Hermitian Lattices and Bounds in K-Theory

of Algebraic Integers

To Sasha Merkurjev, for his sixtieth birthday

Eva Bayer-Fluckiger^{1}, Vincent Emery^{2} and Julien Houriet^{1}

Received: July 17, 2014

Abstract. Elaborating on a method of Soulé, and using better estimates for the geometry of hermitian lattices, we improve the upper bounds for the torsion part of the K-theory of the rings of integers of number fields.

2010 Mathematics Subject Classification: 11R70 (primary); 11E39 (secondary) Keywords and Phrases: K-theory of number fields, hermitian lattices

1 Introduction

Let F be a number field of degree d, with ring of integers O_{F} and discriminant D_{F}. We denote by K_{n}(O_{F}) the n-th K-theory group of O_{F}, which was defined by Quillen and showed by him to be finitely generated. The rank of K_{n}(O_{F}) has been computed by Borel in [4]. In this article we consider the problem of finding an upper bound – in terms of n, d and D_{F} – for the order of the torsion part K_{n}(O_{F})_{tors} . Such general bounds have been obtained by Soulé in [11]. Our Theorem 1.1 below sharpens Soulé’s results. As in Soulé’s paper, our inequalities hold “up to small torsion”. To state this precisely, for a finite abelian group A let us write card_{ℓ}(A) for the order of A/B, where B ⊂ A is the subgroup generated by elements of order ≤ ℓ.

1First and third authors partially supported by SNSF, Project number 200020-121647

2Second author supported by SNSF, Project number PZ00P2-148100

Documenta Mathematica · Extra Volume Merkurjev (2015) 71–83

Theorem 1.1. Let F ≠ ℚ be a number field of degree d and with discriminant D_{F}. Then for any n ≥ 2 we have

log card_{ℓ}K_{n}(O_{F})_{tors} ≤ (2n + 1)^{71n^{4}d^{3}} · d^{293n^{5}d^{5}} · |D_{F} |^{528n^{5}d^{4}} ,

where ℓ = max(d + 1,2n + 2).

To improve the readability, we have not tried to state here the best possible bounds that one could get with the method we use. We refer to the PhD thesis of the third author [7, Theorem 4.3] – in which the result was originally obtained – for slightly better estimates. However, it does not change the fact that the upper bounds are huge, and – although explicit – certainly unusable for practical computation. We shall insist here on the qualitative aspect of our result, which could be stated as follows.

Corollary 1.2. There exist α and β, both polynomials in n and d, such that for any number field F of degree d ≥ 2 we have

log card_{ℓ}K_{n}(O_{F})_{tors} ≤ (nd)^{α} |D_{F} |^{β},

where ℓ = max(d + 1,2n + 2) and n ≥ 2.

Compared to [11], our result improves the bound by an exponential factor: the previous bound for logcard_{ℓ}K_{n}(O_{F}) was at least exp(α|D_{F} |^{1/2} ), for some polynomial α = α(n,d) (see Proposition 4 in loc. cit. for the precise statement). The strategy is the following. The group K_{n}(O_{F}) can be related – via the Hurewicz map – to the integral homology H_{n}(GL(O_{F})), and an upper bound (up to small torsion) for the order of K_{n}(O_{F}) can then be obtained through the study of the integral homology of GL_{N} (O_{F}), with N = 2n + 1 (cf. Section 3). The proof of Theorem 1.1 follows the method of Soulé, which uses Ash’s well-rounded retract (cf. Section 2) to study these homology groups. This reduces the problem to finding good estimates concerning the geometry of hermitian lattices. Our approach to these estimates differs from that of Soulé (cf. Section 4), leading to the improved bounds in Theorem 1.1. For F = ℚ our method does not bring any improvement, so that [11, Prop. 4 iv)] is still the best available general bound for K_{n}(ℤ). We refer to [6, Theorem 1.3] for a different approach to the same problem for K_{2}, which gives better result than Corollary 1.2 in the case of totally imaginary fields. Note that all these results remain very far from the general bound conjectured by Soulé in [11, Sect. 5.1], which should take the following form for some constant C(n,d):

log cardK_{n}(O_{F})_{tors} ≤ C(n,d) log|D_{F} |.  (1.1)

Acknowledgement

We thank Christophe Soulé for suggesting that an improvement of his bound in [11] could be possible with the methods of the paper of the first named author [3].

