A Bound for the Torsion in the K-Theory

of Algebraic Integers

Christophe Soulé

Received: November 6, 2002 Revised: May 30, 2003

A Kazuya Kato, qui marche sous la lune.

Abstract. Let m be an integer bigger than one, A a ring of algebraic integers, F its fraction field, and K_{m}(A) the m-th Quillen K-group of A. We give a (huge) explicit bound for the order of the torsion subgroup of K_{m}(A) (up to small primes), in terms of m, the degree of F over Q, and its absolute discriminant.

Let F be a number field, A its ring of integers and K_{m}(A) the m-th Quillen K-group of A. It was shown by Quillen that K_{m}(A) is finitely generated. In this paper we shall give a (huge) explicit bound for the order of the torsion subgroup of K_{m}(A) (up to small primes), in terms of m, the degree of F over Q, and its absolute discriminant. Our method is similar to the one developed in [13] for F = Q. Namely, we reduce the problem to a bound on the torsion in the homology of the general linear group GL_{N}(A). Thanks to a result of Gabber, such a bound can be obtained by estimating the number of cells of given dimension in any complex of free abelian groups computing the homology of GL_{N}(A). Such a complex is derived from a contractible CW -complex W̃ on which GL_{N}(A) with compact quotient. We shall use the construction of W̃ given by Ash in [1] . It consists of those hermitian metrics h on A^{N} which have minimum equal to one and are such that their set M(h) of minimal vectors has rank equal to N in F^{N}. To count cells in W̃/GL_{N}(A), one will exhibit an explicit compact subset of A^{N} ⊗_{Z} R which, for every h ∈ W̃, contains a translate of M(h) by some matrix of GL_{N}(A) (Proposition 2). The proof of this result relies on several arguments from the geometry of numbers using, among other things, the number field analog of Hermite’s constant [4].

The bound on the K-theory of A implies a similar upper bound for the étale cohomology of Spec (A[1/p]) with coefficients in the positive Tate twists of Z_{p}, for any (big enough) prime number p. However, this bound is quite large since it is doubly exponential both in m and, in general, the discriminant of F. We expect the correct answer to be polynomial in the discriminant and exponential in m (see 5.1). The paper is organized as follows. In Section 1 we prove a few facts on the geometry of numbers for A, including a result about the image of A^{∗} by the regulator map (Lemma 3), which was shown to us by H. Lenstra. Using these, we study in Section 2 hermitian lattices over A, and we get a bound on M(h) when h lies in W̃. The cell structure of W̃ is described in Section 3. The main Theorems are proved in Section 4. Finally, we discuss these results in Section 5, where we notice that, because of the Lichtenbaum conjectures, a lower bound for higher regulators of number fields would probably provide much better upper bounds for the étale cohomology of Spec (A[1/p]). We conclude with the example of K_{8}(Z) and its relation to the Vandiver conjecture.

1 Geometry of algebraic numbers

1.1

Let F be a number field, and A its ring of integers. We denote by r = [F : Q] the degree of F over Q and by D = |disc (K/Q)| the absolute value of the discriminant of F over Q. Let r_{1} (resp. r_{2}) be the number of real (resp. complex) places of F. We have r = r_{1} + 2 r_{2}. We let Σ = Hom (F, C) be the set of complex embeddings of F. These notations will be used throughout. Given a finite set X we let # (X) denote its cardinal.

1.2

We first need a few facts from the geometry of numbers applied to A and A^{∗}. The first one is the following classical result of Minkowski:

Lemma 1. Let L be a rank one torsion-free A-module. There exists a non zero element x ∈ L such that the submodule spanned by x in L has index

# (L/Ax) ≤ C_{1} ,

where

C_{1} = \frac{r!}{r^{r}} · 4^{r_{2}} π^{−r_{2}} \sqrt{D}

in general, and C_{1} = 1 when A is principal.

Proof. The A-module L is isomorphic to an ideal I in A. According to [7], V §4, p. 119, Minkowski’s first theorem implies that there exists x ∈ I the norm of which satisfies

|N(x)| ≤ C_{1} N(I) .

Here |N(x)| = # (A/Ax) and N(I) = # (A/I), therefore # (I/Ax) ≤ C_{1}. The case where A is principal is clear. q.e.d.

1.3

The family of complex embeddings σ : F → C, σ ∈ Σ, gives rise to a canonical isomorphism of real vector spaces of dimension r

F ⊗_{Q} R = (C^{Σ})^{+} ,

where (·)^{+} denotes the subspace invariant under complex conjugation. Given α ∈ F we shall write sometimes |α|_{σ} instead of |σ(α)|.

Lemma 2. Given any element x = (x_{σ}) ∈ F ⊗_{Q} R, there exists a ∈ A such that

∑_{σ∈Σ} |x_{σ} − σ(a)| ≤ C_{2} ,

with

C_{2} = \frac{4^{r_{1}} π^{r_{2}}}{r^{r−2} r!} \sqrt{D}

in general, and

C_{2} = 1/2  if  F = Q .

Proof. Define a norm on F ⊗_{Q} R by the formula

∥x∥ = ∑_{σ∈Σ} |x_{σ}| .

The additive group A is a lattice in F ⊗_{Q} R, and we let μ_{1}, …, μ_{r} be its successive minima. In particular, there exist a_{1}, …, a_{r} ∈ A such that ∥a_{i}∥ = μ_{i}, 1 ≤ i ≤ r, and {a_{1}, …, a_{r}} are linearly independent over Z. Any x ∈ F ⊗_{Q} R can be written

x = ∑_{i=1}^{r} λ_{i} a_{i} , λ_{i} ∈ R .

Let n_{i} ∈ Z be such that |n_{i} − λ_{i}| ≤ 1/2, for all i = 1, …, r, and

a = ∑_{i=1}^{r} n_{i} a_{i} .

Clearly

∥x − a∥ ≤ ∑_{i=1}^{r} |λ_{i} − n_{i}| ∥a_{i}∥ ≤ \frac{1}{2} (μ_{1} + ⋯ + μ_{r}) ≤ \frac{r}{2} μ_{r} .  (1)

On the other hand, we know from the product formula that, given any a ∈ A − {0},

∏_{σ∈Σ} |σ(a)| ≥ 1 .  (2)

By the inequality between arithmetic and geometric means this implies

∥a∥ = ∑_{σ∈Σ} |σ(a)| ≥ r ,

hence

μ_{i} ≥ r  for all i = 1, …, r .  (3)

Minkowski’s second theorem tells us that

μ_{1} … μ_{r} ≤ 2^{r} W 2^{−r_{2}} \sqrt{D}  (4)

([7], Lemma 2, p. 115), where W is the euclidean volume of the unit ball for ∥ · ∥ in F ⊗_{Q} R. (Note that the covolume of A is \sqrt{D}.) The volume W is the euclidean volume of those elements (x_{i}, z_{j}) ∈ R^{r_{1}} × C^{r_{2}} such that

∑_{i=1}^{r_{1}} |x_{i}| + 2 ∑_{j=1}^{r_{2}} |z_{j}| ≤ 1 .

