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An overwiev of a conjecture of Kitaoka

Marcin Mazur

An overwiev of a conjecture of Kitaoka – p. 1/2

V is a finite dimensional Q-vector space.

An overwiev of a conjecture of Kitaoka – p. 2/2

V is a finite dimensional Q-vector space. β : V × V − → Q is a positive definite symmetric bilinear

form (i.e. an inner product).

An overwiev of a conjecture of Kitaoka – p. 2/2

V is a finite dimensional Q-vector space. β : V × V − → Q is a positive definite symmetric bilinear

form (i.e. an inner product).

q is the associated quadratic form.

An overwiev of a conjecture of Kitaoka – p. 2/2

V is a finite dimensional Q-vector space. β : V × V − → Q is a positive definite symmetric bilinear

form (i.e. an inner product).

q is the associated quadratic form.

A lattice is a finitely generated subgroup of V which contains a basis of V .

An overwiev of a conjecture of Kitaoka – p. 2/2

V is a finite dimensional Q-vector space. β : V × V − → Q is a positive definite symmetric bilinear

form (i.e. an inner product).

q is the associated quadratic form.

A lattice is a finitely generated subgroup of V which contains a basis of V . A positive lattice is a pair (L, β), where L is a lattice in a finite dimensional vector space with an inner product β.

An overwiev of a conjecture of Kitaoka – p. 2/2

V is a finite dimensional Q-vector space. β : V × V − → Q is a positive definite symmetric bilinear

form (i.e. an inner product).

q is the associated quadratic form.

A lattice is a finitely generated subgroup of V which contains a basis of V . A positive lattice is a pair (L, β), where L is a lattice in a finite dimensional vector space with an inner product β. It is often more convenient to use q instead of β.

An overwiev of a conjecture of Kitaoka – p. 2/2

Let (L, β) be a positive lattice.

An overwiev of a conjecture of Kitaoka – p. 3/2

Let (L, β) be a positive lattice. Consider a number field k with the ring of integers R.

An overwiev of a conjecture of Kitaoka – p. 3/2

Let (L, β) be a positive lattice. Consider a number field k with the ring of integers R. We can extend β to a symmetric R-bilinear form βR on

RL = R ⊗Z L by setting βR(a ⊗ v, b ⊗ w) = abβ(v, w).

An overwiev of a conjecture of Kitaoka – p. 3/2

Let (L, β) be a positive lattice. Consider a number field k with the ring of integers R. We can extend β to a symmetric R-bilinear form βR on

RL = R ⊗Z L by setting βR(a ⊗ v, b ⊗ w) = abβ(v, w).

• Question. Let (L1, β1), (L2, β2) be positive lattices such that

RL1 and RL2 are isometric. Does it follow that L1 and L2

are isometric?

An overwiev of a conjecture of Kitaoka – p. 3/2

Let (L, β) be a positive lattice. Consider a number field k with the ring of integers R. We can extend β to a symmetric R-bilinear form βR on

RL = R ⊗Z L by setting βR(a ⊗ v, b ⊗ w) = abβ(v, w).

• Question. Let (L1, β1), (L2, β2) be positive lattices such that

RL1 and RL2 are isometric. Does it follow that L1 and L2

are isometric?

Exercise: Consider the positive lattices (Z2, 2x2 + 3y2) and

(Z2, x2 + 6y2). These are not isometric but after tensoring

with the Gaussian integers Z[i] they become isometric via isometry given by the matrix

• 2i,3

1,−i

• .

An overwiev of a conjecture of Kitaoka – p. 3/2

However, if we restrict our attention to totally real number fields k then the situation is much more optimistic.

An overwiev of a conjecture of Kitaoka – p. 4/2

However, if we restrict our attention to totally real number fields k then the situation is much more optimistic.

Conjecture A (Kitaoka) If k is totally real and σ : RL1 −

→ RL2

is an isometry then σ(L1) = L2.

An overwiev of a conjecture of Kitaoka – p. 4/2

First attempt

An overwiev of a conjecture of Kitaoka – p. 5/2

First attempt

Tensor product of positive lattices.

• Definition. Let (L1, β1), (L2, β2) be positive lattices. Define

(L1, β1) ⊗ (L2, β2) := (L1 ⊗ L2, β),

where β(m1 ⊗ m2, n1 ⊗ n2) = β1(m1, n1)β2(m2, n2).

An overwiev of a conjecture of Kitaoka – p. 5/2

First attempt

Tensor product of positive lattices.

• Definition. Let (L1, β1), (L2, β2) be positive lattices. Define

(L1, β1) ⊗ (L2, β2) := (L1 ⊗ L2, β),

where β(m1 ⊗ m2, n1 ⊗ n2) = β1(m1, n1)β2(m2, n2).

