(V). We extend an orthogonal transformation of V_{1} to an orthogonal transformation of V such that it acts trivial on the orthogonal complement. This defines also an embedding O′(V_{1}) ↪ O′(V) and the embeddings N_{1} ↪ N, H_{1} ↪ H, are O′(V_{1})-equivariant.

We will use this in the following situation. Assume that e_{1}, …, e_{4} are four isotropic vectors, such that (e_{1}, e_{2}) = (e_{3}, e_{4}) = 1 and such that any other two scalar products between the four are zero. Hence H_{1} = ℝe_{1} + ℝe_{2} and H_{2} = ℝe_{3} + e_{4} are two orthogonal hyperbolic planes. There sum

V_{1} = H_{1} + H_{2}

can be identified with the space of 2 × 2-matrices,

(x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3} + x_{4}e_{4}) ⟼ X = ( \begin{matrix}−x_{4} & x_{1} \\ x_{2} & x_{3} \end{matrix} ) .

The given quadratic form is x_{1}x_{2} + x_{3}x_{4} = − det X. Now we can apply the second example above (.5.4–.5.6) to obtain an embedding H × H → H.

zwEmb 6.1 Proposition. Let e_{1}, e_{2}, e_{3}, e_{4} be four isotropic vectors of V such that (e_{1}, e_{2}) = (e_{3}, e_{4}) = 1 and such that any other two scalar products between the four are zero. Then we obtain a homomorphism*)

SL(2, ℝ) × SL(2, ℝ) ⟶ O′(V), ((A, B), X) ⟼ AXB′,

where the the matrix X corresponds to

(x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3} + x_{4}e_{4}) ⟷ X = ( \begin{matrix}−x_{4} & x_{1} \\ x_{2} & x_{3} \end{matrix} )

and embeddings

H × H ⟶ H, P^{1}(ℂ) × P^{1}(ℂ) ⟶ N

which —if the choice of the connected component H has been made properly— is defined by

(z_{1}, z_{2}) ⟼ z_{1}e_{1} + z_{2}e_{2} + e_{3} − z_{1}z_{2}e_{4}.

These maps are compatible with the actions of O′(V) on H and the action of SL(2, ℝ) on H by the usual Moebius transformations.

*) The group SL(2, ℝ) × SL(2, ℝ) acts trivial on the orthogonal complement of the space V_{1} = ℝe_{1} + ⋯ ℝe_{4}. Hence the action has only to be described on V_{1}

Chapter II. Orthogonal modular groups

1. Discrete sub-groups of real vector spaces

A subset S of a topological space X is called discrete if it is closed and if it contains no inner points. In the case that X is a locally compact space this means that the intersection with any compact subset is finite.

A group L is called a finitely generated free abelian group, if there exists an integer n ≥ 0 such that L is isomorphic to ℤ^{n}. The number n is uniquely determined. It is called the rank of L. A system of elements e_{1}, …, e_{n} of L is called a ℤ-basis, if the map

ℤ^{n} ⟶ L, x ⟼ ∑_{i} x_{i}e_{i}

is an isomorphism.

Dis 1.1 Remark. A discrete sub-group L of a real vector space V is a finitely generated free abelian group. The following three conditions are equivalent 1. The rank of L equals the dimension of V.

2. L generates V as ℝ-vector space.

3. A ℤ-basis of L is also a basis of the vector space V.

A very useful consequence of this remark is

Kom 1.2 Corollary. Let V be a finite dimensional real vector space and L a discrete subgroup of maximal rank, n := rank L = dim V. Let M ⊂ V be another discrete subgroup such that M ⊃ tL for a suitable number t ≠ 0. Then M is also a free abelian group of rank n.

Two subgroups A, B of a group G are called commensurable, if there intersection has finite index in A and in B.

Com 1.3 Remark. For two discrete subgroups L, M ⊂ V of a finite dimensional real vector space the following two conditions are eqivalent

1. L and M generate the same ℚ-vector space of V.

2. L and M are commensurable, i.e. L/(L ∩ M) and M/(L ∩ M) are finite groups.

§2. Lattices 17

The notion of “commensurability” is connected with the notion of a ℚ-form of real vector space V. A ℚ-form of V is a ℚ-vector space W ⊂ V such that the dim_{ℚ} W = dim_{ℝ} V and such that V is generated by W as real vector space. If this means: If e_{1}, …, e_{n} is a ℚ-basis of W then it is also a ℝ-basis of V. If we use this basis to construct an isomorphism V ≅ ℝ^{n}, the subspace W will correspond to ℚ^{n}. Another way to express this is to say that the canonical map W ⊗_{ℚ} ℝ → V is an isomorphism.

Qfo 1.4 Remark. Let L be a discrete subgroup of maximal rank of the real vector space V. The ℚsubspace generated by L is a ℚ-form of V. The group L is determined up to commensurability by this ℚ-form.

2. Lattices

Let L be a finitely generated free abelian group A bilinear form B on L is a map

L × L ⟶ ℚ

with the properties

B(x + y, z) = B(x, z) + B(y, z), B(z, x + y) = B(z, x) + B(z, y).

The bilinear form is called symmetric if B(x, y) = B(y, x). It is called nondegenerated if

B(a, x) = 0 for all x ∈ L ⟹ a = 0.

Ten 2.1 Remark. A finitely generated free abelian group L can be considered as an additive subgroup of a real vector space V, such that an arbitrary ℤ-basis of L is also a basis of the ℝ-vector space V. Such a vector space is uniquely determined in the following sense. If W is a second vector space with the same property then there exist a unique ismorphism V → W which is the identity on W.

The standard way to construct such a vector space is to use the tensor product L ⊗_{ℤ} ℝ.

A bilinear form B on L extends to a ℝ-bilinear form on V, which we also denote by B. Hence we can apply the results about quadratic spaces to lattices.

LQnd 2.2 Remark. A bilinear from B on a L is non-degenerated if and only if the extended bilinear form B on V is non-degenerated.

Proof. Assume that B is non-degenerated on L. We claim that B is nondegenerated on V. (The converse direction is trivial.) Assume that a ∈ V is a vector with the property B(a, x) = 0 for all x ∈ V. Then this is true for all x ∈ L. Using a ℤ-basis of L, we can interprete a as an element in ℝ^{n} which solves a system of linear equations with coefficients in ℚ. From linear algebra it is known that such a system has a non-zero rational solution, if it has a nonzero real solution. Hence we may assume a ∈ ℚ^{n}. Multiplying with a common

denominator we can get a ∈ ℤ^{n}.  ⊔ ⊓

Latt 2.3 Definition. A lattice is a finitely generated free abelian group L, which is equipped with a non-degenerated symmetric ℤ-bilinear from

L × L ⟶ ℝ

The usual way to construct lattices is to consider a quadratic space (V, q) and to take a subgroup L ⊂ V which is discrete and which generates V as vector space. A lattice which is obtained in this way simply is called a “lattice in (V, q)”.

Two lattices (L_{i}, B_{i}) are called isomorphic if there exists an isomorphism σ : L_{1} → L_{2} such that B_{2}(σ(x), σ(y)) = B_{1}(x, y). Then the enveloping quadratic spaces are ismorphic too.

If (V, B) is a quadratic space and L ⊂ V a discrete subgroup of maximal rank then L equipped with the restriction of B is a lattice. We call (L, B) a lattice in (V, B) Any lattice (L, B) is isomorphic to a lattice in ℝ^{p,q}, where p, q are uniquely determined. We call (p, q) the signature of the lattice.

3. The discrimant group

Let (L, B) be a lattice embedded into a real vector space V as discrete subgroup of maximal rank n = rank L = dim V. Recall that we extend the bilinear form B as ℝ-bilinear form on V. The dual of L in V is

L′ := { a ∈ V; B(a, x) ∈ ℤ for all x ∈ L }.

This a an additive subgroup of V. But even more:

Dua 3.1 Remark. The dual L′ of a lattice is commensurable with L. The following two conditions are equivalent:

1. L is contained in L′.

2. B(x, y) ∈ ℤ for all x, y ∈ L.

§3. The discrimant group 19

DG 3.2 Definition. A lattice (L, B) is called integral if B(x, y) ∈ ℤ for all x, y ∈ L and even if B(x, x) ∈ 2ℤ for all x ∈ L.

An even lattice is also integral because of the polarization formula

2B(x, y) = B(x + y, x + y) − B(x, x) − B(y, y).

We will consider mostly even lattices. But this is not a serious restricion because one can rescale any lattice to get an even one. Rescaling means to replace the bilinear form B by the new one

B̃(x, y) = tB(x, y)

with a suitable rational rescaling factor t which we always assume to be positive. If L sits inside a vector space V as discrete subgroup of maximal rank we get the same effect (i.e. an isomorphic lattice) if we consider \sqrt{t}L instead of L (now equipped with the restriction of the original bilinear form).

FDG 3.3 Definition. The discriminant group of an integral lattice (L, B) is the finite group

L′/L.

The bilinear form B on an integral lattice has integral values but on L′ the values only are rational.

EQ 3.4 Remark. The bilinear form B of an integral lattice (L, B) induces a map

B̄ : (L′/L) × (L′/L) ⟶ ℚ/ℤ, B̄(x̄, ȳ) := B(x, y) mod ℤ. If furthermore L is even, the quadratic form q(x) := B(x, x)/2 induces a map

q̄: L ⟶ ℚ/ℤ; q̄(x̄) = q(x) mod ℤ.

(We denote by x̄ ∈ L′/L the image of an element x ∈ L′.)

DK 3.5 Definition. Let L be an integral lattice. The discrimant kernel is the kernel of the natural homomorphism

O′(L) ⟶ Aut(L′/L).

This is a subgroup of finite index of O′(L).

For any natural number d we can consider the lattice

L(d) := \sqrt{d}L ⊂ V

instead of L. We have

L(d)′ = (1/d)L′

There is a natural (injective) homomorphism

L′/L ⟶ ((1/d)L′)/L.

The orthogonal groups of L and L(d) are the same (considered as subgroups of GL(V)). We see DS 3.6 Remark. The discriminant group of the rescaled lattice L(d) = \sqrt{d}L is a subgroup of finite index of the discriminant group of the discriminant group of L.

Thus we can construct many subgroups of finite index of O(L).

CG 3.7 Definition. A subgroup Γ ⊂ O′(V) is called a congruence group with respect to L, if there exists a natrual number d such the Γ contains the discrimant group of L(d).

A simple observation states:

SCg 3.8 Remark. The set of congruence groups only depends on the ℚ-form L ⊗_{ℤ} ℚ of V.

A deep theorem states that upto some low dimensional cases every subgroup of finite index of O′(L) is a congruence group. This theorem is true for example if the signature is (2, n), n ≥ 3. A counter example is the elliptic modular group which belongs to n = 1.

Chapter III. The Baily Borel compactification

1. The simplest case

The simplest case is the group Γ = SL(2, ℤ) which acts on the usual upper half plane H by means of the formula

τ ⟼ Mτ = \frac{aτ + b}{cτ + d}.