One finds ([7], Lemma 3, p. 117)

W = 2^{r_{1}} 4^{−r_{2}}(2π)^{r_{2}}/r! .  (5)

From (3) and (4) we get

μ_{r} ≤ 2^{r} W \sqrt{D} 2^{−r_{2}} r^{−(r−1)} .  (6)

The lemma follows from (1), (5) and (6). q.e.d.

1.4

We also need a multiplicative analog of Lemma 2. Let R(F) be the regulator of F, as defined in [7] V, § 1, p. 109. Let s = r_{1} + r_{2} − 1.

Lemma 3. Let (λ_{σ}), σ ∈ Σ be a family of positive real numbers such that λ\overline{σ} = λ_{σ} when \overline{σ} is the complex conjugate of σ. There exists a unit u ∈ A^{∗} such that

Sup_{σ∈Σ} (λ_{σ} |u|_{σ}) ≤ C_{3} ( ∏_{σ∈Σ} λ_{σ})^{1/r} ,

with

C_{3} = exp (s(4 r (log 3 r)^{3})^{s−1} 2^{r_{2}−1} R(F)) .

Proof. We follow an argument of H. Lenstra. Let H ⊂ R^{r_{1}+r_{2}} be the sdimensional hyperplane consisting of vectors (x_{1}, …, x_{r_{1}+r_{2}}) such that x_{1}+x_{2}+ ⋯ + x_{r_{1}+r_{2}} = 0. Choose a subset {σ_{1}, …, σ_{r_{1}+r_{2}}} ⊂ Σ such that σ_{1}, …, σ_{r_{1}} are the real embeddings of Σ and σ_{i} ≠ \overline{σ}_{j} if i ≠ j. Given λ = (λ_{σ})_{σ∈Σ} as in the lemma, we let

ρ(λ) = (log(λ_{σ_{1}}), …, log(λ_{σ_{r_{1}}} ), 2 log(λ_{σ_{r_{1}+1}}), …, 2 log(λ_{σ_{r_{1}+r_{2}}} )) .

If u ∈ A^{∗} is a unit, and λ = (|u|_{σ}), we have ρ(λ) ∈ H. We get this way a lattice

L = {ρ(|u|_{σ}) , u ∈ A^{∗}}

in H. Define a norm ∥ · ∥ on H by the formula

∥(x_{i})∥ = Sup ( Sup_{1≤i≤r_{1}} |x_{i}|, Sup_{r_{1}+1≤i≤r_{1}+r_{2}} |x_{i}|/2) .

It is enough to show that, for any vector x ∈ H there exists a ∈ L such that

∥x − a∥ ≤ log(C_{3}) .

According to [14], Cor. 2, p. 84, we have (when r ≥ 2)

∥a∥ ≥ ε

where

ε = r^{−1}(log(3 r))^{−3} ,

for any a ∈ L − {0}. Therefore, using Minkowski’s second theorem as in the proof of Lemma 2 we get that, for any x ∈ H there exists a ∈ L with

∥x − a∥ ≤ s 2^{s−1} ε^{1−s} W vol (H/L) ,

where W is the euclidean volume of the unit ball for ∥ · ∥, where we identify H with R^{s} by projecting on the first s coordinates. Clearly W ≤ 2^{s+r_{2}−1} when, by definition (loc.cit.), vol (H/L) is equal to R(F). The lemma follows.

1.5

We now give an upper bound for the constant C_{3} of Lemma 3.

Lemma 4. The following inequality holds

R ≤ 11 r^{2} \sqrt{D} log(D)^{r−1} .

Proof. Let κ be the residue at s = 1 of the zeta function of F. According to [11], Cor. 3, p. 333, we have

κ ≤ 2^{r+1} D^{a} a^{1−r}

whenever 0 < a ≤ 1. Taking a = log(D)^{−1} we get

κ ≤ e 2^{r+1} log(D)^{r−1} .  (7)

On the other hand

κ = 2^{r_{1}}(2π)^{r_{2}} \frac{h(F) R(F)}{w(F) \sqrt{D}} ,  (8)

where h(F) is the class number of F and w(F) the number of roots of unity in F ([7], Prop. 13, p. 300). Since h(F) ≥ 1 we get

R(F) ≤ w(F) 2^{−(r_{1}+r_{2})} π^{−r_{2}} e 2^{r+1} \sqrt{D} log(D)^{r−1} .

Since the degree over Q of Q( ^{n} \sqrt{1}) is φ(n), where φ is the Euler function, we must have

φ(w(F)) ≤ r .

When n = p^{t} is an odd prime power we have

φ(p^{t}) = (p − 1) p^{t−1} ≥ p^{t/2} .

Therefore

w(F) ≤ 2 r^{2} .

Since

2^{−(r_{1}+r_{2})} π^{−r_{2}} 2^{r} = (\frac{2}{π} )^{r_{2}} ≤ 1

and 4 e ≤ 11, the lemma follows.

2 Hermitian lattices

2.1

An hermitian lattice \overline{M} = (M, h) is a torsion free A-module M of finite rank, equipped with an hermitian scalar product h on M ⊗_{Z} C which is invariant under complex conjugation. In other words, if we let M_{σ} = M ⊗_{A} C be the complex vector space obtained from M by extension of scalars via σ ∈ Σ, h is given by a collection of hermitian scalar products h_{σ} on M_{σ}, σ ∈ Σ, such that h\overline{σ}(x, y) = h_{σ}(x, y) whenever x and y are in M. We shall also write

h_{σ}(x) = h_{σ}(x, x)

and

∥x∥_{σ} = \sqrt{h_{σ}(x)} .

Lemma 5. Let \overline{M} be an hermitian lattice of rank N. Assume that M contains N vectors e_{1}, …, e_{N} which are F -linearly independent in M ⊗_{A} F and such that

∥e_{i}∥ ≤ 1

for all i = 1, …, N. Then there exist a direct sum decomposition

M = L_{1} ⊕ ⋯ ⊕ L_{N}

where each L_{i} has rank one and contains a vector f_{i} such that

# (L_{i}/A f_{i}) ≤ C_{1}

and

∥f_{i}∥ ≤ (i − 1) C_{1} C_{2} + C_{1}^{1/r} C_{3} .

Here C_{1}, C_{2}, C_{3} are the constants defined in Lemmas 1, 2, 3 respectively.

Proof. We proceed by induction on N. When N = 1, Lemma 1 tells us that L_{1} = M contains x_{1} such that

# (L_{1}/A x_{1}) ≤ C_{1} .

Let us write

x_{1} = α e_{1}

with α ∈ F^{∗}. Using Lemma 3, we can choose u ∈ A^{∗} such that

Sup_{σ∈Σ} |u α|_{σ} ≤ C_{3} (∏_{σ} |α|_{σ})^{1/r} = C_{3} N(α)^{1/r} ≤ C_{3} C_{1}^{1/r} .

