A new Algorithm for the Computation of logarithmic ℓ-Class Groups of Number Fields^{∗}

Francisco Diaz y Diaz, Jean-François Jaulent, Sebastian Pauli, Michael Pohst, Florence Soriano-Gafiuk

Abstract. We present an algorithm for the computation of logarithmic ℓ-class groups of number fields. Our principal motivation is the effective determination of the ℓ-rank of the wild kernel in the K-theory of number fields.

Résumé. Nous développons un nouvel algorithme pour calculer le ℓ-groupe C̃ℓ_{F} des classes logarithmiques d’un corps de nombres F. Il permet en particulier de déterminer effectivement le ℓ-rang du noyau sauvage WK_{2}(F) du groupe K_{2}(F).

1 Introduction

A new invariant of number fields, called group of logarithmic classes, was introduced by J.-F. Jaulent in 1994 [J1]. The interest in the arithmetic of logarithmic classes is because of its applicability in K-Theory. Indeed this new group of classes is revealed to be mysteriously related to the wild kernel in the K-Theory for number fields. The new approach to the wild kernel is so attractive since the arithmetic of logarithmic classes is very efficient. Thus it provides an algorithmic and original study of the wild kernel. A first algorithm for the computation of the group of logarithmic classes of a number field F was developed by F. Diaz y Diaz and F. Soriano in 1999 [DS]. We present a new much better performing algorithm, which also eliminates the restriction to Galois extensions.

Let ℓ be a prime number. If a number field F contains the 2ℓ-th roots of unity then the wild kernel of F and its logarithmic ℓ-class group have the same ℓ-rank. If F does not contain the 2ℓ-th roots of unity the arithmetic of the logarithmic classes still yields the ℓ-rank of the the wild kernel. More precisely:

• If ℓ is odd [JS1, So] we consider F′ := F (ζ_{ℓ}), where ζ_{ℓ} is the ℓ-th root of unity, and use classic techniques from the theory of semi-simple algebras.

• If ℓ = 2 [JS2] we introduce a new group, which we call the ℓ-group of of the positive divisor classes and which can be constructed from the ℓ-group of logarithmic classes.

In the present article we consider the general situation where F is a number field which does not necessarily contain the 2ℓ-th roots of unity.

∗Experimental. Math. 14 (2005), 67–76.

2 The theoretical background

This section is devoted to the introduction of the main notions of logarithmic arithmetic. We also review the facts that are of interest for our purpose. We do not attempt to give a fully detailed account of the logarithmic language. Most proofs may be found in [J1], pp 303-313.

2.1 Review of the main logarithmic objects

For any number field F, let J_{F} be the ℓ-adified group of idèles of F, i.e. the restricted product

J_{F} = ∏_{𝔭}^{res} R_{𝔭}

of the ℓ-adic compactifications R_{𝔭} = \underleftarrow{lim} F_{𝔭}^{×}/F_{𝔭}^{×}^{ℓ^{n}} of the multiplicative groups of the completions of F at each 𝔭. For each finite place 𝔭 the subgroup Ũ_{𝔭} of R_{𝔭} of the cyclotomic norms (that is to say the elements of R_{𝔭} which are norms at any finite step of the local cyclotomic ℤ_{ℓ}-extension F_{𝔭}^{c}/F_{𝔭}) will be called the group of logarithmic units of F_{𝔭}. The product

Ũ_{F} = ∏_{𝔭} Ũ_{𝔭}

is denominated the group of idelic logarithmic units; it happens to be the kernel of the logarithmic valuations

ṽ_{𝔭} | x ↦ −\frac{Log_{ℓ} (N_{F_{𝔭}/ℚ_{p}}(x))}{deg_{F} 𝔭},

defined on the R_{𝔭} and ℤ_{ℓ}-valued. These are obtained by taking the Iwasawa logarithm of the norm of x in the local extension F_{𝔭}/ℚ_{p} with a normalization factor deg_{F} 𝔭 whose precise definition is given in the next subsection.

The quotient Dℓ_{F} = J_{F} /Ũ_{F} is the ℓ-group of logarithmic divisors of F; via the logarithmic valuations ṽ_{𝔭}, it may be identified with the free ℤ_{ℓ}-module generated by the prime ideals of F

Dℓ_{F} = J_{F} /Ũ_{F} = ⊕_{𝔭} ℤ_{ℓ} 𝔭.