Conjecture B (Kitaoka) Let (M, β), (M1, β1), (M2, β2) be

positive lattices. If (M, β) ⊗ (M1, β1) and (M, β) ⊗ (M2, β2) are isometric then so are (M1, β1) and (M2, β2).

An overwiev of a conjecture of Kitaoka – p. 5/2

Consider the trace form tr = trk/Q on R. The pair (R, tr) is a positive lattice.

An overwiev of a conjecture of Kitaoka – p. 6/2

Consider the trace form tr = trk/Q on R. The pair (R, tr) is a positive lattice. If (L, β) is a positive lattice and βR is the extension of β to

RL then (RL, tr ◦ βR) is a positive lattice naturally isometric

to (R, tr) ⊗ (L, β).

An overwiev of a conjecture of Kitaoka – p. 6/2

Consider the trace form tr = trk/Q on R. The pair (R, tr) is a positive lattice. If (L, β) is a positive lattice and βR is the extension of β to

RL then (RL, tr ◦ βR) is a positive lattice naturally isometric

to (R, tr) ⊗ (L, β). Any isometry of RL1 and RL2 induces isometry of

(R, tr) ⊗ (L1, β1) and (R, tr) ⊗ (L2, β2). Conjecture B implies

that (L1, β1) and (L2, β2) are isometric.

An overwiev of a conjecture of Kitaoka – p. 6/2

Minimal vectors

An overwiev of a conjecture of Kitaoka – p. 7/2

Minimal vectors

• Definition. Let (L, β) be a positive lattice.

An overwiev of a conjecture of Kitaoka – p. 7/2

Minimal vectors

• Definition. Let (L, β) be a positive lattice.

min(L) = min{q(v) : v ∈ L, v = 0}

An overwiev of a conjecture of Kitaoka – p. 7/2

Minimal vectors

• Definition. Let (L, β) be a positive lattice.

min(L) = min{q(v) : v ∈ L, v = 0} M(L) = {v ∈ L : q(v) = min(L)}.

An overwiev of a conjecture of Kitaoka – p. 7/2

Minimal vectors

• Definition. Let (L, β) be a positive lattice.

min(L) = min{q(v) : v ∈ L, v = 0} M(L) = {v ∈ L : q(v) = min(L)}.

• Exercise. M(R, tr) = {−1, 1}, min(R, tr) = [k : Q].

An overwiev of a conjecture of Kitaoka – p. 7/2

Minimal vectors

• Definition. Let (L, β) be a positive lattice.

min(L) = min{q(v) : v ∈ L, v = 0} M(L) = {v ∈ L : q(v) = min(L)}.

• Exercise. M(R, tr) = {−1, 1}, min(R, tr) = [k : Q].
• Definition. A positive lattice (M, γ) is of E-type if

M(M ⊗ L) = M(M) ⊗ M(L) for any positive lattice L.

An overwiev of a conjecture of Kitaoka – p. 7/2

• Theorem. Fix a positive integer n.

An overwiev of a conjecture of Kitaoka – p. 8/2

• Theorem. Fix a positive integer n. Suppose that either (R, tr)

is of E-type or every positive lattice of rank at most n is of

E-type.

An overwiev of a conjecture of Kitaoka – p. 8/2

• Theorem. Fix a positive integer n. Suppose that either (R, tr)

is of E-type or every positive lattice of rank at most n is of

E-type. If L1, L2 are positive lattices of rank at most n and σ : RL1 − → RL2 is an isometry, then σ(L1) = L2.

An overwiev of a conjecture of Kitaoka – p. 8/2

• Theorem. Fix a positive integer n. Suppose that either (R, tr)

is of E-type or every positive lattice of rank at most n is of

E-type. If L1, L2 are positive lattices of rank at most n and σ : RL1 − → RL2 is an isometry, then σ(L1) = L2.

Idea of the proof.

An overwiev of a conjecture of Kitaoka – p. 8/2

• Theorem. Fix a positive integer n. Suppose that either (R, tr)

is of E-type or every positive lattice of rank at most n is of

E-type. If L1, L2 are positive lattices of rank at most n and σ : RL1 − → RL2 is an isometry, then σ(L1) = L2.

Idea of the proof. Let v ∈ M(R ⊗ L1).

An overwiev of a conjecture of Kitaoka – p. 8/2

• Theorem. Fix a positive integer n. Suppose that either (R, tr)

is of E-type or every positive lattice of rank at most n is of

E-type. If L1, L2 are positive lattices of rank at most n and σ : RL1 − → RL2 is an isometry, then σ(L1) = L2.

Idea of the proof. Let v ∈ M(R ⊗ L1). By assumption, v ∈ L1.

An overwiev of a conjecture of Kitaoka – p. 8/2

• Theorem. Fix a positive integer n. Suppose that either (R, tr)

is of E-type or every positive lattice of rank at most n is of

E-type. If L1, L2 are positive lattices of rank at most n and σ : RL1 − → RL2 is an isometry, then σ(L1) = L2.