Hermitian Lattices and Bounds in K-Theory 73

2 Hermitian metrics and the well-rounded retract

2.1 Notation

We keep the notation of the introduction. Let us denote by (r_{1},r_{2}) the signature of the number field F. Let us write F_{ℝ} = ℝ ⊗_{ℚ} F. If Σ denotes the set of field embeddings σ : F → ℂ, then F_{ℝ} can be identified with the subspace (ℂ^{Σ} )^{+} ⊂ ℂ^{Σ} invariant under the involution (x_{σ}) ↦ (\overline{x}\overline{σ}), where \overline{a} denotes the complex conjugation. This also provides an isomorphism F_{ℝ} ≅ ℝ^{r_{1}} × ℂ^{r_{2}} . For x ∈ F_{ℝ}, written as x = (x_{σ})_{σ∈Σ} , we denote by \overline{x} = (\overline{x_{σ}}) the result of the complex conjugation applied component-wise. We denote by Tr the trace map from F_{ℝ} to ℝ, defined by Tr(x) = ∑ _{σ∈Σ} x_{σ}. We will also use the absolute value of the norm map: N (x) = ∏ _{σ∈Σ} |x_{σ}|. We fix a free O_{F} -lattice L of finite rank N ≥ 1. Let V = F ⊗_{O_{F}} L and V_{ℝ} = ℝ ⊗_{ℚ} V, so that V_{ℝ} can be seen as a (left) F_{ℝ}-module. Let Γ be the group GL(L) of automorphism of L. By fixing a basis of L, we have the identification Γ = GL_{N} (O_{F}). Then Γ is a discrete subgroup of the reductive Lie group GL(V_{ℝ}) = GL_{N} (F_{ℝ}). We shall denote the latter by G, and we will let it act on V_{ℝ} on the left (and similarly for Γ on L).

2.2 Hermitian metrics

Let h: V_{ℝ} × V_{ℝ} → F_{ℝ} be a hermitian form on V_{ℝ}, that is, h = (h_{σ})_{σ∈Σ} is F_{ℝ}- linear in the first variable and h(y,x) = \overline{h(x,y)}. The pair (L,h) is called a hermitian lattice. When x = y, we also write h(x) = h(x,x). Note that h(x) has only real components, and we say that h is positive definite if h_{σ}(x,x) > 0 for any nonzero x ∈ V_{ℝ} and all σ ∈ Σ. Let X be the (topological) space of positive definite hermitian forms on V_{ℝ}. The group G = GL(V_{ℝ}) acts transitively on X in the following way: the element γ ∈ G maps the form h ∈ X to

(γ · h)(x,y) = h(γ^{−1} x,γ^{−1} y).  (2.1)

The space X can be identified with the set of positive definite symmetric N ×N matrices with coefficients in F_{ℝ}. Using this identification, it is not difficult to see that X is contractible. To each h ∈ X we associate the real quadratic form q_{h} on V_{ℝ} (seen as a real vector space) defined for x ∈ V_{ℝ} by:

q_{h}(x)= Tr(h(x)).  (2.2)

Such a quadratic form q_{h} : V_{ℝ} → ℝ for h ∈ X will be called a hermitian metric. Given h, we will denote by || · ||_{h} the norm on V_{ℝ} induced by q_{h}.