The degree of a logarithmic divisor 𝔡 = ∑_{𝔭} n_{𝔭} 𝔭 is then defined by

deg_{F} (∑_{𝔭} n_{𝔭} 𝔭) = ∑_{𝔭} n_{𝔭} deg_{F} 𝔭,

inducing a ℤ_{ℓ}-valued ℤ_{ℓ}-linear map on the class group of logarithmic divisors. The logarithmic divisors of degree zero form a subgroup of Dℓ_{F} denoted by

D̃ℓ_{F} = { 𝔡 ∈ Dℓ_{F} | deg_{F} 𝔡 = 0 }.

The image of the map d̃iv_{F} defined via the set of logarithmic valuations from the principal idèle subgroup

R_{F} = ℤ_{ℓ} ⊗_{ℤ} F^{×}

of J_{F} to D̃ℓ_{F} is a subgroup denoted by P̃ℓ_{F}, which will be referred to as the subgroup of principal logarithmic divisors. The quotient

C̃ℓ_{F} = D̃ℓ_{F} /P̃ℓ_{F}

is, by definition, the ℓ-group of logarithmic classes of F. And the kernel

Ẽ_{F} = R_{F} ∩ Ũ_{F}

of the morphism d̃iv_{F} from R_{F} in D̃ℓ_{F} is the group of global logarithmic units.

2.2 Logarithmic ramification and ℓ-adic-degrees.

Next we review the basic notions of the logarithmic ramification, which mimic, as a rule, the classical ones.

Let L/F be any finite extension of number fields. Let p be a prime number. Denote by ℚ̂_{p}^{c} the cyclotomic ℤ̂-extension of ℚ_{p}, that is to say the compositum of all cyclotomic ℤ_{q}-extension of ℚ_{p} on all prime numbers q. Let 𝔭 be a prime of F above (p) and 𝔓 a prime of L above 𝔭. The logarithmic ramification (resp. inertia) index ẽ(L_{𝔓}/F_{𝔭}) (resp. f̃(L_{𝔓}/F_{𝔭})) is defined to be the relative degree

ẽ(L_{𝔓}/F_{𝔭}) = [L_{𝔓} : L_{𝔓} ∩ ℚ̂_{p}^{c}F_{𝔭}] (resp. f̃(L_{𝔓}/F_{𝔭}) = [L_{𝔓} ∩ ℚ̂_{p}^{c}F_{𝔭} : F_{𝔭}]).

As a consequence, L/F is logarithmically unramified at 𝔓, i.e. ẽ(L_{𝔓}/F_{𝔭}) = 1, if and only if L_{𝔓} is contained in the cyclotomic extension of F_{𝔭}. Moreover, for any q ≠ p the classical and the logarithmic indexes have the same q-part (see theorem 5). Hence they are equal as soon as p ∤ [F_{𝔭} : ℚ_{p}].

As usual, in the special case L/F = K/ℚ, the absolute logarithmic indexes of a finite place 𝔭 of K over the prime p are just denoted by ẽ_{𝔭} and f̃_{𝔭}. With these notations, the ℓ-adic degree of 𝔭 is defined by the formula:

deg_{K} 𝔭 = f̃_{𝔭} deg_{ℓ} p with deg_{ℓ} p = \{ \begin{matrix}Log_{ℓ} p & for p ≠ ℓ; \\ ℓ & for p = ℓ ≠ 2; \\ 4 & for p = ℓ = 2. \end{matrix}

The extension and norm maps between groups of divisors, denoted by ι_{L/F} and N_{L/F} respectively, have their logarithmic counterparts, ι̃_{L/F} and Ñ_{L/F} respectively. To be more explicit, ι̃_{L/F} is defined on every finite place 𝔭 of F by

ι̃_{L/F} (𝔭) = ∑_{𝔓|𝔭} ẽ_{L_{𝔓}/F_{𝔭}}𝔓 ,

while Ñ_{L/F} is defined on all 𝔓 lying above 𝔭 by

Ñ_{L/F} (𝔓) = f̃_{L_{𝔓}/F_{𝔭}} 𝔭 .

These applications are compatible with the usual extension and norm maps defined between R_{L} and R_{F}, in the sense that they sit inside the commutative diagrams:

R_{L} ⟶^{d̃iv_{L}} D̃ℓ_{L} ⟶^{deg_{L}} ℤ_{ℓ}  R_{L} ⟶^{d̃iv_{L}} D̃ℓ_{L} ⟶^{deg_{L}} ℤ_{ℓ}
↓^{N_{L/F}}  ↓Ñ_{L/F}  ∥ and ↑ι̃_{L/F}  ↑ι̃_{L/F}  ↑^{[L:F]}
R_{F} ⟶^{d̃iv_{F}} D̃ℓ_{F} ⟶^{deg_{F}} ℤ_{ℓ}  R_{F} ⟶^{d̃iv_{F}} D̃ℓ_{F} ⟶^{deg_{F}} ℤ_{ℓ}.