Idea of the proof. Let v ∈ M(R ⊗ L1). By assumption, v ∈ L1. Clearly

σ(v) ∈ M(R ⊗ L2).

An overwiev of a conjecture of Kitaoka – p. 8/2

• Theorem. Fix a positive integer n. Suppose that either (R, tr)

is of E-type or every positive lattice of rank at most n is of

E-type. If L1, L2 are positive lattices of rank at most n and σ : RL1 − → RL2 is an isometry, then σ(L1) = L2.

Idea of the proof. Let v ∈ M(R ⊗ L1). By assumption, v ∈ L1. Clearly

σ(v) ∈ M(R ⊗ L2). Thus w = σ(v) ∈ L2.

An overwiev of a conjecture of Kitaoka – p. 8/2

• Theorem. Fix a positive integer n. Suppose that either (R, tr)

is of E-type or every positive lattice of rank at most n is of

E-type. If L1, L2 are positive lattices of rank at most n and σ : RL1 − → RL2 is an isometry, then σ(L1) = L2.

Idea of the proof. Let v ∈ M(R ⊗ L1). By assumption, v ∈ L1. Clearly

σ(v) ∈ M(R ⊗ L2). Thus w = σ(v) ∈ L2. Now v⊥ and w⊥ are

positive lattices of lower rank and σ induces isometry of

Rv⊥ and Rw⊥.

An overwiev of a conjecture of Kitaoka – p. 8/2

• Theorem. Fix a positive integer n. Suppose that either (R, tr)

is of E-type or every positive lattice of rank at most n is of

E-type. If L1, L2 are positive lattices of rank at most n and σ : RL1 − → RL2 is an isometry, then σ(L1) = L2.

Idea of the proof. Let v ∈ M(R ⊗ L1). By assumption, v ∈ L1. Clearly

σ(v) ∈ M(R ⊗ L2). Thus w = σ(v) ∈ L2. Now v⊥ and w⊥ are

positive lattices of lower rank and σ induces isometry of

Rv⊥ and Rw⊥. Inductively, we may assume that σ(v⊥) = w⊥.

An overwiev of a conjecture of Kitaoka – p. 8/2

• Theorem. Fix a positive integer n. Suppose that either (R, tr)

is of E-type or every positive lattice of rank at most n is of

E-type. If L1, L2 are positive lattices of rank at most n and σ : RL1 − → RL2 is an isometry, then σ(L1) = L2.

Idea of the proof. Let v ∈ M(R ⊗ L1). By assumption, v ∈ L1. Clearly

σ(v) ∈ M(R ⊗ L2). Thus w = σ(v) ∈ L2. Now v⊥ and w⊥ are

positive lattices of lower rank and σ induces isometry of

Rv⊥ and Rw⊥. Inductively, we may assume that σ(v⊥) = w⊥. This implies that σ(QL1) = QL2.

An overwiev of a conjecture of Kitaoka – p. 8/2

• Theorem. Fix a positive integer n. Suppose that either (R, tr)

is of E-type or every positive lattice of rank at most n is of

E-type. If L1, L2 are positive lattices of rank at most n and σ : RL1 − → RL2 is an isometry, then σ(L1) = L2.

Idea of the proof. Let v ∈ M(R ⊗ L1). By assumption, v ∈ L1. Clearly

σ(v) ∈ M(R ⊗ L2). Thus w = σ(v) ∈ L2. Now v⊥ and w⊥ are

positive lattices of lower rank and σ induces isometry of

Rv⊥ and Rw⊥. Inductively, we may assume that σ(v⊥) = w⊥. This implies that σ(QL1) = QL2. Since Li = QLi ∩ RLi, we get σ(L1) = L2.

An overwiev of a conjecture of Kitaoka – p. 8/2

Which lattices are of E-type?

An overwiev of a conjecture of Kitaoka – p. 9/2

Which lattices are of E-type?

• Theorem. Every positive lattice of rank at most 43 is of

E-type.

An overwiev of a conjecture of Kitaoka – p. 9/2

Which lattices are of E-type?

• Theorem. Every positive lattice of rank at most 43 is of

E-type.

• Theorem. (Steinberg) For any n ≥ 292 there is a positive

lattice (unimodular) of rank n which is not of E-type.

An overwiev of a conjecture of Kitaoka – p. 9/2

Which lattices are of E-type?

• Theorem. Every positive lattice of rank at most 43 is of

E-type.

• Theorem. (Steinberg) For any n ≥ 292 there is a positive

lattice (unimodular) of rank n which is not of E-type.

• Idea. It is easy to see that if L∗ is the dual of L then

min(L ⊗ L∗) ≤ n. If we have a unimodular lattice with min(L) > √n then min(L ⊗ L) < min(L)2 and L is not of E-type. Such lattices exist for n ≥ 292 (Conway &

Thompson).