The lemma follows with f_{1} = u x_{1}. Assume now that N ≥ 2, and let L_{1} = M ∩ F e_{1} in M ⊗_{A} F. As above, we choose f_{1} = a_{11} e_{1} in L_{1} with [L_{1} : A f_{1}] ≤ C_{1} and

Sup_{σ∈Σ} |a_{11}|_{σ} ≤ C_{3} C_{1}^{1/r} .

The quotient M′ = L/L_{1} is torsion free of rank N − 1. We equip M′ with the quotient metric induced by h, we let p : M → M′ be the projection, and e′_{i} = p(e_{i}), i = 2, …, N. Clearly

∥e′_{i}∥ ≤ 1

for all i = 2, …, N.

768 Christophe Soulé We assume by induction that M′ can be written

M′ = L′_{2} ⊕ ⋯ ⊕ L′_{N}

and that L′_{i} contains a vector f′_{i} such that

n_{i} = # (L′_{i}/A f′_{i}) ≤ C_{1}

with

f′_{i} = ∑_{2≤j≤i} a_{ij} e′_{j} ,  (9)

a_{ij} ∈ F, and, for all σ ∈ Σ,

|a_{ij}|_{σ} ≤ C_{1} C_{2} if 2 ≤ j < i ≤ N ,

and

|a_{ii}|_{σ} ≤ C_{1}^{1/r} C_{3} , 2 ≤ i ≤ N .

Let s : M′ → M be any section of the projection p. From (9) it follows that there exists μ_{i} ∈ F such that

s(f′_{i}) − ∑_{2≤j<i} a_{ij} e_{j} = μ_{i} e_{1} .

Applying Lemma 2, we can choose b_{i} ∈ A such that

∑_{σ∈Σ} |\frac{μ_{i}}{n_{i}} − b_{i}|_{σ} ≤ C_{2} .

Define t : M′ → M by the formulae

t(x) = s(x) − a(x) b_{i} e_{1}

whenever x ∈ L′_{i}, hence

n_{i} x = a(x) f′_{i} ,

for some a(x) ∈ A, 2 ≤ i ≤ N. If we take f_{i} = t(f′_{i}), we get

f_{i} = s(f′_{i}) − n_{i} b_{i} e_{1} = ∑_{2≤j<i} a_{ij} e_{j} + (μ_{i} − n_{i} b_{i}) e_{1}

and, for all σ ∈ Σ,

|μ_{i} − n_{i} b_{i}|_{σ} ≤ n_{i} C_{2} ≤ C_{1} C_{2} .

We define a_{i1} = μ_{i} − n_{i} b_{i} and L_{i} = t(L′_{i}). Then

M = L_{1} ⊕ ⋯ ⊕ L_{N}

Documenta Mathematica · Extra Volume Kato (2003) 761–788

satisfies our induction hypothesis:

# (L_{i}/A f_{i}) ≤ C_{1}

f_{i} = ∑_{1≤j≤i} a_{ij} e_{j} ,

|a_{ij}|_{σ} ≤ C_{1} C_{2} when j < i ,

and

|a_{ii}|_{σ} ≤ C_{1}^{1/r} C_{3} , for all σ ∈ Σ , i = 1, …, N .

Since

∥e_{i}∥ ≤ 1 for all i = 1, …, N ,

this implies

∥f_{i}∥ ≤ (i − 1) C_{1} C_{2} + C_{1}^{1/r} C_{3} .

q.e.d.

2.2

Lemma 6. Let I ⊂ A be a nontrivial ideal. There exists a set of representatives R ⊂ A of A modulo I such that, for any x in R,

∑_{σ∈Σ} |x|_{σ} ≤ C_{2} (\frac{r + 3}{4}) N(I) ,

where N(I) = # (A/I) and C_{2} is the constant in Lemma 2.

Proof. According to the proof of Lemma 2, the Z-module A contains a basis of r elements e_{1}, …, e_{r} such that

∑_{σ} |e_{i}|_{σ} ≤ μ_{r} ≤ \frac{2}{r} C_{2} .

Therefore, by Lemma 5 applied to the field Q, in which case C_{1} = C_{3} = 1 and C_{2} = 1/2, there exists a basis (f_{i}) of A over Z such that,

∑_{σ∈Σ} |f_{i}|_{σ} ≤ \frac{2}{r} C_{2} (\frac{i − 1}{2} + 1) .

Since the integer n = N(I) belongs to I, the map A/I → A/n A is injective and we can choose R among those

x = ∑_{i=1}^{r} x_{i} f_{i}

such that x_{i} ∈ Z and |x_{i}| ≤ n/2. In that case, if x ∈ R, we have

∑_{σ∈Σ} |x|_{σ} ≤ \frac{n}{2} ∑_{σ,i} |f_{i}|_{σ} ≤ n C_{2} \frac{r + 3}{4} .

q.e.d.

2.3

Lemma 7. Let \overline{M} be an hermitian lattice and assume that M = L_{1} ⊕ L_{2} is the direct sum of two lattices of rank one. Let f_{i} ∈ L_{i} be a non zero vector, and n_{i} = # (L_{i}/A f_{i}), i = 1, 2. Then there exists a vector e_{1} ∈ M, and an isomorphism

ψ : A e_{1} ⊕ L → M

such that L contains a vector e_{2} with

# (L/A e_{2}) ≤ n_{1} n_{2} ,

∥e_{1}∥ ≤ n_{2} C_{2} ∥f_{1}∥ + (1 + C_{2}^{2} \frac{r + 3}{4} n_{1}^{r}) ∥f_{2}∥ ,

and

∥ψ(e_{2})∥ ≤ n_{2} ∥f_{1}∥ + C_{2} \frac{r + 3}{4} n_{1}^{r} ∥f_{2}∥ .

Proof. The algebraic content of this lemma is [9], Lemma 1.7, p. 12. To control the norms in this proof we first define an isomorphism

u_{i} : L_{i} → I_{i}

where I_{i} is an ideal of A. If x ∈ L_{i}, u_{i}(x) ∈ A is the unique element such that

n_{i} x = u_{i}(x) f_{i} , i = 1, 2 .

In particular n_{i} = u_{i}(f_{i}). Next, we choose an ideal J_{1} in the class of I_{1} which is prime to I_{2}. According to [9], proof of Lemma 1.8, we can choose

J_{1} = \frac{x_{0}}{a_{0}} I_{1} ,

where a_{0} is any element of I_{1} − {0} and x_{0} belongs to a set of representatives of A modulo I_{1} J, where I_{1} J = a_{0} A. According to Lemma 6 we can assume that

∑_{σ∈Σ} |x_{0}|_{σ} ≤ C_{2} (\frac{r + 3}{4}) N(I_{1} J) = C_{2} (\frac{r + 3}{4}) N(a_{0}) .