It is well known that the quotient H/Γ can be compactified by one point and that the result is a Rieman sphere. To extend this compactification also to subgroups of finite index Γ ⊂ SL(2, ℤ), it is convenient to extend the action of SL(2, ℤ). First of all SL(2, ℝ) acts on the whole Riemann sphere ℂ̄ by means of the above formula. The subgroup SL(2, ℚ) acts on

H^{∗} := H ∪ ℚ ∪ {∞}.

The elements of

ℚ̄ := ℚ ∪ {∞}

are called cusps in this connection. A basic result is:

FiC 1.1 Lemma. Let Γ ⊂ SL(2, ℚ) be a subgroup which is commensurable with SL(2, ℤ), then ℚ̄/Γ is a finite set.

It is sufficient to prove this for SL(2, ℤ) and in this special case one has even more:

TraC 1.2 Lemma. The group SL(2, ℤ) acts transitively on ℚ.

The elements of

S_{Γ} := ℚ̄/Γ

are called “cusp classes”. The set X_{Γ} := H^{∗}/Γ is the disjoint union of H/Γ and the set of cusp classes,

X_{Γ} = H/Γ ∪ S_{Γ}.

We introduce now a topolgy on H^{∗} which is not induced by the topology of the Riemann sphere. The typical neighbourhouds of ∞ in H^{∗} are the unions of the sets

U_{C} := { τ ∈ H; Im τ > C } (C > 0).

with {∞}. A basic fact for theese sets is:

UnSt 1.3 Remark. Let M ∈ SL(2, ℚ) a substitution such that M∞ = ∞. (This means c = 0. The system of sets {U_{C}} is mapped by M onto itselves (i.e. M(U_{C}) = U_{C′} for some other C′).

Corollary. Let κ ∈ ℚ be a cusp and M ∈ SL(2, ℚ) be a substitution with the property M∞ = κ. The system of sets

M(U_{C}) ∪ {κ}

is independent of the choice of M.

By the way, M(U_{C}) is a disc in the upper half plane tangent at κ to the real axis.

STe 1.4 Remark. There is a unique topolopy on H^{∗} with the following properties:

1. H is open in H^{∗}

2. The induced topolgy on H is the usual one.

3. If κ is a cusp and M ∈ SL(2, ℚ) a substitution with M∞ = κ then the system of sets {M(U_{C}) ∪ {κ}} is a fundamental system of neighbourhoods of κ.

The basic result of this construction is:

BBe 1.5 Proposition. Let Γ ⊂ SL(2, ℚ) be a subgroup which is commensurable with SL(2, ℤ). The space X_{Γ} equipped with the quotient topology of H^{∗} is compact.

It is not quite trivial that X_{Γ} is a Hausdorff space. We omit this. Then it is easy to see that the natural map X_{Γ′} → X_{Γ} is proper for a subgroup of finite index. Hence it is is sufficient to prove II.1.5 this for Γ = SL(2, ℤ) and in this case it is an immediate consequence of the shape of the fundamental domain of the elliptic modular group. It is enough to know that the set

{ τ ∈ H; Re τ < ε^{−1}, Im τ > ε }

contains a system of representatives of SL(2, ℤ) if ε > 0 is small enough.

We want to determine the local structure of X_{Γ} around a cusp class. It is quite clear that the map H^{∗} → H^{∗}, τ ↦ Mτ is topological for every M ∈ SL(2, ℚ). It induces a toplogical map

X_{Γ} ⟶^{∼} X_{MΓM^{−1}}, (τ ⟼ Mτ).

Therefore it is sufficient to consider the case of the cusp ∞. We have to consider the stabilizor

Γ_{∞} := { M ∈ Γ; M∞ = ∞ }.

§2. Boundary components 23

If M ∈ Γ_{∞} then M is an upper triangular matrix. One can show that the diagonal elements are ±1. We skip this because it is true for subgroups of SL(2, ℤ) which would be general enough for our purpose. Hence M ∈ Γ_{∞} acts as translation

Mτ = τ + a.

The set of all a is a non-zero discrete subgroup of ℤ, hence cyclic. We consider a generator N. The local structure of X_{Γ} is given by

LS 1.6 Lemma. Let C > 0 be big enough. Then the map

U_{C}/Γ_{∞} ∪ [∞] ⟶ X_{Γ}

is an open embedding. The assignement

τ ⟼ e^{2πiτ/C}, ∞ ⟼ 0

maps U_{C}/Γ_{∞} ∪ [∞] topologically to a disc in ℂ around the origin.

It is not difficult to prove that X_{Γ} carries a natural structure as Riemann surface. Every compact Riemann surface carries a structure as algebraic curve. If one removes finitely many points from an algebraic curve one still has an algebaic curve:

The quotient H/Γ carries a structure as algebraic curve.

Let L be a lattice of signature (2, n), n > 0 and Γ ⊂ O′(L) a subgroup of finite index. One may conjecture that H/Γ also carries a natural structure as algebraic variety. To prove this one needs a topological compactifiaction of H/Γ. To get a hint how this can be done we translate the notion of a cusp into the orthogonal framework.

We look again at the exceptional isogeny SL(2, ℝ) → O(2, 1). We have to consider the lattice L of all integral symmetric 2×2-matrices with the quadratic form − det X. We identify the image of the zero quadric in P (L ⊗_{ℤ} ℂ) with P^{1}ℂ. The rational boundary points (elements of ℚ̄) can be identified in this context. They correspond to the one dimensional subspaces of L ⊗_{ℤ} ℂ, which are generated by an isotropic element, i.e.

The cusps correspond to the one-dimensional isotropic subspaces of L ⊗_{ℤ} ℚ.

This is the very special case of signature (2, 1). In the case of signature (2, n), n > 1, the situation is more complicated, beacuse isotropic subspaces of dimension > 1 may occur.

2. Boundary components

Let (V, q) be a quadratic space (over a field K of charcateristic ≠ 2. We want to consider isotropic spaces A ⊂ V, i.e. subspaces such that q is zero on A. From the theorem of Witt I.2.2 follows that two isotropic spaces of the same dimension are equivalent under O(V). From this follows that two maximal isotropic spaces have the same dimension (and are equivalent under O(V)). The dimension r of a maximal isotropic space is an important invariant of (V, q). Sometimes it is called the rank of V.

HyPI 2.1 Lemma. Let F be an isotropic subspace of a quadratic space (V, q) (over a field of characteristic ≠ 2). Then there exists another isotropic space F′ of the same dimension d such that F + F′ is the orthogonal sum of d hyperbolic planes. More precisely: There exist basises e_{1}, …, e_{d} of F and e′_{1}, …, e′_{d} of F′ such that (e_{i}, e′_{i}) = 1 for i = 1, …, d and such that the scalar products of any other two of the 2d vectors e_{1}, …, e_{d}, e′_{1}, …, e′_{d} are zero.

Proof. The proof is given by induction. We restrict to the beginning of the induction. i.e. we treat only the case d = 1. By definition a quadratic space is non-degenerated. Therefore there exists a vector a such that (a, e_{1}) ≠ 0. A suitable linear combination te_{1} + sa has the desired property. ⊔ ⊓

As a consequence of .2.1 th inequality 2 rank(V) ≤ dim V holds. It is easy to see that the rank of a complex quadratic space (V, q) of dimension d is

r = \{ \begin{matrix}d/2 & if d is even \\ (d − 1)/2 & if d is odd. \end{matrix}

Another simple consequence is that the rank of a real quadratic space of signature (p, q) is |q − p|.

By the way the rank of (V, q) is the same as of (V, −q).

We have to consider lattices (L, q). The quadratic spaces L⊗_{ℤ} ℚ and L⊗_{ℤ} ℝ can be different. What one can say immediately is

rank(L ⊗_{ℤ} ℚ, q) ≤ rank(L ⊗_{ℤ} ℝ, q).

Therefore we use sometimes the notation K-rank instead of K to make clear which ground field is in consideration.

A simple example is given by the ternary form q(x_{1}, x_{2}, x_{3}) = x_{1}^{2} + x_{2}^{2} − 7x_{2}^{2}. The signature of this form is (2, 1). Hence the rank of this quadratic form considered on ℝ^{3} is 1. But there is no non-zero rational solution of x_{1}^{2} + x_{2}^{2} = 7x_{2}^{2}. Hence the rank of the rational vector space ℚ^{2} equipped with this quadratic form is 0.

This is a very important case but it is nevertheless an exceptional because of IsG 2.2 Proposition. Let (V, q) be a quadratic space over ℚ such that the signature of V ⊗_{ℚ} ℝ is (2, n). Assume n ≥ 5. Then the ℚ-rank of (V, q) is 2.

This is an easy consequence of a famous result of Hermite:

HER 2.3 Proposition. Let q(x_{1}, …, x_{n}) be a rational quadratic form on ℚ^{n} which is not definite. Assume n ≥ 5. Then there exists a non-trivial solution of q(x) = 0 (x ∈ ℚ^{n}, x ≠ 0).

We want to introduce boundary components now and consider a quadratic space (V, q) of signature (2, n), n > 0. Recall that we introduced the half plane H ⊂ N ⊂ P (L ⊗_{ℤ} ℂ) as one of the two connected components of K which is defined by the conditions ⟨z, z⟩ = 0 and ⟨z, z̄⟩ > 0. We describe the boundary of H inside the zero-quadric N.

SB 2.4 Remark and Definition. Let [a] ∈ P (V ⊗_{ℝ} ℂ) be a point which is represented by a real isotropic vector a ∈ V. Then [a] is contained in the boundary of H. A boundary point of this type is called a special boundary point. A set consisting of one special boundary component is called a zerodimensional boundary component.

Proof. We choose two sequences x_{n}, y_{n} ∈ V of vectors of positive norm (q(x_{n}), q(y_{n}) > 0), which are orthogonal and which both converge to a. It is easy to prove the existence of such sequences. Then x_{n} + iy_{n} defines a point in K. We may assume that it is contained in H. The limit of this sequence in the projective space is [a + ia] = [(1 + i)a] = [a]. ⊔ ⊓

A non-special boundary point is called a generic boundary point. We are going to describe the generic part of the boundary. For this purpose we have to consider two-dimensional isotropic subspaces of V = L ⊗_{ℤ} ℝ. Let F be such a subspace. We can consider the set of all boundary points which can be represented by elements of F ⊗_{ℝ} ℂ. This set contains special boundary points. We are interested at the moment in the generic ones:

OneB 2.5 Definition. Let F ⊂ V be a two-dimensional isotropic subspace of V. The set of all generic boundary points which can be represented by an element of V ⊗_{ℝ} ℂ is called the one-dimensional boundary component attached to F.

By a boundary component we understand a one- or two dimensional boundary component.

BiIB 2.6 Lemma. There is a bijective correspondence between boundary components and non-zero isotropic subspaces of V.