An overwiev of a conjecture of Kitaoka – p. 9/2

Which lattices are of E-type?

• Theorem. Every positive lattice of rank at most 43 is of

E-type.

• Theorem. (Steinberg) For any n ≥ 292 there is a positive

lattice (unimodular) of rank n which is not of E-type.

• Idea. It is easy to see that if L∗ is the dual of L then

min(L ⊗ L∗) ≤ n. If we have a unimodular lattice with min(L) > √n then min(L ⊗ L) < min(L)2 and L is not of E-type. Such lattices exist for n ≥ 292 (Conway &

Thompson).

• Question. Is (R, tr) always of E-type?

An overwiev of a conjecture of Kitaoka – p. 9/2

Which lattices are of E-type?

• Theorem. Every positive lattice of rank at most 43 is of

E-type.

• Theorem. (Steinberg) For any n ≥ 292 there is a positive

lattice (unimodular) of rank n which is not of E-type.

• Idea. It is easy to see that if L∗ is the dual of L then

min(L ⊗ L∗) ≤ n. If we have a unimodular lattice with min(L) > √n then min(L ⊗ L) < min(L)2 and L is not of E-type. Such lattices exist for n ≥ 292 (Conway &

Thompson).

• Question. Is (R, tr) always of E-type? No counterexample is

known and the answer is positive if k is contained in a Galois extension with nilpotent Galois group.

An overwiev of a conjecture of Kitaoka – p. 9/2

Second attempt

An overwiev of a conjecture of Kitaoka – p. 10/2

Second attempt

• Theorem. Let k/Q be totally real and Galois, Γ = Gal(k/Q).

FAE:

• 1. Conjecture A holds for k.
• 2. If G ⊆ GLn(R) is a finite, Γ-stable subgroup then

G ⊆ GLn(Z).

An overwiev of a conjecture of Kitaoka – p. 10/2

Second attempt

• Theorem. Let k/Q be totally real and Galois, Γ = Gal(k/Q).

FAE:

• 1. Conjecture A holds for k.
• 2. If G ⊆ GLn(R) is a finite, Γ-stable subgroup then

G ⊆ GLn(Z).

Idea of the proof.

An overwiev of a conjecture of Kitaoka – p. 10/2

Second attempt

• Theorem. Let k/Q be totally real and Galois, Γ = Gal(k/Q).

FAE:

• 1. Conjecture A holds for k.
• 2. If G ⊆ GLn(R) is a finite, Γ-stable subgroup then

G ⊆ GLn(Z).

Idea of the proof.

1 ⇒ 2

An overwiev of a conjecture of Kitaoka – p. 10/2

Second attempt

• Theorem. Let k/Q be totally real and Galois, Γ = Gal(k/Q).

FAE:

• 1. Conjecture A holds for k.
• 2. If G ⊆ GLn(R) is a finite, Γ-stable subgroup then

G ⊆ GLn(Z).

Idea of the proof.

1 ⇒ 2 Let P =

• g∈G

gtg.

An overwiev of a conjecture of Kitaoka – p. 10/2

Second attempt

• Theorem. Let k/Q be totally real and Galois, Γ = Gal(k/Q).

FAE:

• 1. Conjecture A holds for k.
• 2. If G ⊆ GLn(R) is a finite, Γ-stable subgroup then

G ⊆ GLn(Z).

Idea of the proof.

1 ⇒ 2 Let P =

• g∈G
• gtg. Since G is Γ-invariant, we gave

τ(P) = P for all τ ∈ Γ

An overwiev of a conjecture of Kitaoka – p. 10/2

Second attempt

• Theorem. Let k/Q be totally real and Galois, Γ = Gal(k/Q).

FAE:

• 1. Conjecture A holds for k.
• 2. If G ⊆ GLn(R) is a finite, Γ-stable subgroup then

G ⊆ GLn(Z).

Idea of the proof.

1 ⇒ 2 Let P =

• g∈G
• gtg. Since G is Γ-invariant, we gave

τ(P) = P for all τ ∈ Γ Thus P ∈ Mn(Z).

An overwiev of a conjecture of Kitaoka – p. 10/2

Second attempt

• Theorem. Let k/Q be totally real and Galois, Γ = Gal(k/Q).

FAE:

• 1. Conjecture A holds for k.
• 2. If G ⊆ GLn(R) is a finite, Γ-stable subgroup then

G ⊆ GLn(Z).

Idea of the proof.

1 ⇒ 2 Let P =

• g∈G
• gtg. Since G is Γ-invariant, we gave

τ(P) = P for all τ ∈ Γ Thus P ∈ Mn(Z). It is easy to see that P is symmetric and positive definite.

An overwiev of a conjecture of Kitaoka – p. 10/2

Second attempt

• Theorem. Let k/Q be totally real and Galois, Γ = Gal(k/Q).