The composite isomorphism

v_{1} : L_{1} → J_{1} → I_{1}

maps f_{1} to n_{1} x_{0}/a_{0}. We choose a_{0} = n_{1}, hence v_{1}(f_{1}) = x_{0} and

∑_{σ∈Σ} |x_{0}|_{σ} ≤ C_{2} (\frac{r + 3}{4}) n_{1}^{r} .

A Bound for the Torsion … 771

The direct sum of the inverses of v_{1} and u_{2} is an isomorphism

φ : J_{1} ⊕ I_{2} ⟶^{∼} L_{1} ⊕ L_{2} = M .

Since J_{1} and I_{2} are prime to each other we have an exact sequence (as in [9] loc.cit.)

0 ⟶ J_{1} I_{2} ⟶ J_{1} ⊕ I_{2} ⟶^{p} A ⟶ 0

where p is the sum in A. Let

s : A ⟶ J_{1} ⊕ I_{2}

be any section of p and let α ∈ J_{1} be such that

s(1) = (α, 1 − α) .

Let α = λ n_{2} x_{0} with λ ∈ F. Applying Lemma 2, we choose a ∈ A such that

∑_{σ∈Σ} |λ − a|_{σ} ≤ C_{2} .

Since n_{2} x_{0} lies in J_{1} I_{2} the element

β = α − a n_{2} x_{0} = (λ − a) n_{2} x_{0}

lies in J_{1}, and 1 − β lies in I_{2}. Since

β = v_{1}((λ − a) n_{2} f_{1})

and

1 − β = u_{2} (\frac{1}{n_{2}} − (λ − a) x_{0} f_{2})

we get

∥φ(β, 1 − β)∥ ≤ ∥(λ − a) n_{2} f_{1}∥ + ∥(\frac{1}{n_{2}} − (λ − a) x_{0}) f_{2}∥
≤ n_{2} C_{2} ∥f_{1}∥ + (1 + C_{2}^{2} (\frac{r + 3}{4}) n_{1}^{r}) ∥f_{2}∥ .

We let e_{1} = φ(β, 1 − β). On the other hand we define

L = J_{1} I_{2}(≃ L_{1} ⊗ L_{2})

and map L to M by the composite map

L ⟶^{i} J_{1} ⊕ I_{2} ⟶^{φ} M

where i(x) = (x, −x). We choose

e_{2} = n_{2} x_{0} ∈ L

Documenta Mathematica · Extra Volume Kato (2003) 761–788

so that φ ∘ i (e_{2}) = (n_{2} v_{1}(f_{1}), x_{0} u_{2}(f_{2})) has norm

∥φ ∘ i (e_{2})∥ ≤ n_{2} ∥f_{1}∥ + C_{2} (\frac{r + 3}{4}) n_{1}^{r} ∥f_{2}∥ .

Furthermore we have isomorphisms

L ⊕ A ⟶^{(i,s)} J_{1} ⊕ I_{2} ⟶^{φ} M

and

# (L/A e_{2}) = # (J_{1} I_{2}/A e_{2}) ≤ # (J_{1}/A x_{0}) × # (I_{2}/A n_{2}) = n_{1} n_{2} .

q.e.d.

2.4

Proposition 1. Let \overline{M} be a rank N hermitian free A-module such that its unit ball contains a basis of M ⊗_{A} F. Then M has a basis (e_{1}, …, e_{N}) such that

∥e_{i}∥ ≤ B_{i}

with B_{i} = (i − 1) C_{2} + C_{3}, i = 1, …, N, when A is principal and

B_{i} = (1 + C_{1} C_{2})(N C_{2} + C_{3}) (1 + C_{2} \frac{r + 3}{4})^{log_{2}(N)+2} C_{1}^{2(r+1)N/i}

in general. Here log_{2}(N) is the logarithm of N in base 2.

Proof. When A is principal, C_{1} = 1 and Proposition 1 follows from Lemma 5. In general Lemma 5 tells us that

M = L_{1} ⊕ ⋯ ⊕ L_{N}

and L_{i} contains a vector f_{i} with # (L_{i}/A f_{i}) ≤ C_{1} and

∥f_{i}∥ ≤ C_{1}((i − 1) C_{2} + C_{3}) ≤ C_{1} (N C_{2} + C_{3}) .

Let k > 1 be an integer and λ > 0 be a real number. We shall prove by induction N that, if M has a decomposition as above with

# (L_{i}/A f_{i}) ≤ k

and

∥f_{i}∥ ≤ k λ ,

then M has a basis (e_{1}, …, e_{N}) such that

∥e_{i}∥ ≤ λ (\frac{1}{k} + C_{2}) (1 + C_{2} \frac{r + 3}{4})^{t} k^{(r+1)(1+2+···+2^{t})} ,  (10)

for all i = 1, …, N, where t ≥ 1 is such that

\frac{N}{2^{t}} < i ≤ \frac{N}{2^{t−1}} .

The case N ≤ 2 follows from Lemma 7. If N > 2, let N′ be the integral part of N/2. Applying Lemma 7 to every direct sum L_{i} ⊕ L_{N−i}, N/2 < i ≤ N, we get

M = M′ ⊕ ( ⨁_{i=N′+1}^{N} A e_{i})

with

∥e_{i}∥ ≤ kλ (1 + C_{2} k + C_{2}^{2} \frac{r + 3}{4} k^{r})
≤ λ (\frac{1}{k} + C_{2}) (1 + C_{2} \frac{r + 3}{4}) k^{r+1}

and M′ is free, M′ = ⊕_{i=0}^{N′} L′_{i}, and each L′_{i} contains a vector f′_{i} such that

[L′_{i} : A f′_{i}] ≤ k^{2}

and

∥f′_{i}∥ ≤ λ (1 + C_{2} \frac{r + 3}{4}) k^{r+1} .

By the induction hypothesis, M′ has a basis (e_{i}), 1 ≤ i ≤ N′, such that

∥e_{i}∥ ≤ (1 + C_{2} \frac{r + 3}{4}) k^{(}r+1) (\frac{1}{k^{2}} + C_{2}) (1 + C_{2} \frac{r + 3}{4})^{t} (k^{2})^{(r+1)(1+···+2^{t})}

whenever

\frac{N′}{2^{t}} < i ≤ \frac{N′}{2^{t−1}} .

If

\frac{N}{2^{t+1}} < i ≤ \frac{N}{2^{t}} ,

this inequality implies

∥e_{i}∥ ≤ λ (\frac{1}{k} + C_{2}) (1 + C_{2} \frac{r + 3}{4})^{t+1} k^{(r+1)(1+···+2^{t+1})} .

Therefore M satisfies the induction hypothesis (10). Since

1 + 2 + ⋯ + 2^{t} = 2^{t+1} − 1 ≤ \frac{2N}{i}

and t ≤ log_{2}(N) + 1, Proposition 1 follows by taking k = C_{1} and

λ = C_{1}(N C_{2} + C_{3})

in (10). q.e.d.