When L/F is a Galois extension with Galois group Gal(L/F), one deduces from the very definitions the unsurprising and obvious properties:

Ñ_{L/F} ∘ ι̃_{L/F} = [L : F] and ι̃_{L/F} ∘ Ñ_{L/F} = ∑_{σ∈Gal(L/F)} σ .

2.3 Ideal theoretic description of logarithmic classes.

By the weak density theorem every class in J_{F} /Ũ_{F} R_{F} may be represented by an idele with trivial components at the ℓ-adic places, that is to say that every class in Dℓ_{F} /Pℓ_{F} comes from a ℓ-divisor 𝔡 = ∑_{𝔭∤ℓ} α_{𝔭} 𝔭.

The canonical map from R_{F} to Dℓ_{F} maps a ∈ R_{F} to d̃iv_{F} (a) = ∑_{𝔭} ṽ_{𝔭}(a)𝔭. Now for each finite place 𝔭 ∤ ℓ, the quotient ẽ_{𝔭}/e_{𝔭} = f_{𝔭}/f̃_{𝔭} of the classical and logarithmical indexes associated to 𝔭 is a ℓ-adic unit (theorem 5), say λ_{𝔭} (which is 1 for almost all 𝔭), and one has the identity:

ṽ_{𝔭} = λ_{𝔭}v_{𝔭}

between the logarithmic and the classical valuations. So every ℓ-divisor 𝔡 comes from a ℓ-ideal 𝔞 by the formula:

𝔞 = ∏_{𝔭∤ℓ} 𝔭^{α_{𝔭}} ↦ 𝔡_{F} (𝔞) = ∑_{𝔭∤ℓ} λ_{𝔭} α_{𝔭} 𝔭 .

This gives the following ideal theoretic description of logarithmic classes:

Definition & Proposition 1. Let Id_{F} = {𝔞 = ∏_{𝔭∤ℓ} 𝔭^{α_{𝔭}}} be the group of ℓ-

ideals, Ĩd_{F} = {𝔞 ∈ Id_{F} | deg_{F} 𝔡_{F} (𝔞) = 0} be the subgroup of ℓ-ideals of degree 0 and P̃r_{F} = {∏_{𝔭∤ℓ} 𝔭^{v_{𝔭}(a)}|ṽ_{𝔭}(a) = 0 ∀𝔭 | ℓ} the subgroup of principal ℓ-ideals generated by principal ideles a having logarithmic valuations 0 at every ℓ-adic places. Then one has:

C̃ℓ_{F} ≃ Ĩd_{F} /P̃r_{F}

Proof. As explained above, the surjectivity follows from the weak approximation theorem. So let us consider the kernel of the canonical map ϕ_{F} : Ĩd_{F} ↦ C̃ℓ_{F}. Clearly we have: ker ϕ_{F} = {𝔞 ∈ Ĩd_{F} | ∃a ∈ R_{F} 𝔡_{F} (𝔞) = d̃iv_{F} (a)}. The condition 𝔡_{F} (𝔞) = d̃iv_{F} (a) with 𝔞 ∈ Ĩd_{F} implies ṽ_{𝔭}(a) = 0 ∀𝔭 | ℓ; and then gives (a) ∈ P̃r_{F} as expected.

The generalized Gross conjectures (for the field F and the prime ℓ) asserts that the logarithmic class group C̃ℓ_{F} is finite (cf. [J1]). This conjecture, which is a consequence of the p-adic Schanuel conjecture was only proved in the abelian case and a few others (cf. [FG, J5]). Nevertheless, since C̃ℓ_{F} is a ℤ_{ℓ}-module of finite type (by the ℓ-adic class field theory), the Gross conjecture just claims the existence of an integer m such that ℓ^{m} kills the logarithmic class group. As in numerical situations it is rather easy to compute such an exponent m (when the classical invariants of the number field are known), this give rise to a more suitable description of C̃ℓ_{F} in order to carry on numerical computations:

Proposition 2. Assume the integer m to be large enough such that the logarithmic class group C̃ℓ_{F} is annihilated by ℓ^{m}. Thus introduce the group:

Ĩd_{F}^{(ℓ^{m})} = {𝔞 ∈ Id_{F} | deg_{F} 𝔡_{F} (𝔞) ∈ ℓ^{m} deg_{F} Dℓ_{F} } = Ĩd_{F} Id_{F}^{ℓ^{m}}

Thus, denoting P̃r_{F}^{(ℓ^{m})} = P̃r_{F} Ĩd_{F}^{ℓ^{m}}, one has: C̃ℓ_{F} ≃ Ĩd_{F}^{(ℓ^{m})}/P̃r_{F}^{(ℓ^{m})} .