Proof. The proof will follow from the following description: Let F ⊂ L ⊗_{ℤ} ℝ be a two dimensional isotropic subspace. Then there exists (.2.1) a complementary isotropic space F′ ⊂ L ⊗_{ℤ} ℝ such that F + F′ is the sum of two orthogonal hyperbolic planes. There exists a basis e_{2}, e_{4} of F and a basis e_{1}, e_{3} of F′ such that

\begin{aligned}(e_{1}, e_{2}) & = 1, \\ (e_{3}, e_{4}) & = 1, \\ (e_{i}, e_{i}) & = 0 (j = 1, …, 4), \\ (e_{i}, e_{j}) & = 0 (i = 1, 2, j = 3, 4). \end{aligned}

We briefly write (z_{1}, z_{2}, z_{3}, z_{4}) instead of z_{1}e_{1} + ⋯ + z_{4}e_{4}.

It is very useful now to keep in mind that the vectors e_{1}, …, e_{4} give rise to embeddings

H × H ⟶ H, P^{1}(ℂ) × P^{1}(ℂ) ⟶ N

and an equivariant homomorphism

SL(2, ℝ) × SL(2, ℝ) ⟶ O′(V).

The embeddings are given by the formula

(z_{1}, z_{2}) ↦ [z_{1}, z_{2}, 1, −z_{1}z_{2}],

if the connected components fit together. What we need is that [i, i, 1, 1] lies in H. This we may assume because we can replace e_{1} by −e_{1} and e_{3} by −e_{3}.

The elements of F are of the form (0, z_{2}, 0, z_{4}). We assume that this is not a multiple of a real point. Then z_{2}, z_{4} both must be different from 0, and we may normalize such that z_{2} = 1, i.e. the point is of the form (0, 1, 0, τ).

We have to clearify whether it is in the boundary of H. For this we use the embedding

H × H ⟶ H, (z_{1}, z_{2}) ⟼ [z_{1}, z_{2}, 1, −z_{1}z_{2}].

We have (in the projective space)

lim_{t→∞}[τ, it, 1, −τit] = lim_{t→∞} [\frac{τ}{it}, 1, \frac{1}{it}, −τ] = [0, 1, 0, −τ].

This means that the point [0, 1, 0, −τ] belongs to the boundary of H if the imaginary part of τ is positive. Thus the one dimensional boundary components can be considered as usual complex upper half planes.

raHa 2.7 Proposition. Let F ⊂ V be a two dimensional isotropic space. After the choice of an isotropic space F′ ⊂ V such that F + F′ is the orthogonal sum of two hyperbolic planes and basises e_{2}, e_{4} of F and e_{1}, e_{3} of F′ with the above properties one obtains a holomorphic embedding of the usual upper half plane H into the zero quadric N, such that the image is precisely the one-dimensional boundary component belonging to F. Additional Remark. The embedding H → N extends to a topological map of H ∪ ℝ ∪ {∞} onto the set of all boundary points, which can be represented by a point of F ⊗_{ℤ} ℂ (including the special ones).

It is worth while to consider the case O(2, 2) separately. vergS 2.8 Remark. We use the homomorphism

SL(2, ℝ) × SL(2, ℝ) ⟶ O′(2, 2)

and the identification .5.4–.5.6,

H × H ⟶^{∼} H, P^{1}(ℂ) × P^{1}(ℂ) ⟶^{∼} N .

The boundary components in this picture are: The zero dimensional boundary components are points in ℝ̄ × ℝ̄. The one-dimensional boundary components are of the form {a}×H or H ×{a}, where a ∈ ℝ̄.

Our explicit description of boundary components shows:

interB 2.9 Remark. Let (V, q) be a quadratic space of signature (2, n) and V_{1} a subspace of signature (2, n_{1}), 0 < n_{1} ≤ n. We denote the corresponding zero quadrics by N_{1} ⊂ N and the half planes by H_{1} ⊂ H, where we assume of course that the connected components have been chosen properly. The boundary components of H_{1} are precisely the intersections of the boundary components of H with N_{1}.

We now introduce the notion of a rational boundary component. Let L ⊂ V be a discrete subgroup of maximal rank such that (L, q) is even. The notion of a rational boundary component depends not really on L but only on the ℚ-form

V_{ℚ} := L ⊗_{ℤ} ℚ ⊂ V.

We consider V_{ℚ} as a quadratic space over ℚ equipped with the restriction of q. of V.

The notation “boundary component” is justified by the following lemma.

DisB 2.10 Lemma. The boundary of H in the zero quadric N is the disjoint union of the boundary components.

The notation “boundary component” is justified by

DisB 2.11 Lemma. The boundary of H in the zero quadric N is the disjoint union of the boundary components.

There is another very natural question. Consider a one-dimensional boundary component which is defined by the isotropic subspace F. The normalizer

N_{G}(F) := { g ∈ G; g(F) = F }

acts on the boundary component and hence on the upper half plane using the identification .2.7. We want to define a homomorphism N_{G}(F) → SL(2, ℝ)/±E such that the action of N_{G}(F) on the boundary component factors through the natural action of SL(2, ℝ) on H (Moebius transformations). Therefore we consider also the centralizor

Z_{G}(F) := { g ∈ N_{G}(F); g(x) = x ∀x ∈ F }.

Assume that F′ has been chosen. Using Witt’s theorem it is easy to see that every element of N_{G}(F)/Z_{G}(F) can be represented by an element which normalizes F + F′. Hence it is sufficient for our purpose to consider the normalizer of F inside O(F +F′). This case can be easily treated using .2.8. For example the normalizer of the boundary component of the boundary component {a}×H in SL(2, ℝ)×SL(2, ℝ) is the direct product of the stabilizer of a and the full SL(2, ℝ). Projection to the second factor gives the desired homomorphism. We obtain:

ranSL 2.12 Lemma. Let F ⊂ V be a two dimensional isotropic subspace. After the choice of the complementary space F and the basis vectors e_{1}, …, e_{4} one obtains a homomorphism

N_{G}(F) ⟶ SL(2, ℝ)/{±E}

such that the action of N_{G}(F) on the boundary component corresponding to F is compatible with the standard action of SL(2, ℝ) on H. The restriction of this homomorphism to N_{O′(F +F′)}(F) is already surjective.

3. Rational boundary components

DeRB 3.1 Definition. A boundary component is called rational, if the corresponding isotropic subspace F ⊂ V is defined over ℚ. This means nothing else but that V is generated over ℝ by F_{ℚ} := F ∩ V_{ℚ}.

Hence the rational boundary components are in 1-1-correspondence to the isotropic subspaces of V_{ℚ}.

NoSt 3.2 Notation. The union of H with all rational boundary components is denoted by H^{∗}.

The orthogonal group O′(V_{ℚ}), hence also the integral orthogonal group O(L) acts on H^{∗} and we can introduce as set

X_{Γ} := H^{∗}/Γ

§3. Rational boundary components 29

for any subgroup of finite index of O′(L) or more generally of for any subgroup of O′(V_{ℚ}) := O(V_{ℚ}) ∩ O′(V) which is commensurable with O′(L).

The main purpose of compactification theory is to show that this space carries a natural structure as projective algebraic variety.

It may happen that V_{ℚ} is anistropic, i.e. that it contains no non-zero isotropic vector (recall the example q(x) = x_{1}^{2} + x_{2}^{2} − 7x_{3}^{2}). In this case there are no rational boundary components and in this case on has:

OhnC 3.3 Theorem. Let V_{ℚ} be anistropic then X_{Γ} is compact for every subgroup of O′(V_{ℚ}) which is commensurable with O′(L).

This is a very special case of a the compactification theory which we are going to describe in the case where V_{ℚ} is of rank two, i.e. admits an isotropic subspace of dimension 2. This means that we skip some low-dimensional cases (which are very interesting and deserve particular interest.)

The Satake-Baily-Borel topology

As mentioned we assume no that V_{ℚ} has rank two. Then we can choose for rational vectors e_{1}, …, e_{4} such that (e_{1}, e_{2}) = (e_{3}, e_{4}) = 0 an such that all other products of the e_{i} are 0. Recall that

H′ := { (z_{1}, z_{2}, z) ∈ ℂ × ℂ × W (ℂ); y_{1}y_{2} + q(𝔶) > 0, y_{1} > 0 }

is a model of H. We can assume that the map defined in .4.6 defines a biholomorphic map H′ → H (s. .4.7).

Sdom 3.4 Definition. Let t > 0 the Siegel domain F′(t) is the subset of H′ defined by the inequalities

\begin{aligned}x_{1}^{2} + x_{2}^{2} + |q(𝔵)| & < t^{2}, \\ 1/t & < y_{1}, \\ y_{1}^{2} & < t^{2}q(y), \\ |q(𝔶)| & < t^{2}y_{1}^{2}. \end{aligned}

The image in H is denoted by F(t)

The set F(t) depends on the choice of the e_{1}, …, e_{4}.

A fundamental result —essentially due to Siegel— states

FundB 3.5 Theorem. Let Γ ⊂ O′(V_{ℚ} be a subgroup which is commensurable with the integral orthogonal group O′(L) of some lattice L ⊂ V_{ℚ}.

1. The set of all γ ∈ Γ with

γ(F(t)) ∩ F(t) ≠ ∅

is finite for every t > 0.

2. Assume that t is big enough. There exist finitely many g_{1}, …, g_{h} ∈ Γ such

that

F := ⋃_{1≤j≤h} g_{j}(F(t))

is a fundamental set of Γ, which means that H is the union of all translates γ(F), γ ∈ Γ.

We also have to consider the closure \overline{F(t)} of F(t) in H^{∗} with respect to the topology induced by the usual topology of N. We are going to define a quite different topology on H^{∗}.

SatT 3.6 Theorem. There exists a unique topology T on H^{∗}, which does not depend on the choice of the e_{1}, …, e_{4} and which has the following properties:

1. The set H is open and dense in H^{∗} and the induced topology is the usual one.

2. The induced topologies on \overline{F(t)} coming from T and from the usual topology of N agree.

3. The group O′(V_{ℚ}) acts as group of topological automorphisms on (H, T ).

4. Let Γ ⊂ O′(V_{ℚ}) be a subgroup which is commensurable with some O′(L), where L ⊂ V_{ℚ} is a lattice. Every point a ∈ H^{∗} admits a fundamental system of open neighbourhoods, which are invariant under the stabilzer Γ_{a} and such that

U(a)/Γ_{a} ⟶ H^{∗}/Γ

is injective (and then an open embedding).

The basic result about this topology is

KomP 3.7 Theorem. The space (equipped with the quotient topology)

X_{Γ} = H^{∗}/Γ

is compact.

The compactness of X_{Γ} is related to some finiteness results which we formulate separately:

endO 3.8 Theorem. Let Γ ⊂ O′(V_{ℚ}) be an arithmetic group as defined above: Let F be any quadratic space over ℚ There only finitely many Γ-orbits of isometric embeddings F ↪ V_{ℚ}. Especially there are only finitely many Γ-orbits of isotropic subspaces of V_{ℚ}.