2.5

Let \overline{M} be a rank N hermitian free A-module. We let

m(h) = Inf {h(x) , x ∈ M − {0}}

be the minimum value of h on M − {0} and

M(h) = {x ∈ M/h(x) = m(h)}

be the (finite) set of minimal vectors of M. Let ω_{N} be the standard volume of the unit ball in R^{N}.

Proposition 2. Let \overline{M} = (M, h) be as above. Assume that m(h) = 1 and that M(h) spans the F -vector space M ⊗_{A} F. Then M has a basis f_{1}, …, f_{N} such that any x ∈ M(h) is of the form

x = ∑_{i=1}^{N} y_{i} f_{i}

with

∑_{σ∈Σ} |y_{i}|_{σ}^{2} ≤ T_{i} ,

T_{i} = r^{rN} C_{3}^{2rN+2} γ^{N} ∏_{j≠i} B_{j}^{2} ,

and

γ = 4^{r_{1}+r_{2}} ω_{N}^{−2r_{1}/N} ω_{2N}^{−2r_{2}/N} D .

Proof. From Proposition 1 we know that M has a basis (e_{1}, …, e_{N}) with ∥e_{i}∥ ≤ B_{i}. Let x ∈ M(h) be a minimal vector and (x_{i}) its coordinates in the basis (e_{i}). Fix i ∈ {1, …, N} and σ ∈ Σ. Consider the square matrix

H_{i} = (h_{σ}(v_{k}, v_{ℓ})) ,

where v_{k} = e_{k} if k ≠ i and v_{i} = x. Furthermore, let

H_{σ} = (h_{σ}(e_{k}, e_{ℓ})) .

Since

|x_{i}|_{σ}^{2} = det(H_{i}) det(H_{σ})^{−1}

the Hadamard inequality implies

|x_{i}|_{σ}^{2} ≤ h_{σ}(x) ∏_{j≠i} h_{σ}(e_{j}) det(H_{σ})^{−1} .

For any unit u ∈ A^{∗} we can replace e_{i} by u^{−1} e_{i}, and x_{i} by y_{i} = u x_{i}. We then have

∑_{σ∈Σ} |y_{i}|_{σ}^{2} ≤ ∑_{σ∈Σ} h_{σ}(x) ∏_{j≠i} h_{σ}(e_{j}) |u|_{σ}^{2} det(H_{σ})^{−1} .  (11)

Applying Lemma 3 to λ_{σ} = det(H_{σ})^{−1/2} we find u such that, for all σ ∈ Σ,

|u|_{σ}^{2} det(H_{σ})^{−1} ≤ C_{3}^{2} ∏_{σ∈Σ} det(H_{σ})^{−1} .  (12)

Since ∑_{σ} h_{σ}(x) = 1 and h_{σ}(e_{j}) ≤ ∥e_{j}∥^{2} ≤ B_{j}^{2}, we deduce from (11) and (12)

that

∑_{σ∈Σ} |y_{i}|_{σ}^{2} ≤ C_{3}^{2} · ∏_{j≠i} B_{j}^{2} · ∏_{σ∈Σ} det(H_{σ})^{−1} .  (13)

According to Icaza [4], Theorem 1, there exists z ∈ L such that

∏_{σ∈Σ} h_{σ}(z) ≤ γ ∏_{σ∈Σ} det(H_{σ})^{1/N}

with

γ = 4^{r_{1}+r_{2}} ω_{N}^{−2r_{1}/N} ω_{2N}^{−2r_{2}/N} D .

Using Lemma 3 again and the fact that m(h) = 1, we find v ∈ A^{∗} such that

1 ≤ h(vz) ≤ r C_{3}^{2} ∏_{σ∈Σ} h_{σ}(z)^{1/r}
≤ r C_{3}^{2} γ^{1/r} ∏_{σ∈Σ} det(H_{σ})^{1/rN} .

From this it follows that

∏_{σ∈Σ} det(H_{σ})^{−1} ≤ (r C_{3}^{2})^{rN} γ^{N}  (14)

and Proposition 2 follows from (13) and (14).

2.6

To count the number of vectors in M(h) using Proposition 2 we shall apply the following lemma :

Lemma 8. The number of elements a in A such that

∑_{σ∈Σ} |a|_{σ}^{2} ≤ T

is at most

B(T) = Sup (T^{r/2} 2^{r+3}, 1) .

Proof. When r_{2} > 0, this follows from [7], V § 1, Theorem 0, p. 102, by noticing that one can take C_{3} = 2^{r+3} in loc.cit. When r_{2} = 0, the argument is similar.

3 Reduction theory

3.1

Fix an integer N ≥ 2. Let

Γ = GL_{N}(A)

and

G = GL_{N}(F ⊗_{Q} R) .

On the standard lattice L_{0} = A^{N} consider the hermitian metric h_{0} defined by

h_{0}(x, y) = ∑_{σ∈Σ} ∑_{i=1}^{N} x_{iσ} \overline{y_{iσ}}

for all vectors x = (x_{iσ}) and y = (y_{iσ}) in L_{0} ⊗_{Z} C = (C^{N})^{Σ}. Any g ∈ G defines an hermitian metric h = g(h_{0}) on L_{0} by the formula

g(h_{0}) (x, y) = h_{0}(g(x), g(y)) .

Let K be the stabilizer of h_{0} and G and X = K\G . We can view each h ∈ X as a metric on L_{0}. Following Ash [1], we say that a finite subset M ⊂ L_{0} is well-rounded when it spans the F -vector space L_{0} ⊗_{A} F. We let W̃ ⊂ X be the space of metrics h such that m(h) = 1 and M(h) is well-rounded. Given a well-rounded set M ⊂ L_{0} we let C(M) ⊂ W̃ be the set of metrics h such that

• h(x) = 1 for all x ∈ M

• h(x) > 1 for all x ∈ L_{0} − (M ∪ {0}).

As explained in [1], proof of (iv), pp. 466-467, C(M) is either empty or topologically a cell, and the family of closed cells \overline{C(M)} gives a Γ-invariant cellular decomposition of W̃, such that

\overline{C(M)} = ∐_{M′⊃M} C(M′) .

Furthermore W̃/Γ is compact, of dimension dim(X) − N.

3.2

Proposition 3. i) For any integer k ≥ 0, the number of cells of codimension k in W̃ is at most

c(k, N) = \binom{a(N)}{N + k}

where

a(N) = 2^{N(r+3)} (∏_{i=1}^{N} T_{i})^{r/2} ,

and T_{i} is as in Proposition 2. ii) Given a cell in W̃, its number of codimension one faces is at most a(N)^{N+1}.

Proof. Let Φ be the set of vectors x = (x_{i}) in A^{N} such that, for all i = 1, …, N,

∑_{σ∈Σ} |x_{i}|_{σ}^{2} ≤ T_{i} .