Proof. The hypothesis gives Ĩd_{F}^{ℓ^{m}} ⊂ P̃r_{F} and by a straightforward calculation:

\begin{aligned}Ĩd_{F}^{(ℓ^{m})}/P̃r_{F}^{(ℓ^{m})} & = Ĩd_{F} Id^{ℓ^{m}}/P̃r_{F} Id^{ℓ^{m}} ≃ Ĩd_{F} /(Ĩd_{F} ∩ P̃r_{F} Id^{ℓ^{m}}) \\ & ≃ Ĩd_{F} /P̃r_{F} Ĩd_{F}^{ℓ^{m}} = Ĩd_{F} /P̃r_{F} ≃ C̃ℓ_{F} . \end{aligned}

Remark 3. A lower bound for m which will be required for a sufficient precision of the p-adic calculations will be given after lemma 12.

3 The algorithms

Throughout this section a finite abelian group G is presented by a column vector g ∈ G^{m}, whose entries form a system of generators for G, and by a matrix of relations M ∈ ℤ^{n×m} of rank m, such that v^{T} g = 0 for v ∈ ℤ^{m} if and only if v^{T} is an integral linear combination of the rows of M. We note that for every a ∈ G there is a v ∈ ℤ^{m} satisfying a = v^{T} g. If g_{1}, …, g_{m} is a basis of G, M is usually a diagonal matrix. Algorithms for calculations with finite abelian groups can be found in [C2]. If G is a multiplicative abelian group, then v^{T} g is an abbreviation for g_{1}^{v_{1}} ⋯ g_{m}^{v_{m}}.

One of the steps in the computation of the logarithmic class group is the computation of the ideal class group of a number field. Algorithms for this can be found in [C1, He, PZ]. One tool used in these algorithms are the 𝔰-units, which we will also use directly in our algorithm.

Definition 4 (𝔰-units). Let 𝔰 be an ideal of a number field F. We call the group

{α ∈ F^{×} | v_{𝔭}(α) = 0 for all 𝔭 ∤ 𝔰}

the 𝔰-units of F.

For this section let F be a fixed number field. We denote the ideal class group of F by Cℓ = Cℓ_{F}. We also write C̃ℓ for C̃ℓ_{F}, D̃ℓ for D̃ℓ_{F}, and so on.

3.1 Computing deg_{F}(𝔭) and ṽ_{𝔭}(·)

We describe how invariants of the logarithmic objects can be computed. Some of the tools presented here also applied directly in the computation of the logarithmic class group.

Definition & Proposition 5. Let p be a prime number. Let F be a number field. Let 𝔭 be a prime ideal of F over p. For a ∈ ℚ_{p}^{×} ≅ p^{ℤ} × 𝔽_{p}^{×} × (1 + 2pℤ_{p}) denote by ⟨a⟩ the projection of a to (1 + 2pℤ_{p}). Let F_{𝔭} be the completion of F with respect to 𝔭. For α ∈ F define

h_{𝔭}(α) := \frac{Log_{p}⟨N_{F_{𝔭}/ℚ_{p}}(α)⟩}{[F_{𝔭} : ℚ_{p}] · deg_{p} p}.

The p-part of the logarithmic ramification index ẽ_{𝔭} is [h_{𝔭}(F_{𝔭}^{×}) : ℤ_{p}]. For all primes q with q ≠ p the q-part of ẽ_{𝔭} is the q-part of the ramification index e_{𝔭} of 𝔭.

For a proof see [J1].

In section 2.2 we have seen that the degree deg_{F} (𝔭) of a place 𝔭 can be computed as deg_{F} (𝔭) = f̃_{𝔭} deg_{ℓ} p. From section 2.3 we know that ẽ_{𝔭}f̃_{𝔭} = e_{𝔭}f_{𝔭}. We have

ṽ_{𝔭}(x) = −\frac{Log_{ℓ}(N_{F_{𝔭}/ℚ_{p}}(x))}{deg_{F} (𝔭)}.