We describe the boundary components of X_{Γ}. Let F_{ℚ} ⊂ V_{ℚ} be a two dimensional isotropic space. We consider the biholomorphic map defined in .2.7 from the usual upper half plane H onto the boundary component related to F. This extends to an embedding

H^{∗} ⟶ H^{∗},

whose image consists of all rational boundary points which can be represented by a one dimensional subspace of F ⊗_{ℚ} ℂ (including special boundary points). Her H^{∗} means of course the extended halfplane H^{∗} = H ∪ ℚ̄. We consider the group

Γ_{0} := N_{F} (Γ)/Z_{F} (Γ).

This is a subgroup of SL(2, ℚ), which is commensurable with SL(2, ℤ): Hence we can consider H^{∗}/Γ_{0}. We have an obvious map

H^{∗}/Γ_{0} ⟶ H^{∗}/Γ.

We call Γ_{0} the projection of Γ to the boundary component related to F. This projection depends on certain choices of a basis, but the change of such a basis only means conjugation of Γ_{0} inside SL(2, ℚ).

RandK 3.9 Theorem. Let F ⊂ V_{ℚ} be a two dimensional isotropic space and Γ_{0} ⊂ SL(2, ℚ) the projected group. The natural map

H^{∗}/Γ_{0} ⟶ H^{∗}/Γ

is an embedding.

Let F_{1} …, F_{h} a system of representatives of the Γ-orbits of the two dimensional isotropic subspaces of V_{ℚ} and let Γ_{j} denote the corresponding projected groups. The images of the H^{∗}/Γ_{j} cover the “boundary”,

H^{∗}/Γ − H/Γ =: S_{Γ} = ⋃_{1≤j≤h} H^{∗}/Γ_{j}.

We give an example: Let M be an even positive definite lattice of rank n, which is selfdual (M = M^{∗}). Such a lattice exists if and only if n is divisible by n. The number of isomorphy-classes h(n) of such lattices is finite. One knows h(8) = 1, h(16) = 2, h(24) = 24, h(32) > 600 000. We consider the orthogonal sum of two hyperbolic planes over ℤ and M,

L = H ⊕ H ⊕ (−M),

where H = ℤ^{2} with quadratic form q(X_{1}, x_{2}) = x_{1}x_{2}. Then L is an even selfdual lattice of signature (2, n + 2). It is a known result that two unimodular selfdual indefinite lattices of the same dimension and signature are isomorphic (in contrast to the definite case). unCu 3.10 Proposition. Let n be a natural number which is divisible by 8 and L an even selfdual lattice of signature (2, n + 2): Then the following holds:

1. There is only one O′(L)-orbit of isotropic lines F ⊂ L ⊗_{ℤ} ℚ.

2. The number of O′(L)-orbits of isotropic planes F ⊂ L ⊗_{ℤ} ℚ is h(n).

A system of representatives of isotropic planes can be obtained as follows. Let M_{1}, …, M_{h(n)} be a system of representatives of the even selfdual positive definit lattices. As we mentioned there exist isometric embeddings −M_{j} ↪ L. It is nor very difficult to prove that the orthogonal complement of −M_{j} is isomorphic H ×H = ℤ^{4} (s. above). The vectors (1, 0, 0, 0), (0, 0, 1, 0) generated an isotropic plane F_{j}. This gives a system of representatives of isotropic planes.

Now we see the picture for the boundary. The standard basis e_{1}, …, e_{4} of H × H = ℤ^{4} gives an embedding of H^{∗} into H. It is easy to see that in this case the projected group Γ_{0} of Γ = O′(L) is SL(2, ℤ)/{±E}. Keeping in mind that H^{∗}/ SL(2, ℤ) is a Riemann sphere ℂ̄ = P^{1}(ℂ) we see:

sdBo 3.11 Proposition. Let L be an even self-dual lattice of signature (2, n + 2). The boundary S_{Γ} of H^{∗}/Γ, Γ = O′(L) is the disjoint union of h(n) Riemann spheres which intersect in one common point.

Chapter IV. Basic facts about orthogonal modular forms

1. The notion of an orthogonal modular form

We consider an even lattice (L, q) of signature (2, n), n > 1 and use the notation V_{ℚ} = L ⊗_{ℤ} ℚ, V = L ⊗_{ℤ} ℝ. We fix some subgroup Γ ⊂ O′(V_{ℚ}) which is commensurable with O′(L). We will call such a subgroup simply a modular group. The notion of a modular group only depends on V_{ℚ} and not on the particular choice of the lattice L. We recall also the notations

H ⊂ K ⊂ N ⊂ P (V (ℂ)).

It is useful to consider also the inverse image of H, K, N in V (ℂ) − {0}. We denote them by

H̃, K̃, Ñ , ⊂ V (ℂ) − {0}.

We can introduce now the notion of an orthogonal modular form:

defMo 1.1 Definition. Let r be an integer and v : Γ → ℂ^{•} a character. A modular form of weight r and character v on Γ is a holomorphic function f : H̃ → ℂ with the properties

1. f(γ(z)) = v(γ)f(z) for γ ∈ Γ,

2. f(tz) = t^{−r}f(z) for t ∈ ℂ^{•}.

It is worth while to work out what this means in the simplest case O(2, 1). Let’s take for V the space of all real symmetric matrices X with q(X) = − det(X) as quadratic form. Recall that we have an embedding

H ⟶ H̃, τ ⟼ ( \begin{matrix}τ^{2} & τ \\ τ & 1 \end{matrix} ) .

The composition of this embedding with the natural projection is a biholomorphic map H → H. Hence a modular form f in the sense of I.1.1 is determined by the function

F : H ⟶ ℂ, F (τ) := f ( \begin{matrix}τ^{2} & τ \\ τ & 1 \end{matrix} ) .

Recall that we have a natural homomorphism

SL(2, ℚ) ⟶ O′(V_{ℚ}).

We denote the inverse image of Γ by Γ′ ⊂ SL(2, ℚ). We determine the transformation properties of F under Γ′. Using the formula used for the proof of .5.2 we see orEl 1.2 Remark. In the case O(2, 1) described above, the modular forms f on Γ in the sense of I.1.1 are in 1-1-correspondence to holomorphic functions F on the usual upper half plane with the transformation property

F (M(τ)) = (cτ + d)^{2r}v′(M)F (τ) for all M ∈ Γ′.

Here v′ denotes the composition of the projection Γ′ → Γ with the character v.

In a similar way the case of signature (2, 2) can be treated. Here we use for V the model all 2 × 2-matrices X (not necessarily symmetric) with q(X) = − det(X) as quadratic form. Here we can use the embedding

H × H ⟶ H̃, (τ_{1}, τ_{2}) ⟼ ( \begin{matrix}τ_{1}τ_{2} & τ_{1} \\ τ_{2} & 1 \end{matrix} ) .

The composition with the projection H̃ → H defines a biholomorphic map H × H → H. Hence

F (τ_{1}, τ_{2}) := f ( \begin{matrix}τ_{1}τ_{2} & τ_{1} \\ τ_{2} & 1 \end{matrix} )

determines F. No we have a homomorphism

SL(2, ℝ) × SL(2, ℝ) ⟶ O′(V).

Again we consider the inverse image Γ′ of Γ and the pull-back v′ of the character v. As in I.1.2 on proves:

orEll 1.3 Remark. In the case O(2, 2) described above, the modular forms f on Γ in the sense of I.1.1 are in 1-1-correspondence to holomorphic functions F on the product of two usual upper half planes with the transformation property

F (M_{1}τ_{1}, M_{2}τ_{2}) = (c_{1}τ_{1}+d_{1})^{r}(c_{2}τ_{2}+d_{2})^{r}v′(M_{1}, M_{2})F (τ_{1}, τ_{2}), (M_{1}, M_{2}) ∈ Γ′.

Here v′ denotes the composition of the projection Γ′ → Γ with the character v.

The group Γ′ depends on the choice of the lattice. Our main interest that this lattice splits over ℚ in the sense that V_{ℚ} is the space of rational matrices.

zerfE 1.4 Remark. Assume that V_{ℚ} is the space of rational symmetric 2×2-matrices resp. the space of all rational 2 × 2-matrices. Then the group Γ′ defined in I.1.2 resp. .1.3 is commensurable with SL(2, ℤ) resp. SL(2, ℤ) × SL(2, ℤ).

We go back to the general case. Assume that (V_{1})_{ℚ} ⊂ V_{ℚ} is a subspace also of signature (2, n_{1}). Recall that we embedded O′((V_{1})_{ℚ}) into O′(V_{ℚ}). Let Γ_{1} be the intersection of Γ with O′((V_{1})_{ℚ}). This is also a modular group in our sense (i.e. commensurable with O′(L_{1}), L_{1} = L ∩ V_{1}). We also have a natural embedding H_{1} ↪ H, if the connected components are correctly chosen. It is clear that the restriction of a modular form on Γ is a modular form of the same weight and an obvious character. We formulate this in a slightly more general context.

§2. The automorphy factor 35

puLL 1.5 Remark. Let (V_{1})_{ℚ} ↪ V_{ℚ} be an isometric embedding of two quadratic spaces over ℚ of signature (2, n_{1}) resp. (2, n). Let Γ ⊂ O′(V_{ℚ}) be a modular group and Γ_{1} = Γ∩O′((V_{1})_{ℚ}). If the connected components are properly chosen we have a natural embedding H_{1} ↪ H. The pull-back of a modular form f on Γ is a modular form on Γ_{1} of the same weight and an obvious character.

It is worth while to point out the particular case V_{1} = V. In this case an isometric embedding is simply an orthogonal transformation g ∈ O(V_{ℚ}).

trFO 1.6 Remark. Let g ∈ O(V_{ℚ}) be an orthogonal transformation The assignment f(z) ↦ f(gz) transforms modular forms on Γ to modular forms on the conjugate group gΓg^{−1}. Corollary. The normalizer of Γ in O′(V_{ℚ}) acts on the space of all modular forms on Γ of a given weight and trivial character.

2. The automorphy factor

As in the cases O(2, 1) and O(2, 2) we restrict a modular form f to the halfplane model H′ of H. We recall the setting.

We consider a hyperbolic plane in V spanned by two isotropic vectors e_{1}, e_{2} with an isotropic e_{2}. For simplicity we choose e_{1} such that e_{1} is also isotropic and such that (e_{1}, e_{2}) = 1. The domain K′ is an open subset of V_{1}(ℂ), where V_{1} is the orthogonal complement of the e_{i} in V (s. .4.4). There is a biholomorphic map K′ → K. This map lifts to a map

K′ → K̃, Z ⟼ (1, −q(Z), Z).

This means that we constructed a section of K̃′ → K using the tube domain model (.4.4).

We consider an element g ∈ O(V). We write

g(1, −q(Z), z) = (w_{1}, w_{2}, w).

The element w_{1} has to be different from 0 but not necessarily one. We denote this element by J(g, Z) := w_{1}.

autO 2.1 Remark. The map

J : O(V) × K′ ⟶ ℂ^{•}

defined by

g(1, ∗, z) = (J(g, z), ∗, w)

is holomorphic in the second variable and satisfies J(gh, z) = J(g, hz)J(h, z).