Given h ∈ W̃, Proposition 2 says that we can find a basis (f_{i}) of L_{0} such that any x in M(h) has its coordinates (x_{i}) bounded as above. If γ ∈ Γ is the matrix mapping the standard basis of A^{N} to (f_{i}), this means that M(γ(h)) = γ^{−1}(M(h)) is contained in Φ. Let \overline{C(M)} be a nonempty closed cell of codimension k in W̃. For any x ∈ L_{0}, the equation h(x) = 1 defines a real affine hyperplane in the set of N × N hermitian matrices with coefficients in (F ⊗_{Q} C)^{+}. The equations h(x) = 1, x ∈ M, may not be linearly independent, but, since \overline{C(M)} has codimension k, M has at least N + k elements. And since M ⊂ M(h) for some h ∈ W̃, there exists γ ∈ Γ such that γ^{−1}(M) is contained in Φ. Therefore, modulo the action of Γ, there are at most \binom{card(Φ)}{N+k} cells \overline{C(M)} of codimension k. From Lemma 7 we know that

card(Φ) ≤ a(N) ,

therefore i) follows. To prove ii), consider a cell \overline{C(M)} and a codimension one face \overline{C(M′)} of \overline{C(M)}. We can write M′ = M ∪{x} for some vector x and there exists γ ∈ Γ such that γ(M′) ⊂ Φ. Since M is well-rounded, the matrix γ is entirely determined by the set of vectors γ(M), i.e. there are at most card(Φ)^{N} matrices γ such that γ(M) ⊂ Φ. Since γ(x) ∈ Φ, there are at most card(Φ)^{N+1} vectors x as above.

q.e.d.

3.3

Lemma 9. Let γ ∈ Γ − {1} and p be a prime number such that γ^{p} = 1. Then

p ≤ 1 + Sup(r, N) .

Proof. Since γ is non trivial we have P (γ) = 0 where P is the cyclotomic polynomial

P (x) = X^{p−1} + X^{p−2} + ⋯ + 1 .

If F does not contain the p-th roots of one, P is irreducible, and therefore it divides the characteristic polynomial of the matrix γ over F, hence p − 1 ≤ N. Otherwise, F contains Q(μ_{p}), which is of degree p − 1, therefore p − 1 ≤ r.

4 The main results

4.1

For any integer n > 0 and any finite abelian group A we let card_{n}(A) be the largest divisor of the integer #(A) such that no prime p ≤ n divides card_{n}(A). Let N ≥ 2 be an integer. We keep the notation of § 3 and we let

w̃ = dim(X) − N = r_{1} \frac{N(N + 1)}{2} + r_{2} N^{2} − N

be the dimension of W̃. For any k ≤ w̃ we define

h(k, N) = a(N)^{(N+1)c(_{w}−k−1,N)} ,

where c(·, N) and a(N) are defined in Proposition 3.

Theorem 1. The torsion subgroup of the homology of GL_{N}(A) is bounded as follows

card_{1+sup(r,N)}H_{k}(GL_{N}(A), Z)_{tors} ≤ h(k, N) .

Proof. We know from [1] that W̃ is contractible and the stabilizer of any h ∈ W̃ is finite. From Lemma 9 it follows that, modulo S_{1+sup(r,N)}, the homology of Γ = GL_{N}(A) is the homology of a complex (C_{·}, ∂), where C_{k} is the free abelian group generated by a set of Γ-representatives of those k-dimensional cells c in W̃ such that the stabilizer of c does not change its orientation ([2], VII). According to Proposition 3, the rank of C_{k} is at most c (w̃−k, N) and any cell of W̃ has at most a(N)^{N+1} faces. Theorem 1 then follows from a general result of Gabber ([13], Proposition 3 and equation (18)).

4.2

For any integer m ≥ 1 let

k(m) = h(m, 2 m + 1) .

Denote by K_{m}(A) the m-th algebraic K-group of A.

Theorem 2. The following inequality holds

card_{sup(r+1,2m+2)}K_{m}(A)_{tors} ≤ k(m) .

Proof. As in [13], Theorem 2, we consider the Hurewicz map

H : K_{m}(A) → H_{m}(GL(A), Z) ,

the kernel of which lies in S_{n}, n ≤ (m + 1)/2. Since, according to Maazen and Van der Kallen,

H_{m}(GL(A), Z) = H_{m}(GL_{N}(A), Z)

when N ≥ 2 m + 1, Theorem 2 is a consequence of Theorem 1.

4.3

Let p be an odd prime and n ≥ 2 an integer. For any ν ≥ 1 denote by Z/p^{ν}(n) the étale sheaf μ_{p^{ν}}^{⊗n} on Spec(A[1/p]), and let

H^{2}(Spec(A[1/p]), Z_{p}(n)) = \underleftarrow{lim}_{ν}H^{2}(Spec(A[1/p]), Z_{p^{ν}} (n)) .

From [12], we know that this group is finite and zero for almost all p.

Theorem 3. The following inequality holds

∏_{{p≥4n−1 \atop p≥r+2}} card H^{2}(Spec(A[1/p]), Z_{p}(n)) ≤ k(2n − 2) .

Proof. According to [12], the cokernel of the Chern class

c_{n,2} : K_{2n−2}(A) → H^{2}(Spec(A[1/p]), Z_{p}(n))

lies in S_{n+1} for all p. Furthermore, Borel proved that K_{2m−2}(A) is finite. Therefore Theorem 3 follows from Theorem 2.

4.4

By Lemmas 1 to 7 and Propositions 1 to 3, the constant k(m) is explicitly bounded in terms of m, r and D. We shall now simplify this upper bound.

Proposition 4. i) log log k(m) ≤ 220 m^{4} log(m) r^{4r} \sqrt{D} log(D)^{r−1} ii) If F has class number one,

log log k(m) ≤ 210 m^{4} log(m) r^{4r} \sqrt{D} log(D)^{r−1}

780 Christophe Soulé iii) If F = Q(\sqrt{−D}) is imaginary quadratic

log log k(m) ≤ 1120 m^{4} log(m) log(D) ;

if furthermore F has class number one

log log k(m) ≤ 510 m^{4} log(m) log(D) .

iv) When F = Q and m ≥ 9

log log k(m) ≤ 8 m^{4} log(m) ;

furthermore

log log k(7) ≤ 40 545

and

log log k(8) ≤ 70 130 .

Proof. By definition

k(m) = h(m, 2 m + 1) = a(N)^{(N+1)c(_{w}−m−1,N)}

with N = 2 m + 1 and

c(w̃ − m − 1, N) = \binom{a(N)}{N + w̃ − m − 1} .