2.3 Ash’s well-rounded retract

For h ∈ X we set m(L,h) ∈ ℝ_{>0} to be the minimum of q_{h}(x) over the nonzero x ∈ L ⊂ V, and define

M(L,h) = {x ∈ L | q_{h}(x) = m(L,h)} .  (2.3)

Definition 2.1. We say that h ∈ X (or (L,h)) is well rounded if m(L,h) = 1 and M(L,h) generates V (as a vector space over F).

Let W̃ ⊂ X be the subspace of well-rounded hermitian forms. Note that the action defined by (2.1) restricts to an action of Γ = GL_{N} (O_{F}) on W̃. In [2, p. 466–467], Ash defined a CW-complex structure on W̃ that has the following properties: two points h and h′ of W̃ belong to the interior of the same cell C(h) = C(h′) if and only if M(L,h) = M(L,h′), and C(h′) ⊂ C(h) if and only if M(L,h′) ⊃ M(L,h). Moreover, we have that M(L,γ · h) = γM(L,h). Then the action of Γ on W̃ is compatible with the cell structure: an element γ ∈ Γ maps the cell C(h) to C(γ · h).

Theorem 2.2 (Ash). W̃ is a deformation retract of X on which Γ acts with finite stabilizer Γ_{σ} for each cell σ of W̃. The quotient Γ\W̃ is compact, of dimension dim(X) − N.

Proof. The proof of this statement follows from the argument of Ash given in the proof of the main theorem of [2], page 462. More precisely, the argument to prove that W̃ is a deformation retract of X is the same as Ash uses for W = W̃/Γ, in [2, §3 (i)]. The compactness of Γ\W̃ is proved in §3 (ii) of loc. cit., and the dimension is computed on page 466.

Let C_{•} be the complex of cellular chains on Γ\W̃. We can decompose it as C_{•} = C_{•}^{+} ∪ C_{•}^{−} , where Γ preserves (resp. does not preserve) the orientation of each σ ∈ C_{•}^{+} (resp. σ ∈ C_{•}^{−} ). It then follows from the spectral sequence described in [5, VII (7.10)] that up to prime divisors of the finite stabilizers Γ_{σ}, the homology of C_{•}^{+} computes H_{•}(Γ). In particular, one has the following (cf. [11, Lemma 9]). See Section 1 for the definition of card_{ℓ}.

Corollary 2.3. Let ℓ = 1 + max(d,N). Then for any n we have:

card_{ℓ}H_{n}(Γ)_{tors} = card_{ℓ}H_{n}(C_{•}^{+} )_{tors} ,

where H_{n}(·)_{tors} denotes the torsion part of the integral homology.

3 Bounding torsion homology and K-theory

3.1 The Hurewicz map

For any n > 1 we consider the n-th Quillen K-group K_{n}(O_{F}) = π_{n}(B GL(O_{F})^{+} ) (“plus construction”). The Hurewicz map relating homotopy groups to homology provides a map K_{n}(O_{F}) → H_{n}(GL(O_{F})^{+} ) = H_{n}(GL(O_{F})). We know (see for instance [1, Theorem 1.5]) that its kernel does not contain elements of order p for p > \frac{n+1}{2} . Moreover, by a stability result of van der Kallen and Maazen (cf. [12, Theorem 4.11]) the homology of GL(O_{F}) = \underrightarrow{lim}GL_{N} (O_{F}) is equal to the homology of GL_{N} (O_{F}) for any N ≥ 2n + 1. Let then N = 2n + 1, and consider Γ = GL_{N} (O_{F}). We deduce from Corollary 2.3 that for ℓ = max(d + 1,2n + 2) we have:

card_{ℓ}K_{n}(O_{F})_{tors} ≤ card_{ℓ}H_{n}(C_{•}^{+} )_{tors} .  (3.1)

3.2 Gabber’s lemma

The abstract result that allows to obtain a bound for the right hand side of (3.1) is the following lemma. It was discovered by Gabber, and first appeared in Soulé [10, Lemma 1]. See Sauer [9, Lemma 3.2] for a more elementary proof.

