A Bound for the Torsion in the K-Theory of Algebraic Integers

Let m be an integer bigger than one, A a ring of algebraic integers, F its fraction field, and Km(A) the m-th Quillen K-group of A. We give a (huge) explicit bound for the order of the torsion subgroup of Km(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 Km(A) the m-th Quillen K-group of A. It was shown by Quillen that Km(A) is finitely generated. In this paper we shall give a (huge) explicit bound for the order of the torsion subgroup of Km(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 GLN (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 GLN (A). Such a complex is derived from a contractible CW -complex W̃ on which GLN (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 which have minimum equal to one and are such that their set M(h) of minimal vectors has rank equal to N in F . To count cells in W̃/GLN (A), one will exhibit an explicit compact subset of A ⊗ZR which, for every h ∈ W̃ , contains a translate of M(h) by some matrix of GLN (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]. Documenta Mathematica · Extra Volume Kato (2003) 761–788 762 Christophe Soulé 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 Zp, 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 K8(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 r1 (resp. r2) be the number of real (resp. complex) places of F . We have r = r1 + 2 r2. 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) ≤ C1 , where C1 = r! rr · 42 π2 √ D in general, and C1 = 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)| ≤ C1 N(I) . Documenta Mathematica · Extra Volume Kato (2003) 761–788 A Bound for the Torsion . . . 763 Here |N(x)| = #(A/Ax) and N(I) = # (A/I), therefore # (I/Ax) ≤ C1. 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) , 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 = (xσ) ∈ F ⊗Q R, there exists a ∈ A such that ∑ σ∈Σ |xσ − σ(a)| ≤ C2 , with C2 = 41 π2 rr−2 r! √ D in general, and C2 = 1/2 if F = Q . Proof. Define a norm on F ⊗Q R by the formula ‖x‖ = ∑