Since

N + w̃ − m − 1 = r_{1} \frac{N(N + 1)}{2} + r_{2} N^{2} − m − 1
≤ 2 r m^{2} + 3 r m + r − 2 m − 1

and since a(2 m + 1) is very big, we get

log log k(m) ≤ (2 r m^{2} + 3 r m + r − 2 m − 1) log a(2 m + 1)
+ log(2 m + 2) + log log a(2 m + 1)
≤ r(2 m^{2} + 3 m + 1) log a(2 m + 1) .  (15)

From Proposition 3 and Proposition 2 we get

a(N) = 2^{N(r+3)} (∏_{i=1}^{N} T_{i})^{r/2} ,  (16)

and

∏_{i=1}^{N} T_{i} = (r^{rN} γ^{N} C_{3}^{2rN+2})^{N} ∏_{i=1}^{N} B_{i}^{N−1} .  (17)

Documenta Mathematica · Extra Volume Kato (2003) 761–788

A Bound for the Torsion … 781

According to Proposition 1

∏_{i=1}^{N} B_{i} = [(1 + C_{1} C_{2})(N C_{2} + C_{3}) (1 + C_{2} \frac{r + 3}{4})^{log_{2}(N)+2}]^{N} · C_{1}^{2(r+1)NH_{N}} , (18)

where

H_{N} = ∑_{i=1}^{N} \frac{1}{i} ≤ 1 + log(N) .

Assume s ≠ 0. Then the upper bound C_{3}^{∗} we get from Lemmas 3 and 4 for C_{3} is much bigger than C_{2}. Therefore

log(N C_{2} + C_{3}) ≤ log(N) + log(C_{3}^{∗}) .  (19)

We deduce from (15), (16), (17), (18), (19) that

log log k(m) ≤ X_{1} + X_{2}

with

X_{1} = r(2 m^{2} + 3 m + 1) \frac{r}{2} (N(2 r N + 2) + N(N − 1)) log(C_{3}^{∗})

and

X_{2} = r(2 m^{2} + 3 m + 1)(N(r + 3) log(2)
+ \frac{r}{2} N(N log(γ) + (N − 1)[log(1 + C_{1} C_{2}) + log(N)
+ (log_{2}(N) + 2) log(1 + C_{2} \frac{r + 3}{4})
+ 2(r + 1)(1 + log(N)) log(C_{1})])) .  (20)

Since s ≤ r − 1, Lemma 3 and Lemma 4 imply

log(C_{3}^{∗}) ≤ 11r^{2}(r − 1)(4 r(log 3 r)^{3})^{r−2} 2^{r−1} \sqrt{D} log(D)^{r−1} ,

from which it follows that

X_{1} ≤ 208 log(m) m^{4} r^{4r} \sqrt{D} log(D)^{r−1}

when m ≥ 2 and r ≥ 2. To evaluate X_{2} first notice that

4 ω_{N}^{−2/N} ≤ 1 + N/4

Documenta Mathematica · Extra Volume Kato (2003) 761–788

782 Christophe Soulé by [10], II, (1.5), Remark, hence

log(γ) ≤ r_{1} log (1 + \frac{N}{4} ) + 2 r_{2} log (1 + \frac{N}{2} ) + log(D)
≤ r log(N) + log(D)  (21)

since N ≥ 5. By the Stirling formula and Lemma 1, if r ≥ 2,

log(C_{1}) = log(r!) − r log(r) + r_{2} log (\frac{4}{π} ) + \frac{1}{2} log(D)
≤ 1 + \frac{1}{2} log(r) + \frac{1}{2} log(D) ,  (22)

log (1 + C_{2} \frac{r + 3}{4}) ≤ Sup (log(C_{2}) + log (\frac{r + 3}{4}) + 1, log(2)) , where

log(C_{2}) + log(\frac{r + 3}{4}) ≤ r log(4) − (r − 2) log(r) − log(r!)
+ log(\frac{r + 3}{4}) + \frac{1}{2} log(D)
≤ 2.4 + \frac{1}{2} log(D) ,

so that

log (1 + C_{2} \frac{r + 3}{4}) ≤ 3.4 + \frac{1}{2} log(D) .  (23)

We also have

log(1 + C_{1} C_{2}) ≤ Sup(1 + log(C_{1}) + log(C_{2}), log(2))

and

log(C_{1}) + log(C_{2}) ≤ −r log(r) + r − (r − 2) log(r) + r log(4) + log(D)
≤ 3.4 + log(D) ,

so that

log(1 + C_{1} C_{2}) ≤ 4.4 + log(D) .  (24)

From (20), (21), (22), (23), (24) we get

X_{2} ≤ a log(D) + b

Documenta Mathematica · Extra Volume Kato (2003) 761–788

with

a = r(2 m^{2} + 3 m + 1)(2 m + 1)(\frac{r}{2} ((2 m + 1) + 2 m + m log_{2}(2 m + 1) + m
+ 2 m (r + 1)(1 + log(2 m + 1)))) ≤ 75 r^{3} m^{4} log(m)

if r ≥ 2 and m ≥ 2. Finally

b = r(2 m^{2} + 3 m + 1)(2 m + 1)((r + 3) log(2) + \frac{r}{2} (2 m + 1) r(log(r) + log(2 m + 1))
+ \frac{r}{2} (2 m)(4.4 + log(2 m + 1) + 3.4(log_{2}(2 m + 1) + 2)
+ 2 (r + 1) (1 + log(2 m + 1)) (1 + \frac{1}{2} log(r)) ))≤ 148 r^{4} m^{4} log(m)

when r ≥ 2 and m ≥ 2. Therefore

log log k(m) ≤ 208 log(m) m^{4} r^{4r} \sqrt{D} log(D)^{r−1} + 75 r^{3} m^{4} log(m) log(D)
+ 148 r^{4} m^{4} log(m) ≤ 220 m^{4} log(m) r^{4r} \sqrt{D} log(D)^{r−1}

when m, r and D are at least 2. This proves i). If we assume that A is principal, we can take C_{1} = 1 in Lemma 1 and B_{i} = (i − 1) C_{2} + C_{3} in Proposition 1. Since C_{2} < C_{3} we get

log (∏_{i=1}^{N} B_{i}) ≤ log(N!) + N log(C_{3})

and

log log k(m) ≤ X_{1} + X_{3}

where

X_{3} = r(2 m^{2} + 3 m + 1)[(r + 3)(2 m + 1) log(2) + \frac{r^{2}}{2} (2 m + 1)^{2} log(r)
+ \frac{r}{2} (2 m + 1)^{2} log(γ) + \frac{r}{2} (2 m) log((2 m + 1)!)]
≤ 6 m^{4} r^{2} log(D) + 2 r^{4r} m^{4} log(m) .

Therefore

X_{1} + X_{3} ≤ 210 m^{4} log(m) r^{4r} \sqrt{D} log(D)^{r−1} .

Assume now that r_{1} + r_{2} = 1. Then C_{3} = 1 and the term X_{1} disappears from the above computation. Assume first that F = Q(\sqrt{−D}). Since r_{2} = 1 and r_{1} = 0 we get

log log k(m) ≤ (4 m^{2} + 3 m + 1) log a(2 m + 1) .