Lemma 3.1 (Gabber). Let A = ℤ^{a} with the standard basis (e_{i})_{i=1,...,a} and B = ℤ^{b}, so that B ⊗ ℝ is equipped with the standard Euclidean norm ∥ · ∥. Let ϕ: A → B be a ℤ-linear map and let α ∈ ℝ such that ∥ϕ(e_{i})∥ ≤ α for each i = 1,...,a. If we denote by Q the cokernel of ϕ, then

|Q_{tors} | ≤ α^{min (a,b)} .

Corollary 3.2. Suppose that the cellular complex Γ\W̃ has at most α_{k} faces for any k ≥ 0, and that any k-cell has at most β codimension 1 faces. Then

H_{k}(C_{•}^{+} )_{tors} ≤ β^{\frac{1}{2} min (α_{k+1} ,α_{k})} .

Proof. For a cell c ∈ C_{k+1}^{+} , its image by the boundary map ∂ is a sum of at most β k-cells, so that ||∂c|| ≤ \sqrt{β}. Thus, by Lemma 3.1 coker(∂)_{tors} is bounded by β^{\frac{1}{2} min (α_{k+1} ,α_{k})} and a fortiori so is H_{k}(C_{•}^{+} )_{tors} .

3.3 Counting the cells

Suppose that the finite subset Φ ⊂ L has the following property:

for any well-rounded pair (L,h), there exists γ ∈ Γ = GL_{N} (O_{F}) such that γM(L,h) ⊂ Φ.

In other words, Φ contains a representative of every element of Γ\W̃. Since C(h) has codimension j, where N + j is the cardinality of M(L,h), it follows immediately that the number of cells of codimension j in Γ\W̃ is bounded by the binomial coefficient \binom{card (Φ)}{N+j}. For large card(Φ) we lose little by bound-

ing this binomial coefficient by card(Φ)^{N+j} . Recall that Γ\W̃ has dimension dim(X)−N, so that for a k-cell of codimension j we have N +j = dim(X)−k. For the dimension of X we have (where (r_{1},r_{2}) is the signature of F):

dim(X) = r_{1} \frac{N(N + 1)}{2} + r_{2}N^{2}  (3.2)
≤ d \frac{N(N + 1)}{2} .  (3.3)

Thus, for the number of k-cells in Γ\W̃ we can use the following upper bound:

α_{k} = card(Φ)^{d· \frac{N(N+1)}{2} −k}  (3.4)

By a similar counting argument, Soulé shows in [11, proof of Prop. 3] that there are at most β = card(Φ)^{N+1} faces of codimension 1 in any given cell (not necessarily top dimensional) on Γ\W̃.

3.4 Bounds for K-theory in terms of Φ

Let ℓ = max(d + 1,2n + 2). By Corollary 3.2 and (3.1) we have that card_{ℓ}K_{n}(O_{F})_{tors} is bounded by β^{\frac{1}{2} α_{n+1}} , where α_{n+1} and β can be chosen as in Section 3.3 (with N = 2n + 1). This gives (using now logarithmic notation):

logcard_{ℓ}K_{n}(O_{F})_{tors} ≤ (n + 1) log(card(Φ))card(Φ)^{e(d,n)} ,  (3.5)

where Φ ⊂ L has the property defined in Section 3.3, and

e(d,n) = d(2n^{2} + 3n + 1) − n − 1.  (3.6)

This reduces the problem to finding such a set Φ ⊂ L of size as small as possible. In [11] Soulé constructed a suitable set Φ using the geometry of numbers. In what follows, we will exhibit a smaller Φ by using better estimates on hermitian lattices.

4 Hermitian lattices and bounded bases

The goal of this section is to construct in any well-rounded lattice (L,h) a basis whose vectors have bounded length, with respect to the norm induced by h. The method in an adaptation of the idea used by Soulé in [11] (see Section 4.3 below), in which we incorporate the results from [3], corresponding to the rank one case.