FAE:

• 1. Conjecture A holds for k.
• 2. If G ⊆ GLn(R) is a finite, Γ-stable subgroup then

G ⊆ GLn(Z).

Idea of the proof.

1 ⇒ 2 Let P =

• g∈G
• gtg. Since G is Γ-invariant, we gave

τ(P) = P for all τ ∈ Γ Thus P ∈ Mn(Z). It is easy to see that P is symmetric and positive definite.

An overwiev of a conjecture of Kitaoka – p. 10/2

Second attempt

• Theorem. Let k/Q be totally real and Galois, Γ = Gal(k/Q).

FAE:

• 1. Conjecture A holds for k.
• 2. If G ⊆ GLn(R) is a finite, Γ-stable subgroup then

G ⊆ GLn(Z).

Idea of the proof.

1 ⇒ 2 Let P =

• g∈G
• gtg. Since G is Γ-invariant, we gave

τ(P) = P for all τ ∈ Γ Thus P ∈ Mn(Z). It is easy to see that P is symmetric and positive definite.

An overwiev of a conjecture of Kitaoka – p. 10/2

Second attempt

• Theorem. Let k/Q be totally real and Galois, Γ = Gal(k/Q).

FAE:

• 1. Conjecture A holds for k.
• 2. If G ⊆ GLn(R) is a finite, Γ-stable subgroup then

G ⊆ GLn(Z).

Idea of the proof.

1 ⇒ 2 Let P =

• g∈G
• gtg. Since G is Γ-invariant, we gave

τ(P) = P for all τ ∈ Γ Thus P ∈ Mn(Z). It is easy to see that P is symmetric and positive definite. Thus P defines an

inner product β on L = Zn.

An overwiev of a conjecture of Kitaoka – p. 10/2

Second attempt

• Theorem. Let k/Q be totally real and Galois, Γ = Gal(k/Q).

FAE:

• 1. Conjecture A holds for k.
• 2. If G ⊆ GLn(R) is a finite, Γ-stable subgroup then

G ⊆ GLn(Z).

Idea of the proof.

1 ⇒ 2 Let P =

• g∈G
• gtg. Since G is Γ-invariant, we gave

τ(P) = P for all τ ∈ Γ Thus P ∈ Mn(Z). It is easy to see that P is symmetric and positive definite. Thus P defines an

inner product β on L = Zn. Any g ∈ G defines an isometry

• f RL = Rn.

An overwiev of a conjecture of Kitaoka – p. 10/2

Second attempt

• Theorem. Let k/Q be totally real and Galois, Γ = Gal(k/Q).

FAE:

• 1. Conjecture A holds for k.
• 2. If G ⊆ GLn(R) is a finite, Γ-stable subgroup then

G ⊆ GLn(Z).

Idea of the proof.

1 ⇒ 2 Let P =

• g∈G
• gtg. Since G is Γ-invariant, we gave

τ(P) = P for all τ ∈ Γ Thus P ∈ Mn(Z). It is easy to see that P is symmetric and positive definite. Thus P defines an

inner product β on L = Zn. Any g ∈ G defines an isometry

• f RL = Rn. By 1, g takes L to itself, i.e. g ∈ GLn(Z).

An overwiev of a conjecture of Kitaoka – p. 10/2

2 ⇒ 1

An overwiev of a conjecture of Kitaoka – p. 11/2

2 ⇒ 1 Let (L1, β1), (L2, β2) be positive lattices and σ : RL1 − → RL2 an isometry.

An overwiev of a conjecture of Kitaoka – p. 11/2

2 ⇒ 1 Let (L1, β1), (L2, β2) be positive lattices and σ : RL1 − → RL2 an isometry. Let L = L1 ⊥ L2.

An overwiev of a conjecture of Kitaoka – p. 11/2

2 ⇒ 1 Let (L1, β1), (L2, β2) be positive lattices and σ : RL1 − → RL2 an isometry. Let L = L1 ⊥ L2. σ induces an

isometry σ of RL onto itself, where σ acts as σ on RL1 and as σ−1 on RL2.

An overwiev of a conjecture of Kitaoka – p. 11/2

2 ⇒ 1 Let (L1, β1), (L2, β2) be positive lattices and σ : RL1 − → RL2 an isometry. Let L = L1 ⊥ L2. σ induces an

isometry σ of RL onto itself, where σ acts as σ on RL1 and as σ−1 on RL2. Let G be the group of all self-isometries of

RL.

An overwiev of a conjecture of Kitaoka – p. 11/2

2 ⇒ 1 Let (L1, β1), (L2, β2) be positive lattices and σ : RL1 − → RL2 an isometry. Let L = L1 ⊥ L2. σ induces an

isometry σ of RL onto itself, where σ acts as σ on RL1 and as σ−1 on RL2. Let G be the group of all self-isometries of

• RL. A choice of a basis of L allows to identify G with a

Γ-invariant subgroup of GLn(R).