We call J(g, z) the canonical factor of automorphy. orEL 2.2 Remark. The modular forms f on Γ in the sense of I.1.1 are in 1-1-correspondence to holomorphic functions F on the half plane model H′ with the transformation property

F (γ(Z)) = J(γ, Z)^{−r}v(γ)F (Z) for all γ ∈ Γ.

Here γ(Z) means the action of O′(V) on H′ which is induced by the natural action on H using the identification H′ → H.

The canonical automorphy factor can be determined as follows:

autJa 2.3 Proposition. Let j(g, Z) denote the Jacobian determinant of g ∈ O(V) considered as substition on K′. Then

j(g, Z) = det(g)J(g, Z)^{n}.

Proof. It is sufficient to prove this proposition for reflections,

z ⟼ g(z) := z − \frac{(a, z)}{q(a)}a (q(a) ≠ 0),

because they generate O(V). We write z = (1, −q(Z), Z) and a = (a_{1}, a_{2}, A) and obtain

φ(Z) := \frac{(a, z)}{q(a)} = \frac{−a_{1}q(Z) + a_{2} + (A, Z)}{a_{1}a_{2} + q(A)}

and as a consequence

(1, −q(Z), Z) ⟼ (1 − a_{1}φ(Z), ∗, Z − Aφ(Z)).

We have J(g, Z) = 1 − a_{1}φ(Z) and have to compare this with the Jacobian of

Z ⟼ \frac{Z − Aφ(Z)}{1 − a_{1}φ(Z)}.

We introduce a basis of V_{1}(ℂ). The Jacobian is independent of such a choice. Hence we can assume Z = (Z_{1}, …, Z_{n}), A = (A_{1}, …, A_{n}) and q(Z) = Z_{1}^{2} + ⋯ + Z_{n}^{2}. It is convenient to express the Jacobian as a Wronsky determinant. Recall that the Wronskian of function f_{0}, …, f_{n} in n variables x_{1}, …, x_{n} is defined as

W (f_{0}, …, f_{n}) := det ( \begin{matrix}f_{0} & … & f_{n} \\ ∂f_{0}/∂x_{1} & … & ∂f_{n}/∂x_{1} \\ ⋮ & & ⋮ \\ ∂f_{0}/∂x_{n} & … & ∂f_{n}/∂x_{n} \end{matrix} ) .

§3. The Koecher principle 37

It is known and easy to prove that

W (f_{0}, …, f_{n}) = f_{0}^{−n−1} j(\frac{f_{1}}{f_{0}} , …, \frac{f_{n}}{f_{0}} ),

where j denotes the Jacobian. Hence we have to compute the Wronsky determinant of

(1 − a_{1}φ(Z), Z_{1} − A_{1}φ(Z), …, Z_{n} − A_{n}φ(Z)).

This determinant can be computed: We assume a_{1} ≠ 0. One adds suitable multiples of the first column to the other columns to get a matrix of the shape

( \begin{matrix}γ & α_{1} & … & α_{n} \\ β_{1} & 1 & & 0 \\ ⋮ & ⋮ & ... & ⋮ \\ β_{n} & 0 & … & 1 \end{matrix} )

The determinant of this matrix is γ − ∑_{j} α_{j}β_{j}. In our case

γ = 1 − a_{1}φ(Z), , β_{j} = −a_{1} \frac{∂φ(Z)}{∂Z_{j}}, α_{j} = Z_{j} − \frac{A_{j}}{a_{1}} .

A straight forward calculation now gives

γ − ∑_{j} α_{j}β_{j} = −(1 − a_{1}φ(Z)).  ⊔ ⊓

We express the transformation of modular forms in the sense of .1.6 on the tube domain model.

traTu 2.4 Remark. Let f be a modular form and F the corresponding function on the tube domain model. Let be g ∈ O′(V_{ℚ}). The modular form f(g(Z)) corresponds to the function J(g, Z)^{−r}F (gZ).

3. The Koecher principle

We now assume L is a lattice of signature (2, n) such that V_{ℚ} = L ⊗_{ℤ} ℚ has rank ≥ 1. We can and will choose rational vectors e_{1}, e_{2} with the property (e_{2}, e_{2}) = 1, (e_{1}, e_{2}) ≠ 0. We consider a modular form f and the corresponding function F on the half plane model H′. Recall that this is a tube domain in V_{1}(ℂ), where V_{1} is the orthogonal complement of the e_{i} in V. The space V_{1} is also defined over ℚ, which means that it is generated as real vector space by (V_{1})_{ℚ} = V_{ℚ} ∩ V_{1}. We also consider a modular group Γ ⊂ O′(V_{ℚ}), i.e. a subgroup which is commensurable with O′(L). We introduce so-called Eichler elements.

Eichl 3.1 Remark. Let u ∈ V be an isotropic element, i.e. q(u) = 0, and v an element which is orthogonal to u, i.e. (u, v) = 0. The transformation

E(u, v)(a) = a − (a, u)v + (a, v)u − q(v)(a, u)u ((v, v) = 2q(v))

is contained in O′(V). In the case u, v ∈ L this element is contained in the discriminant kernel.

The Eichler transformations generated an abelian group for fiexed u:

Eichgr 3.2 Lemma. One has

E(u, v_{1})E(u, v_{2}) = E(u, v_{1} + v_{2}).

We consider a special Eichler transformation:

sEic 3.3 Remark. The transformation

(x_{1}, x_{2}, X) ⟼ (x_{1}, x_{2} − (X, H) − q(H)x_{1}, X + x_{1}H)

is an Eichler transformation ((E(e_{2}, −H)). Its effect on the tube domain is

Z ⟶ Z + H.

The automorphy factor for this Eichler transformation is 1. We consider all H ∈ (V_{1})_{ℚ} such that the Eichler transformation (E(e_{2}, −H) is contained in Γ. This is a lattice. This follows from the following trivial remark: Let H ⊂ G be groups and G_{1} ⊂ G a subgroup of finite index. Then H_{1} = G_{1}∩H has finite index in H. We obtain that modular forms considered on the tube domain model are periodic functions.

FouR 3.4 Lemma. There exists a lattice

T ⊂ (V_{1})_{ℚ}

such that every modular form f with trivial character v. considered as function on H′ is periodic with respect to T ,

f(Z + T) = f(Z) for all T ∈ T .

Hence f admits a Fourier expansion

f(Z) = ∑_{T ∈T ′} a(T)e^{2πi(T,Z)}.

Here T′ denotes the dual lattice of T with respect to the given bilinear form on V_{1} (restriction of the bilinear form on V).

We consider elements g ∈ O′(V) with the property g(e_{1}) = e_{1}, g(e_{2}) = e_{2}. The can be considered as orthogonal transformations of the hyperbolic space V_{1}. Obviously J(g, Z) = 1. traFO 3.5 Lemma. Let

f(Z) = ∑_{T} a(T)e^{2πi(T,Z)}.

be a modular form considered as function on H and g ∈ O′((V_{1})_{ℚ}) a rational orthogonal transformation (extended by g(e_{1}) = e_{1}, g(e_{2}) = e_{2}). Then

f(g(Z)) = ∑ a(g^{−1}T)e^{2πi(T,Z)}

is the Fourier expansion of the transformed modular form. Especially

a(g(T)) = v(g)a(T) for all g ∈ Γ ∩ O′(V_{1}).

We denote by P the positive cone in V_{1}, which means

H′ = V_{1} + iP

and by P̄ its closure

KoeC 3.6 Theorem. Let f(Z) = ∑_{T ∈T ′} a(T)e^{2πi(T,Z)} be the Fourier expansion of a modular form. Then

a(T) ≠ 0 ⟹ T ∈ P̄,

especially q(T) ≥ 0 up to the following two exceptional cases:

O(2, 1) with rank(V_{ℚ}) = 1 resp. O(2, 2) with rank(V_{ℚ}) = 2.

Proof. Firstly we mention that this result only depends on the isptropic vector e_{2} and not on e_{1}. Hence we can assume that e_{2} is isotropic to and that (e_{1}, e_{2}) = 0. We will prove this theorem only in the case n ≥ 3 and ℚ-rank two. We need some special features in the case of ℚ-rank two. Recall that we refined the tube domain model in this case (s. .4.6 and .4.7). We have to consider four vectors e_{1}, …, e_{4} ∈ V_{ℚ} such that (e_{1}, e_{2}) = (e_{3}, e_{4}) = 1 and such that all other products between the four are zero. Let W be the orthogonal complement of the four vectors in V. This is a negative definite space which is defined over ℚ, i.e. generated as real vector space by W_{ℚ} = W ∩ V_{ℚ}. The role of V_{1} now plays ℝe_{1} + ℝe_{2} + W and the role of the distinguished hyperbolic plane ℝe_{3} + ℝe_{4}. The tube domain now lies in ℂe_{1} + ℂe_{2} + W (ℂ). We denote the elements of this space by

Z = (z_{1}, z_{2}, 𝔷) (z_{1}, z_{2} ∈ ℂ, 𝔷 ∈ W (ℂ)).

The space K′ is given by y_{1}y_{2} > −q(𝔶). A connected component is defined by y_{1} > 0 and we assume this connected component to be H′.

The following two transformations are special Eichler transformations which fix e_{3}, e_{4}. The image of (x_{1}, x_{2}, x_{3}, x_{4}, 𝔵) is:

1. (x_{1} + (h, 𝔵) − q(h)x_{2}, x_{2}, x_{3}, x_{4}, 𝔵 − x_{2}h) (h ∈ W).

2. (x_{1}, x_{2} + (h, 𝔵) − q(h)x_{1}, x_{3}, x_{4}, 𝔵 − x_{1}h) (h ∈ W).

An easy calculation shows: The image of (z_{1}, z_{2}, Z) with respect to the transformations 1–2 (in the same ordering) is:

1. (z_{1} + (h, 𝔷) − q(h)z_{2}, z_{2}, 𝔷 − z_{2}h),

2. (z_{1}, z_{2} + (h, 𝔷) − q(h)z_{1}, 𝔷 − z_{1}h).

We use the notations of .3.4. The indices T now are of the form T = (t_{1}, t_{2}, 𝔱).

unIN 3.7 Lemma. Assume n ≥ 3. The Fourier coefficients a(T) satisfy

\begin{aligned}a(t_{1}, t_{2}, 𝔱) & = a(t_{1} + (h, 𝔱) − q(h)t_{2}, t_{2}, 𝔱 − t_{2}h) \\ & = a(t_{1}, t_{2} + (h, 𝔱) − q(h)t_{1}, 𝔱 − t_{1}h) \end{aligned}

for all h in a suitable lattice in W (ℚ).