4.1 Geometry of ideal lattices

Let I ⊂ F_{ℝ} be a nonzero O_{F} -submodule of the form I = x𝔞, where x ∈ F_{ℝ} and 𝔞 is a fractional ideal of F. We define the norm of I by the rule N (I) = N (x)N (𝔞). Let q_{0} be the standard (positive definite) hermitian metric on F_{ℝ}, i.e., for x ∈ F_{ℝ}: q(x) = Tr(x\overline{x}) . The pair (I,q_{0}) is an ideal lattice (over F) in the sense of [3, Def. 2.2]. Its determinant is given by (see [3, Cor. 2.4]):

det(I,q_{0}) = N (I)^{2} |D_{F} |.  (4.1)

Let us denote by || · || the norm on F_{ℝ} induced by the hermitian metric q_{0}. Estimates for the geometry of ideal lattices have been studied in [3]. For our particular case (I,q_{0}), Proposition 4.2 in loc. cit. takes the following form.

Proposition 4.1. Let F of degree d, with discriminant D_{F}, and consider the ideal lattice (I,q_{0}). Then for any x ∈ F_{ℝ} there exists y ∈ I such that ||x−y|| ≤ R, where

R = \frac{\sqrt{d}}{2}|D_{F} |^{1/d}N (I)^{1/d}.

4.2 Three consequences

From Proposition 4.1 we deduce the three following lemmas.

Lemma 4.2. Given x = (x_{σ}) ∈ F_{ℝ}, there exists a ∈ O_{F} such that

∑_{σ∈Σ} |x_{σ} − σ(a)| ≤ C_{2},

where

C_{2} = \frac{d}{2}|D_{F} |^{1/d}.  (4.2)

Proof. First note the general inequality (∑ _{i=1}^{d} b_{i})^{2} ≤ d·∑ _{i=1}^{d} b_{i}^{2}, which follows from applying the summation ∑ _{i,j} on both sides of

2b_{i}b_{j}≤ b_{i}^{2} + b_{j}^{2}.

This implies that for a = y ∈ I as in Proposition 4.1 with I = O_{F}, we have

∑_{σ∈Σ} |x_{σ} − σ(a)| ≤ \sqrt{d · ∑_{σ∈Σ} |x_{σ} − σ(a)|^{2}}

= \sqrt{d}||x − a||

≤ \frac{d}{2}|D_{F} |^{1/d}.

Lemma 4.3. Given x = (x_{σ}) ∈ F_{ℝ}, there exists a ∈ O_{F} such that

sup_{σ∈Σ} |σ(a)x_{σ}| ≤ C_{3} N (x)^{1/d},

where

C_{3} = \sqrt{d}· |D_{F} |^{1/d}.  (4.3)

Proof. We can suppose that x ≠ 0. We consider the ideal lattice (I,q_{0}) with I = xO_{F}. For R as in Proposition 4.1, we have that F_{ℝ} = I + B_{R}(0), where B_{R}(0) is the closed ball of radius R with respect to || · ||. In particular, the smallest (nonzero) vector xa ∈ I = xO_{F} has length ≤ 2R. That is, there exists a ∈ O_{F} such that

sup_{σ∈Σ} |σ(a)x_{σ}| ≤ ||xa||
≤ 2R;

and the result follows.

Lemma 4.4. Let 𝔞 be an ideal of O_{F}. Then there exists a set R ⊂ O_{F} of representatives of O_{F} /𝔞 such that for any x ∈ R we have

∑_{σ∈Σ} |σ(x)| ≤ C_{2} N (𝔞)^{1/d}.

Proof. Let us consider the ideal lattice (I,q) = (𝔞,q_{0}), and let R be as in Proposition 4.1. Then for any x ∈ O_{F} ⊂ F_{ℝ}, there exists y ∈ I such that ||x−y|| ≤ R. But x−y ≡ x (I), so that the closed ball B_{R}(0) contains a representative of each class of O_{F} /𝔞. The inequality is then obtained as in the proof of Lemma 4.2.