An overwiev of a conjecture of Kitaoka – p. 11/2

2 ⇒ 1 Let (L1, β1), (L2, β2) be positive lattices and σ : RL1 − → RL2 an isometry. Let L = L1 ⊥ L2. σ induces an

isometry σ of RL onto itself, where σ acts as σ on RL1 and as σ−1 on RL2. Let G be the group of all self-isometries of

• RL. A choice of a basis of L allows to identify G with a

Γ-invariant subgroup of GLn(R). The different embeddings

• f k into R allow to embed G as a discrete subgroup of a

product of [k : Q] copies of the orthogonal group On(R), which is compact.

An overwiev of a conjecture of Kitaoka – p. 11/2

2 ⇒ 1 Let (L1, β1), (L2, β2) be positive lattices and σ : RL1 − → RL2 an isometry. Let L = L1 ⊥ L2. σ induces an

isometry σ of RL onto itself, where σ acts as σ on RL1 and as σ−1 on RL2. Let G be the group of all self-isometries of

• RL. A choice of a basis of L allows to identify G with a

Γ-invariant subgroup of GLn(R). The different embeddings

• f k into R allow to embed G as a discrete subgroup of a

product of [k : Q] copies of the orthogonal group On(R), which is compact. Thus G is finite.

An overwiev of a conjecture of Kitaoka – p. 11/2

2 ⇒ 1 Let (L1, β1), (L2, β2) be positive lattices and σ : RL1 − → RL2 an isometry. Let L = L1 ⊥ L2. σ induces an

isometry σ of RL onto itself, where σ acts as σ on RL1 and as σ−1 on RL2. Let G be the group of all self-isometries of

• RL. A choice of a basis of L allows to identify G with a

Γ-invariant subgroup of GLn(R). The different embeddings

• f k into R allow to embed G as a discrete subgroup of a

product of [k : Q] copies of the orthogonal group On(R), which is compact. Thus G is finite. By 2, G ⊆ GLn(Z).

An overwiev of a conjecture of Kitaoka – p. 11/2

2 ⇒ 1 Let (L1, β1), (L2, β2) be positive lattices and σ : RL1 − → RL2 an isometry. Let L = L1 ⊥ L2. σ induces an

isometry σ of RL onto itself, where σ acts as σ on RL1 and as σ−1 on RL2. Let G be the group of all self-isometries of

• RL. A choice of a basis of L allows to identify G with a

Γ-invariant subgroup of GLn(R). The different embeddings

• f k into R allow to embed G as a discrete subgroup of a

product of [k : Q] copies of the orthogonal group On(R), which is compact. Thus G is finite. By 2, G ⊆ GLn(Z). This implies that σ(L1) = L2.

An overwiev of a conjecture of Kitaoka – p. 11/2

Finite Galois stable subgroups of GLn.

An overwiev of a conjecture of Kitaoka – p. 12/2

Finite Galois stable subgroups of GLn.

Let k be now an arbitrary (i.e. not necessarily totally real) number field, Galois over Q. What can be said about finite

Γ-stable subgroups G of GLn(R)?

An overwiev of a conjecture of Kitaoka – p. 12/2

Finite Galois stable subgroups of GLn.

Let k be now an arbitrary (i.e. not necessarily totally real) number field, Galois over Q. What can be said about finite

Γ-stable subgroups G of GLn(R)?

Conjecture C. Any such G is pointwise fixed by the

commutator of Γ. In other words, matrices of G have entries in a cyclotomic field.

An overwiev of a conjecture of Kitaoka – p. 12/2

Finite Galois stable subgroups of GLn.

Let k be now an arbitrary (i.e. not necessarily totally real) number field, Galois over Q. What can be said about finite

Γ-stable subgroups G of GLn(R)?

Conjecture C. Any such G is pointwise fixed by the

commutator of Γ. In other words, matrices of G have entries in a cyclotomic field. Since Conjecture A is true for k abelian over Q, Conjecture C implies Conjecture A.

An overwiev of a conjecture of Kitaoka – p. 12/2

Finite Galois stable subgroups of GLn.

Let k be now an arbitrary (i.e. not necessarily totally real) number field, Galois over Q. What can be said about finite

Γ-stable subgroups G of GLn(R)?

Conjecture C. Any such G is pointwise fixed by the

commutator of Γ. In other words, matrices of G have entries in a cyclotomic field. Since Conjecture A is true for k abelian over Q, Conjecture C implies Conjecture A. Conjecture C implies a much more detailed description of all Γ-stable G. Roughly speaking, any such G is build from matrices in GLn(Z) and diagonal matrices of finite order.

An overwiev of a conjecture of Kitaoka – p. 12/2

One can try to get various special cases of Conjecture C by specializing n, k or G.

An overwiev of a conjecture of Kitaoka – p. 13/2

One can try to get various special cases of Conjecture C by specializing n, k or G.