Proof of .3.6. The closure of the positive cone is given by q(T) ≥ 0 and t_{1} ≥ 0. We want to argument by contradiction and assume that a(T) ≠ 0 but q(T) < 0 or t_{1} < 0 or t_{2} < 0. By Witt’s theorem t there exists an element g ∈ O′(ℚ ⊕ ℚ ⊕ W_{ℚ}) such that g(T) = (t_{1}, t_{1}, 𝔱), t_{1} < 0. Replacing f(Z) by f(g(Z)) we can assume there exists a coefficient a(t_{1}, t_{2}, 𝔱) ≠ 0 with t_{1} < 0. From .3.7 we deduce that the series

∑_{h} exp 2πi{((i, i, 0), (t_{1}, t_{2} + (h, 𝔱) − q(h)t_{1}, 𝔱 − t_{1}h))} =

∑_{h} exp −2πi{t_{1} + t_{2} + (h, 𝔱) − q(h)t_{1}}

converges. But this is not the case because in the exponent the term −q(h) occurs, which is positive definit.

ENT 3.8 Definition. A modular form is called entire at infinity if for every choice of the e_{1}, e_{2} the Fourier coefficients a(T) satisfy

a(T) ≠ 0 ⟹ T ∈ P̄.

We denote the space of all entire modular forms on Γ of weight r and character v by

[Γ, r, v], or by [Γ, r],

the latter if the character v is trivial.

The Koecher principle tells that in case n ≥ 3 all modular forms are entire at infinity. (We proved it for n ≥ 5.) We are usually only interested in entire modular forms, even in the exceptional cases. Hence we assume in the following that modular forms are entire modular forms until we do not tell anything else.

§4. The Siegel operator 41

4. The Siegel operator

We consider a boundary component defined by an isotropic subspace F ⊂ V_{ℚ}. We start with a zero dimensional boundary component, dim F = 1. We choose a non-zero element e_{2} ∈ F and an additional iotropic element e_{1} ∈ V_{ℚ} such that (e_{1}, e_{2}) = 1. As usual we write (V_{1})_{ℚ} for the orthogonal complement of the hyperbolic plane ℚe_{1} + ℚe_{2} and write (z_{1}, z_{2}, Z) instead of z_{1}e_{1} + z_{2}e_{2} + Z (Z ∈ V_{1}(ℂ)). The boundary point F (ℂ) is [0, 1, 0]. Let Z ∈ H′ be an arbitrary pint, i.e. (1, −q(Z), Z) ∈ H. In the Satake topology

lim_{t→+∞}[1, −q(tZ), tZ)] = [1, 0, 1].

Hence it seems to be natural to consider the limit

lim_{t→+∞} f(tZ) = a(0)

for an entire modular form (considered as function on H′), which nothing else but the constant Fourier coefficient. Therefore it is natural to consider this Fourier coefficient as boundary value of f and to denote it by f(F).

But this value depends on the choice of F. For example if one replaces e_{1} ↦ te_{1} for some nonzero t ∈ ℚ, then a(0) will be multiplied by t^{r}. Replacements of e_{2} will not change a(0). So all together we can say: Let f := (f_{0}, … f_{m}) be a m+1-tuple of modular forms of the same weight. The vector of constant Fourier coefficients (a_{0}(0), …, a_{m}(0) depends on the choice of e_{1}, e_{2} only up to a common factor. Especially the space

[a_{1}(0), …, a_{m}(0)] := ℂ(a_{1}(0), …, a_{m}(0)

is well defined. If the vector is non zero this can be considered as an element of the projective space

f(F) := [a_{1}(0), …, a_{m}(0)] ∈ P^{m}(ℂ).

This point only depends on the Γ-orbit of F, if the f_{j} are modular forms on Γ all with the same character. Let now F ⊂ V_{ℚ} be a two dimensional isotropic space. Now we have to choice vectors e_{1}, …, e_{4} such that F = ℚe_{2} + ℚe_{4}, (e_{1}, e_{2}) = (e_{3}, e_{4}) = 1 and all other products are zero. The coordinates of H′ now are (z_{1}, z_{2}, 𝔷). This point corresponds to [z_{1}, z_{2}, 1, ∗, 𝔷] ∈ H. The point [0, 1, 0, −τ, 0] ∈ N is a boundary point of H if τ is in the upper half plane. We have in the Satake topology

lim_{t→∞}[τ, it, 1, −τit, 𝔷] = [0, 1, 0, −τ, 0].

Hence it is natural to consider for an entire modular form f (considered as function on H′) the limit

lim_{t→+∞} f(τ, it, 𝔷).

Using the Fourier expansion

f(z_{1}, z_{2}, 𝔷) = ∑_{t_{1},t_{2},𝔱} a(t_{1}, t_{2}, 𝔱) exp 2πi{t_{2}z_{1} + t_{1}z_{2} + (𝔱, 𝔷)}

one obtains

lim_{t→+∞} f(τ, it, 𝔷) = ∑_{m=0}^{∞} a(0, m, 0)e^{2πimτ} .

This defines a holomorphic function, which we denote by

f|Φ(τ) := ∑_{m=0}^{∞} a(0, m, 0)e^{2πimτ} .

The Φ is a variant of the so-called Siegel operator. It depends on the choice of e_{1}, …, e_{4}. The function f|Φ does not depend on 𝔷. We may take 𝔷 = 0. Hence f|Φ is determined by the restriction of f to with respect to

H × H ⟶ H′, (τ_{1}, τ_{2}, 0).

This shows:

SiegPh 4.1 Proposition. The operator Φ is a linear mapping from the space [Γ, r, v] into a space of (entire) elliptic modular forms. with respect to some group commensurable with SL(2, ℤ).

The function f|Φ describes certain boundary values of f. Let now

f = (f_{0}, …, f_{m}) ∈ [Γ, r, v]

be a tuple of modular forms. It can be shown that the vector

(f_{1}|Φ(τ), …, f_{m}|Φ(τ)

upto a constant factor only depends on the boundary point and not on the special choice of e_{1}, e_{2}, e_{3}, e_{4}. If [a] ∈ H^{∗} is boundary point the space

f([a]) := ℂ(f_{1}|Φ(τ), …, f_{m}|Φ(tau)

is well defined. If it is zero it can be considered as an element of P^{m}(ℂ). Moreover f([a]) depends only on the Γ-equivalence class of [a]. This means that there are only h essentially different Φ-transforms if h denotes the number of one-dimensional cusp classes (= number of Γ-orbits of two dimensional isotropic subspaces of V_{ℚ}). extSA 4.2 Proposition. Let f = (f_{0}, … f_{m}) be a tuple of modular forms on Γ all with the same character. The set

U := { [z] ∈ H^{∗}/Γ, f([z]) ≠ 0 },

is an open subset of H^{∗}/Γ. The map

U ⟶ P^{m}(ℂ), [z] ⟼ f([z]),

is a well-defined continuous map.

We introduce the notion of a cusp form.

CuDe 4.3 Definition. A modular form is called a cusp form if for every choice of e_{1}, e_{2} the Fourier coefficients are supportet on the positive cone (and not only on its closure),

a(T) ≠ 0 ⟹ T ∈ P.

CuRe 4.4 Remark. A modular form is a cusp form if all boundary values vanish.

Remark. If the ℚ-rank of V_{ℚ} is two this means that all Φ-transforms vanish.

The remark follows from the fact that every one-dimensional isotropic subspace of V_{ℚ} is contained in a two-dimensional if the ℚ-rank is two.

We denote the subspace of cusp forms by

[Γ, r, v]_{0} ⊂ [Γ, r, v], [Γ, r]_{0} ⊂ [Γ, r].

Chapter V. Constructions of modular forms

1. Poincarè series

The half plane H is biholomorphically equivalent to a bounded domain. This can be proven by means of the model H′ = V_{1} + iP. We claim there are three linearly independent linear functions l_{i} : V_{1} → ℝ, which are positive on P. For the proof we can assume that P ⊂ ℝ^{n} is defined by y_{1}^{2} > y_{2}^{2} +⋯ y_{n}^{2}. The linear functions then can be taken as y_{1}, y_{1} − y_{2}, …, y_{n} − y_{1}. We can use this linear functions to embed H′ ↪ H^{n} into the product of n upper halfplanes. They are biholomorphically equivalent to the product E^{n} of n unit-discs. Thus we found a realization of H as bounded domain.

2. Eisenstein series (scetch)

3. Generators of some modular groups

In this section we assume that the lattice L contains to hyperbolic planes over ℤ. This implies that there exist for vectors e_{1}, …, e_{4} ∈ L and a negative definit sublattice L_{0} ⊂ L such that

L = ℤe_{1} + ℤe_{2} + ℤe_{3} + ℤe_{4} + L_{0}
(e_{1}, e_{2}) = (e_{3}, e_{4}) = 1 (all other products 0)
L_{0} orthogonal to the e_{j}.

We will use the notation

L_{1} = ℤe_{1} + ℤe_{2} + L_{0},

hence L_{1} is the orthogonal complement of e_{3}, e_{4}.

We want to construct generators of the discriminant group

Γ_{L} := kernel(O′(L) ⟶ Aut(L′/L)).

§3. Generators of some modular groups 45

We need also

Γ_{L_{1}} := Γ_{L} ∩ O(L_{1}).

We use the homomorphism of SL(2, ℤ) × SL(2, ℤ) into the orthogonal group. The image is contained in Γ_{L}.

geN 3.1 Lemma. Let a ∈ L′ be an arbitrary vector. There exists an element g in the image of

SL(2, ℤ) × SL(2, ℤ) ⟶ Γ_{L},

such that g(a) is contained in L_{1} (the orthogonal complement of e_{3}, e_{4}).

Recall that this homomorphism is defined by means of

(x_{1}e_{1} + x_{2}e_{2} + x_{3}e_{3} + x_{4}e_{4}) ⟼ X = ( \begin{matrix}−x_{4} & x_{1} \\ x_{2} & x_{3} \end{matrix} ) .

The group SL(2, ℤ) × SL(2, ℤ) acts by X ↦ AXB′. From the theorem of elementary divisors follows that any integral X can be transformed into a diagonal matrix, which means that the components of e_{1}, e_{2} will be zero.

transL 3.2 Lemma. Every element a ∈ L can be transformed into the orthogonal complement of e_{1}, e_{2} by means of a transformation coming from SL(2, ℤ) × SL(2, ℤ).

A vector a of a lattice L is called primitive, if it cannot be divided inside L by a natural number > 1. If L = ℤ^{n} this means that the components are coprime (and not all of them 0). The group (a, ℤ) is a subgroup of ℤ and hence a multiple of a number n ≥ 0. This number must be one if a is primitive. This means that there exists a vector a′ ∈ L′ with the property (a, a′) = 1. Recall also that the dual of L′ is L.

bahNen 3.3 Lemma. Two primitive vectors a, b ∈ L′ of the same norm q(a) = q(b) are contained in the same orbit of Γ_{L} iff they have the same image in the discrimant group L′/L.

Proof. We may assume that both vectors a, b are orthogonal to e_{1}, e_{2}. Because both are primitive there exist a′, b′ ∈ L_{1} such that

(a, a′) = (b, b′) = 1.

A simple calculation shows

E(e_{1}, a′)(a) = a − e_{1}, E(e_{1}, b′)(b) = b − e_{1}.

Another straightforward computation gives

E(e_{2}, b − a)(a − e_{1}) = b − e_{1}.  ⊔ ⊓.