4.3 Existence of bounded bases

Lemma 4.5 (Soulé). Let L = L_{1} ⊕··· ⊕L_{N} be a decomposition of the hermitian lattice (L,h) into rank one lattices, and suppose that each L_{i} contains a vector f_{i} with |L_{i}/O_{F} f_{i}| ≤ k and ||f_{i}||_{h} ≤ kλ, for some k,λ > 1. Then L has a basis e_{1},...,e_{N} such that

||e_{i}||_{h} ≤ λ(1 + C_{2}) ^{t+1} k^{(d+1)(4N−1)} ,

where t = ⌊log_{2}(N)⌋ + 1.

Proof. The statement and its proof is essentially contained in the proof of [11, Prop. 1]. The main difference is that our constant C_{2} is now smaller than C_{2} in loc. cit. We can follow verbatim the same proof with the new C_{2} except for the use of Lemma 6 (needed in Lemma 7) of loc. cit., which must be replaced by Lemma 4.4. Accordingly, the factor (1 + C_{2} \frac{r+3}{4} ) (where r = d) is replaced by 1 + C_{2}.

To obtain a bounded basis for (L,h) we need to find elements f_{i} that satisfy the condition of Lemma 4.5. This is done in the following proposition.

Proposition 4.6. Let (L,h) be a free hermitian lattice over O_{F} of rank N, with F ≠ ℚ. We suppose that there exist e_{1},...,e_{N} ∈ L that span V = F ⊗_{O_{F}} L and such that ||e_{i}||_{h} ≤ 1 for i = 1,...,N. Then there exists a decomposition L = L_{1} ⊕ ··· ⊕ L_{N} and elements f_{i} ∈ L_{i} such that:

|L_{i}/f_{i}O_{F} | ≤ C_{1}C_{3}^{d} ;
||f_{i}||_{h} ≤ iC_{1}C_{2}C_{3}^{d},

where C_{1} = |D_{F} |^{1/2} , and C_{2} (resp. C_{3}) is defined in (4.2) (resp. (4.3)).

Proof. The proof proceeds by induction, and follows the line of argument of [11, proof of Lemma 5]. Let N = 1. By Lemma 1 in loc. cit., there exists x ∈ L such that |L/O_{F} x| ≤ C_{1} = |D_{F} |^{1/2} . Let us write x = α · e_{1} for α ∈ F^{×} , where by assumption ||e_{1}||_{h} ≤ 1. By Lemma 4.3 applied to α ∈ F_{ℝ}, there exists a ∈ O_{F} such sup_{σ} |σ(aα)| ≤ C_{3}|N(α)|^{1/d}. In particular,

|N(aα)| ≤ ( sup_{σ∈Σ} |σ(aα)|) ^{d}
≤ C_{3}^{d}|N(α)|,

so that |N(a)| ≤ C_{3}^{d}. We set f_{1} = a · x. Then

|L/O_{F} f_{1}| = |N(a)| · |L/O_{F} x|
≤ C_{3}^{d}C_{1}.

For the norm we have:

||f_{1}||_{h}^{2} = Tr(h(f_{1},f_{1}))

= ∑_{σ∈Σ} |σ(α)|^{2} h_{σ}(e_{1},e_{1})

≤ ( sup_{σ∈Σ} |σ(α)|) ^{2} ||e_{1}||_{h}^{2}

≤ C_{3}^{2} · |N(α)|^{2/d}.