• Theorem. (MM) Conjecture C is true for n ≤ 4.

An overwiev of a conjecture of Kitaoka – p. 13/2

One can try to get various special cases of Conjecture C by specializing n, k or G.

• Theorem. (MM) Conjecture C is true for n ≤ 4.
• Theorem. (MM) Conjecture C is true if |G| has no prime

divisors bigger that 17.

An overwiev of a conjecture of Kitaoka – p. 13/2

One can try to get various special cases of Conjecture C by specializing n, k or G.

• Theorem. (MM) Conjecture C is true for n ≤ 4.
• Theorem. (MM) Conjecture C is true if |G| has no prime

divisors bigger that 17.

• Theorem. (MM) It suffices to prove Conjecture C for G

elementary abelian p-group and under the additional assumption that p is the only prime which ramifies in k/Q.

An overwiev of a conjecture of Kitaoka – p. 13/2

Finite flat group schemes.

An overwiev of a conjecture of Kitaoka – p. 14/2

Finite flat group schemes.

A finite flat group scheme over a ring S is an affine group scheme represented by an S-algebra T which is a finitely generated projective S-module.

An overwiev of a conjecture of Kitaoka – p. 14/2

Finite flat group schemes.

A finite flat group scheme over a ring S is an affine group scheme represented by an S-algebra T which is a finitely generated projective S-module. For any S-algebra A, the set G(A) = HomS−alg(T, A) is a group.

An overwiev of a conjecture of Kitaoka – p. 14/2

Finite flat group schemes.

A finite flat group scheme over a ring S is an affine group scheme represented by an S-algebra T which is a finitely generated projective S-module. For any S-algebra A, the set G(A) = HomS−alg(T, A) is a group. This means that there are S-algebra homomorphisms

m : T − → T ⊗S T (comultiplication), inv : T − → T

(coinversion), and e : T −

→ S (counit) which satisfy the

axioms dual to the group axioms.

An overwiev of a conjecture of Kitaoka – p. 14/2

T

m

• m
• T ⊗S T

id⊗m

• T ⊗S T

m⊗id T ⊗S T ⊗S T

An overwiev of a conjecture of Kitaoka – p. 15/2

T

m

• m
• T ⊗S T

id⊗m

• T ⊗S T

m⊗id T ⊗S T ⊗S T

T ⊗S T

id⊗inv

T

T

e

• m
• S

i

• An overwiev of a conjecture of Kitaoka – p. 15/2

T

m

• m
• T ⊗S T

id⊗m

• T ⊗S T

m⊗id T ⊗S T ⊗S T

T ⊗S T

id⊗inv

T

T

e

• m
• S

i

• T ⊗S T

id⊗(i◦e)

T

T

id

• m
• An overwiev of a conjecture of Kitaoka – p. 15/2

• Theorem. (MM) Let S be a ring and G a finite flat group

scheme over S with coordinate ring T such that T m is a free

S-module for some m. Then there is a closed embedding of G into GLn defined over S, where n is the rank of the S−module T m.

An overwiev of a conjecture of Kitaoka – p. 16/2

• Theorem. (MM) Let S be a ring and G a finite flat group

scheme over S with coordinate ring T such that T m is a free

S-module for some m. Then there is a closed embedding of G into GLn defined over S, where n is the rank of the S−module T m.

In particular, we can consider G(A) as a subgroup of

GLn(A) for any S-algebra A.

An overwiev of a conjecture of Kitaoka – p. 16/2

• Theorem. (MM) Let S be a ring and G a finite flat group

scheme over S with coordinate ring T such that T m is a free

S-module for some m. Then there is a closed embedding of G into GLn defined over S, where n is the rank of the S−module T m.

In particular, we can consider G(A) as a subgroup of

GLn(A) for any S-algebra A.

Suppose that S is a domain with a field of fractions L. Let L be an algebraic closure of L and let S be the integral closure of S in L. Then the group G(L) = G(S) is a finite,

Gal(L/L)−stable subgroup of GLn(S).

An overwiev of a conjecture of Kitaoka – p. 16/2

• Theorem. (MM) With the above notation, assume that S is a

normal domain of characteristic 0 and let G be a finite,

Gal(L/L)−stable subgroup of GLn(S). Then there is a finite

flat closed subgroup scheme G of GLn/S such that

G(S) = G.

An overwiev of a conjecture of Kitaoka – p. 17/2

• Theorem. (MM) With the above notation, assume that S is a

normal domain of characteristic 0 and let G be a finite,

Gal(L/L)−stable subgroup of GLn(S). Then there is a finite

flat closed subgroup scheme G of GLn/S such that

G(S) = G.

Conjecture D. If G is a finite flat commutative group scheme

• ver Z annihilated by a prime p then it is a direct sum of

copies of the multiplicative group scheme µp of order p, the etale (constant) group scheme Z/pZ of order p and, if p = 2, the nontrivial element in Ext(Z/2Z, µ2) (in the category of commutative group schemes annihilated by 2).