The proof shows a little more, namely baHnenH 3.4 Lemma. Two primitive vectors a, b ∈ L′, which are orthogonal to e_{1}, e_{2} and with the same image in the discrimant group are equivalent under the group which is generated by Eichler transformations E(e_{1}, v), E(e_{2}, v), where v ∈ L is orthogonal to e_{1}, e_{2}.

We are now able to prove the following result about generators of Γ_{L}:

erZa 3.5 Proposition. The group Γ_{L} is generated by the stabilizer of e_{4} and by the picture of the image of SL(2, ℤ) × SL(2, ℤ).

Proof. We denote by G the group generated by the two subgroups. So we have to show G = Γ_{L}. Let g ∈ Γ_{L}. We consider b := g(e_{4}). We can aply I.3.2 and can hence assume that b is contained in the orthogonal complement of e_{1}, e_{2}. Using Eichler transformations E(e_{1}, v), E(e_{2}, v), where v is orthogonal to e_{1}, e_{2}, we can transform b to e_{4} and hence g reduce to the stabilizer of e_{4}. It remains to prove that the Eichler transformations are contained in G. To prove this we decompose v = e + v_{0}, where e ∈ ℤe_{3} + ℤe_{4} and v_{0} ∈ L_{0}. We know E(e_{1}, v) = E(e_{1}, e)E(e_{1}, v_{0}). The element E(e_{1}, e) is contained in the image of SL(2, ℤ)×SL(2, ℤ) and the element E(e_{1}, v_{0}) stabilizes e_{4}. The same argument

shows E(e_{2}, v) ∈ G.  ⊔ ⊓

erZb 3.6 Proposition. The group Γ_{L} is generated by

1. the stabilizer of e_{4},

2. the image of (\binom{1 0}{0 1}, \binom{0−1}{1 0}).

Proof. Let G be the group generated by the these transformations. The transformtion corresponding to (\binom{1 0}{0 1}, \binom{1 1}{0 1}) stabilizes e_{4}. Hence the group {±E} × SL(2, ℤ) is contained in G, because SL(2, ℤ) is generated by the two matrices \binom{1 1}{0 1} and \binom{0−1}{1 0}. The transformation τ which acts on X as transposition X ↦ X′ is contained in the stabilizer of e_{4}. Conjugation with τ transforms {±E} × SL(2, ℤ) to SL(2, ℤ) × {±E}. Hence the image of the full

SL(2, ℤ) × SL(2, ℤ) is contained in G and we can apply I.3.5.  ⊔ ⊓

Maxpar 3.7 Lemma. Let L_{1} denote the orthogonal complement of e_{3}, e_{4} in L The stabilizer of e_{4} in Γ_{L} contains the following two subgroups:

1. The group Γ_{L_{1}} := O(L_{1}) ∩ Γ_{L}.

2. The abelian group A consisting of Eichler elements E(e_{4}, v), v ∈ L_{1}.

The full stabilizer is the semidirect product Γ_{L_{1}} · A.

Proof. Let g(e_{4}) = e_{4}. We define b = g(e_{3}). Then (b, b) = 0 and (b, e_{4}) = 1. We have to find an element v orthogonal to e_{3}, e_{4} such that E(e_{4}, v)(b) = e_{3}. The

element v = b − e_{3} − (b, e_{3})e_{4} has this property.  ⊔ ⊓

We reformulate I.3.5 in terms of the corresponding transformations on the tube domain

§4. Gritsenko’s singular modular form 47

erZc 3.8 Remark. The generators in I.3.5 act on the tube domain coordinates Z = (z_{1}, z_{2}, 𝔷) as follows:

1. Z ↦ g(Z) + H, where H ∈ L_{1} and g ∈ Γ_{L_{1}} = O(L_{1}) ∩ Γ_{L}.

2. (z_{1}, z_{2}, 𝔷) ↦ (q(Z)z_{2}^{−1}, −z_{2}^{−1}, z_{2}^{−1}𝔷).

4. Gritsenko’s singular modular form

In this section we assume that L is an even self dual lattice of signature (2, n). This implies n ≡ 2 mod 8. We assume n > 2. It is known that L contains a hyperbolic ℤ-plane ℤe_{1} + ℤe_{2} ((e_{1}, e_{1}) = (e_{2}, e_{2}) = 0, (e_{1}, e_{2}) = 0). This plane splits, L = ℤe_{1} + ℤe_{2} + L_{1}, where L_{1} is the orthogonal complement of e_{1}, e_{2}. This is an even self dual lattice of signature (1, n − 1). We consider the half plane H′ ⊂ V_{1}(ℂ), where V_{1} = L_{1} ⊗_{ℤ} ℝ. Recall that Z ↦ (1, −q(Z), Z) defines the isomorphism from H′ to the projective model H. We have H′ = V_{1} + iP, where P is the distinguished positive cone in V_{1}. We need the number (constant Fourier coefficient of the elliptic Eisenstein series of weight k

γ_{k} = \frac{(k − 1)!ζ(k)}{(2πi)^{k}} (k ≥ 4)

and the finite sum

σ_{k}(T) = ∑_{d|T d>0} d^{k},

where T is a non-zero element of a lattice L_{1} and d runs through all natural numbers with T/d ∈ L_{1}. We denote the boundary of the positive cone by ∂P. It consists of all elements T of the closure of P with the property q(T) = 0.

SinG 4.1 Theorem. The series

f(Z) := γ_{\frac{n−2}{2}} + ∑_{{T ∈L_{1}∩∂P \atop T ≠0}} σ_{\frac{n−4}{2}} (T)e^{2πi(T,Z)}

is a modular form on O′(L) of weight (n − 2)/2 (with trivial character). It does not depend on the choice of e_{1}, e_{2}.

Elements with the property q(T) sometimes are called “singular” (because q(T) is a generalization of the determinant of 2 × 2-matrices), and a modular form whose Fourier coefficients are supported on the set of singular T is called a “singular modular form”. Proof of .4.1. We treat the various generators of O′(L) which equals Γ_{L} in the self-dual case. The only non-trivial generator in .3.8 is

(z_{1}, z_{2}, 𝔷) ↦ (z_{1} − q(𝔷)z_{2}^{−1}, −z_{2}^{−1}, z_{2}^{−1}𝔷).

48 Chapter V. Constructions of modular forms

The stated transformation formula is

f(z_{1} − q(𝔷)z_{2}^{−1}, −z_{2}^{−1}, z_{2}^{−1}𝔷) = z_{2}^{(n−2)/2} f(z_{1}, z_{2}, 𝔷).

This functional equation can seen as a property of the Fourier Jacobi coefficients of a modular form: FoJa 4.2 Definition. The Fourier Jacobi coefficients of a mdoular form

f(Z) = ∑ a(T)e^{2πi(T,Z)}

are defined by

a_{n}(z_{2}, 𝔷) := ∑_{T, t_{1}=n} a(T)e^{2πi(t_{2}z_{2}+(𝔱,𝔷))}.

It is obvious that the series a_{n} converge normally on H × W (ℂ) and that

f(Z) = ∑_{n=0}^{∞} a_{n}(z_{2}, 𝔷)e^{2πinz_{1}}.

This is called a Fourier Jacobi expansion of f. FJF 4.3 Remark. The functional equation

f(z_{1} − q(𝔷)z_{2}^{−1}, −z_{2}^{−1}, z_{2}^{−1}𝔷) = z_{2}^{r} f(z_{1}, z_{2}, 𝔷).

for the series

f(Z) = ∑_{n=0}^{∞} a_{n}(z_{2}, 𝔷)e^{2πinz_{1}}.

is equivalent with

a_{n}(−z_{2}^{−1}, z_{2}^{−1}𝔷) = e^{2πiz_{2}^{−1}q(𝔷)} z_{2}^{r} a_{n}(z_{2}, 𝔷).

§5. Jacobi forms 49

5. Jacobi forms

We have been lead to consider functions in the variables (z_{2}, 𝔷). We want to do this in a more systematic way. If one looks carefully to the previous section on sees that they the coefficients a_{n}(z_{2}, 𝔷) transform under the normalizer of the isotropic space generated by e_{1}, e_{4}. This normalizer contains to essential parts, a Heisenberg group and one embedded SL(2). We start to describe the Heisenberg group:

DefHeis 5.1 Remark. The transformations

[t, 𝔥, 𝔨] := E(e_{4}, te_{1} + 𝔥)E(e_{1}, 𝔨), t ∈ ℝ, 𝔥, 𝔨 ∈ V_{0} = L_{0}ℝ,

perform a group H(V_{0}) with the group law

[t_{1}, 𝔥_{1}, 𝔨_{1}][t_{2}, 𝔥_{2}, 𝔨_{2}] = [t_{1} + t_{2} + (𝔥_{1}, 𝔨_{2}), 𝔥_{1} + 𝔥_{2}, 𝔨_{1} + 𝔨_{2}].

Hence H(V_{0}) is a central extension of ℝ,

\begin{matrix}0 & ⟶ ℝ & ⟶ & H(V_{0}) & ⟶ & V_{0} × V_{0} ⟶ 0 \\ t ⟼ & [t, 0, 0], [t, 𝔥, 𝔨] & ⟼ & (𝔥, 𝔨). \end{matrix}

The action on the tube domain coordinates is given by

(z_{1}, z_{2}, 𝔷) ⟼ (z_{1} − t + (𝔨, 𝔷) − q(𝔨)z_{2}, z_{2}, 𝔷 − z_{2}𝔨 − 𝔥).

We call H(V_{0}) the Heisenberg group of (V_{0}, q). The integral Heisenberg group H(L_{0}) is defined by the additional conditions t ∈ ℤ, 𝔥, 𝔨 ∈ L_{0}.

The Heisenberg group is contained in the normalizer of the plane spanned by e_{1}, e_{4}. Recall that we write sometimes \binom{−x_{4}x_{1}}{x_{2} x_{3}} = x_{1}e_{1} + ⋯ + x_{4}e_{4}. The plane spanned by e_{1}, e_{4} consists of matrices of the form \binom{∗ ∗}{0 0}. This matrices are invariant under multiplication from the left. Hence the embedded group {±E} × SL(2, ℝ) ⊂ O(V) is contained in the normalizer of the plane generated by e_{1}, e_{4}. We denote by [M] the image of (E, M) in the orthogonal group O′(V). The action on the tube domain coordinates is given by

(z_{1}, z_{2}, 𝔷) ⟼ (z_{1} + \frac{cq(𝔷)}{cz_{2} + d}, \frac{az_{2} + d}{cz_{2} + d}, \frac{𝔷}{cz_{2} + d}).

A simple calculation yields the commutation law

[t, 𝔥, 𝔨][M] = [M][t + .

DefJac 5.2 Proposition. The set

J(V_{0}) := ({±E} × SL(2, ℝ) · H(V_{0})

is a group (actually a semidirect product with normal subgroup H(V_{0})) which is contained in the normalizer of the plane generated by e_{1}, e_{4}.

The group J(V_{0}) is called the Jacobi subgroup and

J(L_{0}) := (SL(2, ℤ) × {±E}) · H(L_{0})

the integral Jacobi subgroup.