Moreover, |L/O_{F} x| = |N(α)|·|L/O_{F} e_{1}|, so that |N(α)| ≤ C_{1}. This shows that ||f_{1}||_{h} ≤ C_{3}C_{1}^{1/d} and thus concludes the proof for N = 1. The induction step is done exactly as in loc. cit., adapting the constants when necessary (C_{1} to be replaced by C_{1}C_{3}^{d}), to obtain the desired f_{i} ∈ L_{i}, i.e., with (using F ≠ ℚ in the last inequality, so that C_{2} ≥ 1):

||f_{i}||_{h} ≤ (i − 1)C_{1}C_{3}^{d}C_{2} + C_{3}C_{1}^{1/d}
≤ iC_{1}C_{2}C_{3}^{d}.

We finally obtain the result about the existence of bounded bases. The assumption N ≥ 5 is only here in order to simplify the statement.

Proposition 4.7. Let (L,h) be a free hermitian lattice over O_{F} of rank N ≥ 5, with F ≠ ℚ, and such that the subset {x ∈ L | ||x||_{h} ≤ 1} spans V = F ⊗_{O_{F}} L. Then there exists a basis e_{1},...,e_{N} of L such that ||e_{i}||_{h} ≤ B for every i = 1,...,N, where

B = \frac{4N^{2}}{2^{N}} d^{5Nd^{2}} |D_{F} |^{6N(d+1)} .

Proof. By Proposition 4.6 we can apply Lemma 4.5 with

k = C_{1}C_{3}^{d} ;
λ = N C_{2}.

This shows the existence of a basis e_{1},...,e_{N} with

||e_{i}||_{h} ≤ N C_{2}(1 + C_{2})^{⌊log_{2}(N)⌋+2} (C_{1}C_{3}^{d})^{(d+1)(4N−1)} .

Since C_{2} ≥ 1, we have (1 + C_{2})^{n} ≤ 2^{n}C_{2}^{n}. Moreover, for N ≥ 5 we have ⌊log_{2}(N)⌋ + 3 ≤ N. We deduce:

||e_{i}||_{h} ≤ 4N^{2} C_{2}^{N} (C_{1}C_{3}^{d})^{(d+1)(4N−1)}
= αd^{β}|D_{F} |^{γ},

with (using N ≥ 5 and d ≥ 2):

α = 4 \frac{N^{2}}{2^{N}} ;
β = N + \frac{d}{2}(d + 1)(4N − 1)
≤ 5Nd^{2} ;
γ = \frac{N}{d} + \frac{3}{2}(d + 1)(4N − 1)
≤ 6(d + 1)N.

This finishes the proof.

5 Improved estimates for K-groups

5.1 A Bounded set Φ

The construction of a bounded set Φ ⊂ L will follow from this proposition.

Proposition 5.1. Let (L,h) be a free well-rounded O_{F} -lattice of rank N ≥ 5, with F ≠ ℚ. Let e_{1},...,e_{N} and B be defined as in Proposition 4.7, and for x ∈ M(L,h) write x = ∑ _{i} x_{i}e_{i}, with x_{i} ∈ O_{F}. Then for every i = 1,...,N we have:

∑_{σ∈Σ} |σ(x_{i})|^{2} ≤ T,

where

T = N^{Nd}d^{\frac{3}{2} Nd+1} B^{2(Nd−1)} |D_{F} |^{2N} .

Proof. Let x ∈ M(L,h), i.e., h(x) = 1. For each σ ∈ Σ let us consider the matrix H_{σ} = (h_{σ}(e_{i},e_{j})). Then the first argument in [11, proof of Prop. 2], based on the Hadamard inequality for positive definite matrix, shows that

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

Since h_{σ}(e_{j}) ≤ ||e_{j} ||_{h}^{2} ≤ B^{2}, and similarly h_{σ}(x) ≤ 1, we obtain:

∑_{σ∈Σ} |σ(x_{i})|^{2} ≤ B^{2(N−1)} ∑_{σ∈Σ} det(H_{σ})^{−1} .  (5.1)

For ∑ _{σ} det(H_{σ})^{−1} we can write, using the Hadamard inequality:

∑_{σ∈Σ} det(H_{σ})^{−1} = ∑_{σ∈Σ} ( ∏_{σ′≠σ} det(H_{σ′}) ∏_{σ′∈Σ} det(H_{σ′})^{−1} )

≤ ( ∑_{σ∈Σ} ∏_{σ′≠σ} ∏_{j=1}^{N} h_{σ′}(e_{j})) · ( ∏_{σ∈Σ} det(H_{σ})^{−1} )

≤ d · B^{2N(d−1)} ∏_{σ∈Σ} det(H_{σ})^{−1} .  (5.2)

According to Icaza [8, Theorem 1], there exists z ∈ L such that

∏_{σ∈Σ} det(H_{σ})^{−1} ≤ γ^{N} N (h(z))^{−N} ,  (5.3)

where (cf. [11, Equ. (21)]):

γ ≤ N^{d}|D_{F} |.

By applying Lemma 4.3 to h(z) ∈ F_{ℝ}, we find a ∈ O_{F} such that h_{σ}(az) = σ(a)h_{σ}(z) ≤ C_{3}N (h(z))^{1/d} for every σ ∈ Σ. Since (L,h) is well rounded, this implies:

dC_{3} N (h(z))^{1/d} ≥ h(az) ≥ 1,  (5.4)

so that N (h(z))^{−1} ≤ d^{d}C_{3}^{d} = d^{\frac{3}{2} d}|D_{F} |. Using this with (5.1), (5.2) and (5.3), this concludes the proof.

Corollary 5.2. Let L be a free O_{F} -lattice of rank N ≥ 5, with F ≠ ℚ. Then there exists a subset Φ ⊂ L with the property given in Section 3.3 and such that

card(Φ) ≤ N^{3N^{2}d^{2}} · d^{5N^{3}d^{4}} · |D_{F} |^{9N^{3}d^{3}} .

Proof. Let f_{1},...,f_{N} be any basis of L, and set, for T as in Proposition 5.1:

Φ = { ∑_{i=1}^{N} x_{i}f_{i} | x_{i} ∈ O_{F} with ∑_{σ∈Σ} |σ(x_{i})|^{2} ≤ T } .

According to [11, Lemma 8], the number of elements x_{i} ∈ O_{F} with ∑ _{σ∈Σ} |σ(x_{i})|^{2} ≤ T is at most T^{d/2}2^{d+3} , so that card(Φ) is bounded above by T^{Nd/2} 2^{N(d+3)} . Expanding the constants T and B as in the statements of Propositions 5.1 and 4.7, we obtain the stated upper bound for card(Φ). Let h be a well-rounded hermitian metric on L. We can apply Proposition 5.1 to write every x ∈ M(L,h) as x = ∑ x_{i}e_{i} for a bounded basis e_{1},...,e_{N} of L. The proposition implies that the transformation γ ∈ Γ = GL_{N} (O_{F}) that sends the basis (e_{i}) to (f_{i}) is such that γ ·x ∈ Φ. This means that Φ has the property defined in Section 3.3.

5.2 Upper bounds for K_{n}(O_{F})

We finally come to the bounds for the K-groups of O_{F}, as stated in Theorem 1.1. Let ℓ = max(d + 1,2n + 2). From Equation (3.5) we obtain

log card_{ℓ}K_{n}(O_{F})_{tors} ≤ card(Φ)^{e(d,n)+n+1} ,

and note that for n ≥ 2 we have e(d,n) + n + 1 ≤ \frac{15}{4} n^{2}d. Theorem 1.1 now follows directly from Corollary 5.2, applied with N = 2n + 1 ≤ \frac{5}{2}n.

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SB–MATHGEOM–CSAG EPF Lausanne Bâtiment MA, Station 8 CH-1015 Lausanne Switzerland eva.bayer@epfl.ch vincent.emery@math.ch julien.houriet@gmail.com 84

Documenta Mathematica · Extra Volume Merkurjev (2015)