An overwiev of a conjecture of Kitaoka – p. 17/2

Let D ⊆ GLn(Z) be a finite subgroup consisting of diagonal

• matrices. D corresponds to a finite flat group scheme GD
• ver Z. GD is a diagonalizable group scheme.

An overwiev of a conjecture of Kitaoka – p. 18/2

Let D ⊆ GLn(Z) be a finite subgroup consisting of diagonal

• matrices. D corresponds to a finite flat group scheme GD
• ver Z. GD is a diagonalizable group scheme.

Let G ⊆ GLn(Z) be a finite subgroup such that for each prime p the reduction mod p homomorphism

GLn(Z) − → GLn(Z/p) is injective on G (by a well known

lemma of Minkowski this condition is always satisfied for

• dd primes p). G corresponds to a finite flat group scheme

G over Z. G is a constant group scheme.

An overwiev of a conjecture of Kitaoka – p. 18/2

Abrashkin proved that the only indecomposable objects in the category of finite flat commutative group schemes over

Z annihilated by 2 are the constant group scheme Z/2Z of

• rder 2, the multiplicative group scheme µ2, and the unique

non-trivial extension in Ext(Z/2Z, µ2). This non-trivial extension is given by the subgroup {±I, ±

• i

−i

• } of

GL2(Z).

An overwiev of a conjecture of Kitaoka – p. 19/2

Conjecture D was proved by Abrashkin and (independently) by Fontaine for p ≤ 17 (p ≤ 23 under GRH).

An overwiev of a conjecture of Kitaoka – p. 20/2

Conjecture D was proved by Abrashkin and (independently) by Fontaine for p ≤ 17 (p ≤ 23 under GRH).

• Theorem. (MM) Conjectures C and D are equivalent.

Moreover, they imply the following more detailed description

• f finite flat group schemes over Z.

i) every finite flat group scheme over Z is an extension of a constant group scheme by a diagonalizable group scheme; ii) every finite flat group scheme over Z of odd order is a split extension of a constant group scheme by a diagonalizable group scheme; iii) every finite flat commutative group scheme over Z of odd

• rder is a product of a diagonalizable group scheme and a

constant group scheme.

An overwiev of a conjecture of Kitaoka – p. 20/2

Can one give a similar description of finite flat group schemes over the ring of integers S in any number field L?

An overwiev of a conjecture of Kitaoka – p. 21/2

Can one give a similar description of finite flat group schemes over the ring of integers S in any number field L? Abelian varieties over L with everywhere good reduction provide many finite flat group schemes. They do not exist

• ver Q(i), Q(ζ3), Q(

√ 5). They do exist over cyclotomic fields Q(ζm) for every m ≥ 16.

An overwiev of a conjecture of Kitaoka – p. 21/2

Let K be a number field. Consider a finite extension L of K

• f degree n and let τ1, ..., τn be the embeddings L ֒

→ Q over

• K. Let u1, ..., un be a basis of L/K. Define a matrix

U = (ui,j) by ui,j = uτj

i . Plainly U ∈ GLn(K) and for any

τ ∈ ΓK the matrix UτU−1 is a permutation matrix. Fix a

prime p and denote by P the maximal, elementary abelian

p−subgroup of the diagonal matrices in GLn(K). The group PU = U−1PU is ΓK−stable and the field of definition for PU

equals N = M(ξp), where ξp is a primitive p−th root of 1 and

M is the normal closure of L/K. The only problem with the

above construction is that the group PU does not in general consist of matrices with integral entries.

An overwiev of a conjecture of Kitaoka – p. 22/2

Suppose however that all the ui’s are algebraic integers. Then U has entries in OM. Let d be the determinant of U. Then U−1PU consists of matrices with integral entries provided (1 − ξp)d−1 is an algebraic integer. In particular, if d is a unit of OL then our construction leads to examples with integral entries. It is easy to see that this is the case if OL is a free OK−module and L/K is unramified at all finite primes ( take for the ui’s a basis for OL over OK).

An overwiev of a conjecture of Kitaoka – p. 23/2

Suppose however that all the ui’s are algebraic integers. Then U has entries in OM. Let d be the determinant of U. Then U−1PU consists of matrices with integral entries provided (1 − ξp)d−1 is an algebraic integer. In particular, if d is a unit of OL then our construction leads to examples with integral entries. It is easy to see that this is the case if OL is a free OK−module and L/K is unramified at all finite primes ( take for the ui’s a basis for OL over OK). Take K = Q(

√ 36497). This field has class number 1 and an

everywhere unramified Galois totally real extension L with Galois group A5.

An overwiev of a conjecture of Kitaoka – p. 23/2

Thank You

An overwiev of a conjecture of Kitaoka – p. 24/2