We introduced the Jacobi group as a subgroup of O(V). It is worthwhile to mention, that the given relations defines the Jacobi group as an abstract group related to V_{0} (or L_{0}).

DefJac 5.3 Definition. A Jacobi form of weight r and index m (both integers) with respect to the lattice L_{0} is a holomorphic function f : H × V_{0}(ℂ) → ℂ, which transforms as

f(Mτ) = (cτ + d)^{r} exp(2πim\frac{cq(𝔷)}{cτ + d})f(τ, 𝔷),
f(τ, 𝔷 + 𝔨τ + 𝔥) = exp{−2πim(𝔨, 𝔷) + q(𝔨)τ))}f(τ, 𝔷)

The definition of a Jacobi form needs not the realization of J(L_{0}) is subgroup of O(V). Nevertheless this realization is useful:

andJac 5.4 Remark. A holomorphic function f : H × V_{0}(ℂ) → ℂ is a Jacobi form of weight r and index m if and only if the function

f(z_{2}, 𝔷)e^{2πimz_{1}}

transforms like a modular form with respect to the group J(L_{0}) considered as subgroup of O′(L)

Jacobi forms are periodic and hence admit a Fourier expansion

f(τ, 𝔷) = ∑ b(j, 𝔥)e^{2πi(jτ+(𝔥,gotz)},

where j runs through all integers and 𝔥 through the dual lattice of L_{0}.

deP 5.5 Lemma. The Fourier coefficients b(j, 𝔥) of a Jacobi form of index m only depend on mj + q(𝔥) and on the image of ℌ in the group L′_{0}/mL_{0}.

A Jacobi form is called regular at infinity if

b(j, 𝔥) ≠ 0 ⟹ (m, j, 𝔥) ∈ P̄.

Our interest in Jacobi forms comes from

JacJac 5.6 Remark. The Fourier-Jacobi coefficient a_{m}(z_{2}, 𝔷) (s. .4.2) of a modular form (regular at infinity) on Γ_{L} of weight r and trivial character is a Jacobi form of weight r and index m (regular at infinity).

The proof of .4.1 rested on the following converse theorem:

§6. Hecke operators, generalities 51

UmK 5.7 Proposition. Let a_{m}(z_{2}, 𝔷) be a sequence of Jacobi forms of weigth r and index m such that the series

f(Z) := ∑_{m=0}^{∞} a_{m}(z_{2}, 𝔷)e^{2πimz_{1}}

converges in H′ and is invariant under Γ_{L_{1}}. Then f defines a modular form of weight r on Γ_{L} (with trivial character).

KUR 5.8 Theorem. Let

h(τ, 𝔷) = ∑ b(j, 𝔥)e^{2πi(jτ+(𝔥,𝔷)},

be a Jacobi form of weight r and index 1, which is regular at infinity. Then the function

f(Z) = b(0, 0)γ_{r} + ∑_{T ∈L′, T ≠0, T ∈P̄} ∑_{d|T} d^{k−1}b(\frac{t_{1}t_{2}}{d^{2}} , \frac{𝔱}{d}) e^{2πi(T,Z)}

is a modular form on Γ_{L} of weight k (with trivial character).

6. Hecke operators, generalities

Let G be an abstract group which acts on a complex vector space A from the right. This means that any g induces an linear operator

A ⟶ A, f ⟼ f|g,

such that the unit element acts as identity and such that f|g|h = f|gh. The subspace of G-invariant elements is denoted by A^{G} as usual. Let Γ ⊂ G be a subgroup. Let g ∈ G be an element such that the double coset

gΓg := { gγg; γ ∈ Γ }

contains only finitely many right cosets

gΓg = ⋃_{j=1}^{h} Γg_{j}.

We consider the operator

f ⟼ ∑_{j=1}^{h} f|g_{j}.

We are interested in this operator when f is G-invariant. Then we make two obvious observations:

1) f|g only depends on the coset gΓg and not on the choice of the representatives.

2) The element f|g is G-invariant again.

This means that we obtain an operator

T (g) : A^{G} ⟶ A^{G}.

This is the prototype of a Heckeoperator.

A first example

We take Γ = SL(2, ℤ) and G = GL_{+}(2, ℚ). In this case an arbitrary double coset ΓgΓ for g ∈ G contains only finitely many right cosets. This follows from

SLzer 6.1 Proposition. Let M be an integral matrix of positive determinant δ. The double coset SL(2, ℤ)M SL(2, ℤ) contains a diagonal matrix with positive diagonal entries a, d, such that a divides d (and such that ad = δ)) and this representant is unique.

The set O(δ) of all integral matrices of positive determinant δ decomposes in right cosets as

O(δ) = ⋃_{{ad=δ, a>0 \atop 0≤b<d}} SL(2, ℤ) ( \begin{matrix}a & b \\ 0 & d \end{matrix} ) .

7. Hecke operators for Jacobi forms

We want apply this for elements g of the orthogonal group O′(V). As space A we take the space of all holomorphic functions on H̃ with the property F (tz) = t^{−r}F (z) and the action of O′(V) is given by F |g(z) = F (gz). If we consider H′ instead of H̃ and f(Z) := F (z_{1}, z_{2}, 1, ∗, 𝔷) then the action is given by

f|g(Z) = J(g, Z)^{−r}f(gZ).

For the group Γ we want to take the Jacobi group J(L_{0}). So we need elements g such that J(L_{0})gJ(L_{0}) contains only finitely many right cosets.

Recall that we defined a homomorphism from SL(2, ℝ) × SL(2, ℝ) into O′(V), which was given by the formulae

((A, B), X) ⟼ AXB′.

§7. Hecke operators for Jacobi forms 53

The same formula defines a homomorphism

{ (A, B) ∈ GL_{+}(2, ℝ) × GL_{+}(2, ℝ); det A det B = 1 } ⟶ O′(V).

The image is the same because (A, B) and (A/\sqrt{det A}, B/\sqrt{det B}) define the same orthogonal transformation. But nevertheless this homomorphism is useful, because it gives rise to an embedding

GL_{+}(2, ℝ) ⟶ O′(V), A ⟼ [A] := (( \begin{matrix}det A^{−1} & 0 \\ 0 & 1 \end{matrix} ) , A) .

The elements of GL_{+}(2, ℚ) define rational orthogonal transformations, i.e. transformations which preserve V_{ℚ}.

There action on the tube domain H′ and the automorphy factor is given by

actGL 7.1 Remark. Let M = \binom{a b}{c d} be an element of GL_{+}(2, ℝ). Its action on the tube domain is given by

(z_{1}, z_{2}, 𝔷) ⟼ (\frac{z_{1} + cq(𝔷)}{det M}, \frac{az_{2} + b}{cz_{2} + d}, \frac{𝔷}{cz_{2} + d})

and the automorphy factor is

J([M], (z_{1}, z_{2}, 𝔷)) = (cz_{2} + d).

HeDe 7.2 Lemma. Let M ∈ GL_{+}(2, ℚ) be an integral matrix (not necessarily of determinant one). Then J(L_{0})[M/ det M]J(L_{0}) contains only finitely many right cosets. More precisely: Let

SL(2, ℤ)M SL(2, ℤ) = ⋃_{j=1}^{h} SL(2, ℤ)M_{j},

then

J(L_{0})[M/ det M]J(L_{0}) = ⋃_{j=1}^{h} J(L_{0})[M_{j}/ det M_{j}]

(both understood as disjoint union).

Proof. Let N = M/ det M. We have to consider

J(L_{0})[N]J(L_{0}) = H(L_{0})[SL(2, ℤ)][N]H(L_{0})[SL(2, ℤ)].

Because of H(L_{0})[SL(2, ℤ)] = [SL(2, ℤ)]H(L_{0}) this equals

J(L_{0})[N]J(L_{0}) = H(L_{0})[SL(2, ℤ)N SL(2, ℤ)]H(L_{0}).

Therefore we can assume (0.6.1) that N (and M) are diagonal,

M = ( \begin{matrix}a & 0 \\ 0 & d \end{matrix} ) , m = ab > 0.

Now we write

J(L_{0})[N]J(L_{0}) = [SL(2, ℤ)]H(L_{0})[N]H(L_{0})[SL(2, ℤ)].

We claim that H(L_{0})[N]H(L_{0}) contains only one right coset:

Heiein 7.3 Lemma. Let N be a diagonal matrix with entries 1/a, 1/d with natural numbers a, b. Then

H(L_{0})[N]H(L_{0}) = H(L_{0})[N].

Proof of the Lemma. It is sufficient to show

[N]H(L_{0})[N^{−1}] ⊂ H(L_{0}).

Let [t, 𝔨, 𝔥] ∈ H(L_{0}). We compute the transformation [N][t, 𝔨, 𝔥][N^{−1}] on the tube domain.

Let now f(z_{2}, 𝔷) be a Jacobi form of weight k and index m. We consider it as functions on H′

F (Z) = f(z_{2}, 𝔷)e^{2πimz_{1}}.

This function has the transformation law

F (γ(Z)) = J(γ, Z)^{r}(F (Z), (γ ∈ J(L_{0})).

We apply to this function the operator which comes from the double coset

J(L_{0})[M/ det M]J(L_{0}) = ⋃_{j=1}^{h} J(L_{0})[M_{j}/ det M_{j}]

where M is an integral matrix of positive determinant δ. The matrices M_{j} (s. .7.2) are also integral of determinant δ. The determinant of M_{j}/ det M_{j} is 1/δ. A glance to .7.1 shows that this operator maps F to a function of the form

F̃(Z) = f̃(z_{2}, 𝔷)e^{2πimδz_{1}}.

Here f̃(z_{2}, 𝔷) must be a Jacobi form of weight r and index mδ. We denote this operator by

f̃ = f|V (M).

This is a linear operator, which maps Jacobi forms of weight r and index m to Jacobi forms of weight r and index mδ. This operator depends on the double coset SL(2, ℤ)M SL(2, ℤ).

Let’s denote the space of all Jacobi forms of weight r and index m by [J(L_{0}), r, m]. What we constructed is an operator

V (M) : [J(L_{0}), r, m] ⟶ [J(L_{0}), r, δm].

This operator depends only on the double coset SL(2, ℤ)M SL(2, ℤ). We know (.7.2) that the set of all integral matrices of determinant δ contains only finitely many double cosets. We denote the sum of the corresponding operators by

V (δ) := ∑_{SL(2,ℤ)M SL(2,ℤ), det M=δ} V (M).

This is the Hecke operator which we had in mind.

Using .7.2 we get an explicit formula for this operator. From .7.1 and .7.2 we obtain

§7. Hecke operators for Jacobi forms 55

HeF 7.4 Proposition. The Hecke operator

V (δ) : [J(L_{0}), r, m] ⟶ [J(L_{0}), r, δm]

is given by

f(z_{2}, 𝔷)|V (δ) = ∑_{{ad=δ, a>0 \atop 0≤b<d}} a^{r}δ^{−1}f(\frac{az_{2} + b}{d}, 𝔷).