784 Christophe Soulé Furthermore (18) becomes

∏_{i=1}^{N} B_{i} ≤ [(1 + C_{1} C_{2})(1 + N C_{2}) (1 + \frac{5}{4} C_{2})^{log_{2}(N)+2}]^{N} · C_{1}^{6N(1+log(N))} .

Therefore

log log k(m) ≤ (4 m^{2} + 3 m + 1)[5 N log(2) + 2N^{2} log(2)
+ N^{2} log(γ) + N(N − 1)[log(1 + C_{1} C_{2})
+ log(1 + N C_{2}) + (log_{2}(N) + 2) log (1 + \frac{5}{4} C_{2}) ]
+ 6 N(1 + log N) log(C_{1})] ,

with N = 2 m + 1. We have now

γ ≤ (1 + \frac{N}{2} )^{2} D ,

C_{1} = \frac{2}{π} \sqrt{D} and C_{2} = \frac{π}{2} \sqrt{D} .

This implies log log k(m) ≤ 597 m^{4} log(m)+256 m^{4} log(m) log(D) ≤ 1120 m^{4} log(m) log(D) . If F = Q(\sqrt{−D}) is principal we can take C_{1} = 1 and B_{i} = (i − 1) C_{2} + 1. We get

log log k(m) ≤ 510 m^{4} log(m) log(D) .

Finally, assume that F = Q. Then

B_{i} = \frac{i + 1}{2} since C_{2} = \frac{1}{2} , and γ ≤ 1 + \frac{N}{4} .

Therefore

log log k(m) ≤ (2 m^{2} + 2 m + 1) log a(2 m + 1)
≤ (2 m^{2} + 2 m + 1)[4 N log(2) + \frac{N^{2}}{2} log (1 + \frac{N}{4} )
+ \frac{N − 1}{2} log (∏_{i=1}^{N} \frac{i + 1}{2}) ]
≤ 8 m^{4} log(m)

if m ≥ 9. We can also estimate k(7) and k(8) from this inequality above. This proves iv).

Documenta Mathematica · Extra Volume Kato (2003) 761–788

5 Discussion

5.1

The upper bound in Theorem 2 and Propostition 4 seems much too large. When m = 0, card K_{0}(A)_{tors} is the class number h(F), which is bounded as follows:

h(F) ≤ α \sqrt{D} log(D)^{r−1} ,  (25)

for some constant α(r) [11], Theorem 4.4, p. 153. Furthermore, when F = Q, m = 2 n − 2 and n is even, the Lichtenbaum conjecture predicts that card K_{2n−2}(Z) is the order of the numerator of B_{n}/n, where B_{n} is the n-th Bernoulli number. The upper bound

B_{n} ≤ n^{!} ≈ n^{n}

suggests, since the denominator of B_{n}/n is not very big, that card K_{m}(Z)_{tors} should be exponential in m. We are thus led to the following:

Conjecture. Fix r ≥ 1. There exists positive constants α, β, γ such that, for any number field F of degree r on Q,

card K_{m}(A)_{tors} ≤ α exp(βm^{γ} log D) .

Furthermore, we expect that γ does not depend on r.

5.2

As suggested by A. Chambert-Loir, it is interesting to consider the analog in positive characteristic of the conjecture above. Let X be a smooth connected projective curve of genus g over the finite field with q elements, ζ_{X}(s) its zeta function and

P (t) = ∏_{i=1}^{2g}(1 − α_{i} t) ,

where α_{i} are the roots of Frobenius acting on the fist ℓ-adic cohomology group of X. When n > 1, it is expected that the finite group K_{2n−2}(X) has order the numerator of ζ_{X}(1 − n), i.e. P (q^{n−1}). Since |α_{i}| = q^{1/2} for all i = 1 ⋯ 2g, we get

P (q^{n−1}) ≤ (1 + q^{n−1/2})^{2g} ≤ q^{2ng} .

In the analogy between number fields and function fields, the genus g is known to be an analog of log(D). Therefore the bound above is indeed analogous to the conjecture in §5.1.

5.3

The upper bound for k(m) in Proposition 4 i) is twice exponential in D. One exponential is due to our use of Lemma 3, where C_{3} is exponential in D. Maybe this can be improved in general, and not only when s = 0. The exponential in D occuring in Proposition 4 ii) might be due to our use of the geometry of numbers. Indeed, if one evaluates the class number h(F) by applying naively Minkowski’s theorem (Lemma 1), the bound one gets is exponential in D; see however [8], Theorem 6.5., for a better proof.

5.4

One method to prove (25) consists in combining the class number formula (see (7) and (8)) with a lower bound for the regulator R(F). This suggests replacing the arguments of this paper by analytic number theory, to get good upper bounds for étale cohomology. More precisely, let n ≥ 2 be an integer, and let ζ_{F} (1 − n)^{∗} be the leading coefficient of the Taylor series of ζ_{F} (s) at s = 1 − n. Lichtenbaum conjectured that

ζ_{F} (1 − n)^{∗} = ± 2^{r_{1}} R_{2n−1}(F) \frac{∏_{p} card H^{2}(Spec(A[1/p]), Z_{p}(n))}{∏_{p} card H^{1}(Spec(A[1/p]), Z_{p}(n))_{tors}} , (26)

where R_{2n−1}(F) is the higher regulator for the group K_{2n−1}(F). The equality (26) is known up a power of 2 when F is abelian over Q [5], [6], [3]. The order of the denominator on the right-hand side of (26) is easy to evaluate, as well as ζ_{F} (1 − n)^{∗} (since it is related by the functional equation to ζ_{F} (n)).

Problem. Can one find a lower bound for R_{2n−1}(F)?

If such a problem could be solved, the equality (26) is likely to produce a much better upper bound for étale cohomology than Theorem 3. Zagier’s conjecture suggests that this problem could be solved if one knew that the values of the n-logarithm on F are Q-linearly independent.

5.5

To illustrate our discussion, let F = Q and n = 5. Then we have

H^{2}(Spec(Z[1/p]), Z_{p}(5))/p = C^{(p−5)} ,

where C is the class group of Q( ^{p} \sqrt{1}) modulo p, and C^{(i)} is the eigenspace of C of the i-th power of the Teichmüller character. Vandiver’s conjecture predicts that C^{(p−5)} = 0 when p is odd. It is true when p ≤ 4.10^{6}. Theorem 3 and Proposition 4 tell us that

∏_{p} H^{2}(Spec(Z[1/p], Z_{p}(5)) ≤ k(8) ≤ exp exp (70130).

If one could find either a better upper bound for the order of K_{8}(Z) or a good lower bound for R_{9}(Q), this would get us closer to the expected vanishing of C^{(p−5)}. Notice that, using knowledge on K_{4}(Z), Kurihara has proved that C^{(p−3)} = 0.

References

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Christophe Soulé CNRS and IHES 35 route de Chartres 91440 Bures-sur-Yvette France soule@ihes.fr