Thus we can concentrate on the computation of ẽ_{𝔭} for which we need the completion F_{𝔭} of F at 𝔭 and generators of the unit group F_{𝔭}^{×}.

The Round Four Algorithm was originally conceived as an algorithm for computing integral bases of number fields. It can be applied in three different ways in the computation of logarithmic classgroups. Firstly, it is used for factoring ideals over number fields; secondly, it returns generating polynomials of completions of number fields; and thirdly, it can be used for determining integral bases of maximal orders.

Let Φ(x) be a monic, squarefree polynomial over ℤ_{p}. The algorithm for factoring polynomials over local fields as described in [Pa] returns

• a factorization Φ(x) = Φ_{1}(x) · · · · · Φ_{s}(x) of Φ(x) into irreducible factors Φ_{i}(x) (1 ≤ i ≤ s) over ℤ_{p},

• the inertia degrees e_{i} and ramification indexes f_{i} of the extensions of ℚ_{p} given by the Φ_{i}(x) (1 ≤ i ≤ s), and

• two element certificates (Γ_{i}(x), Π_{i}(x)) with Γ_{i}(x), Π_{i}(x) ∈ F [x] such that v_{i}(Π_{i}(α_{i})) = 1/e_{i} and [𝔽_{p}(\overline{Γ_{i}(α_{i})}) : 𝔽_{p}] = f_{i} where α_{i} is a root of Φ_{i}(x) in F [x]/(Φ_{i}(x)), v_{i} is an extension of the exponential valuation v_{p} of ℚ_{p} to ℚ_{p}[x]/(Φ_{i}(x)) with v_{i}|_{ℚ_{p}} = v_{p}.

The factorization algorithm in [FPR] returns the certificates combined in one polynomial for each irreducible factor. The data returned by these algorithms can be applied in several ways.

• An integral basis of the extension of ℚ_{p} generated by a root α_{i} of Φ_{i}(x) is given by the elements Γ_{i}(α_{i})^{h}Π_{i}(α_{i})^{j} with 0 ≤ h ≤ f_{i} and 0 ≤ j ≤ e_{i}. The local integral bases can be combined to a global integral basis for the extension of ℚ generated by Φ(x).

• For the computation of ṽ_{𝔭} we need to compute the norm of an element in the completions of F. The completions of F are given by the irreducible factors of the generating polynomial of F over ℚ.

Lemma 6 (Ideal Factorization). Let Φ(x) ∈ ℤ_{p}[x] be irreducible over ℚ. Let Φ_{1}(x), …, Φ_{s}(x) ∈ ℤ_{p}[x] be the irreducible factors of Φ(x) with two element certificates (Γ_{i}(x), Π_{i}(x)). Denote by e_{i} the ramification indexes of the extensions of ℚ_{p} given by the Φ_{i}(x) (1 ≤ i ≤ s). The Chinese Remainder Theorem gives polynomials Θ_{1}(x), …, Θ_{s}(x) ∈ ℚ_{p}[x] with

Θ_{i}(x) ≡ Π_{i}(x) mod Φ_{i}(x)
Θ_{i}(x) ≡ 1 mod ∏_{j≠i} Φ_{j}(x).

Let L := ℚ(α) where α is a root of Φ(x) in ℂ. Then

(p) = (p, Θ_{1}(α))^{e_{1}} · · · · · (p, Θ_{s}(α))^{e_{s}}

is a factorization of (p) into prime ideals.

In order to compute [h_{𝔭}(F_{𝔭}^{×}) : ℤ_{p}] it is sufficient to compute the image of a set of generators of F_{𝔭}^{×}. Algorithms for this task were recently developed with respect to the computation of ray class groups of number fields and function fields [C2, HPP], also see [Ha, chapter 15].

Proposition 7.

F_{𝔭}^{×} ≅ π^{ℤ} × (O_{𝔭}/𝔭)^{×} × (1 + 𝔭)

Let 𝔭 be the prime ideal over the prime number p in O_{𝔭}. Let e_{𝔭} be the ramification index and f_{𝔭} the inertia degree of 𝔭. We define the set of fundamental levels

F_{e} :=\{ν | 0 < ν < \frac{pe_{𝔭}}{p−1}, p ∤ ν\}

and let ε ∈ O_{𝔭}^{×} such that p = −π^{e}ε. Furthermore we define the map h_{2} : a + 𝔭 ⟼ a^{p} − εa + 𝔭.

Theorem 8 (Basis of (1+𝔭)). Let ω_{1}, …, ω_{f} ∈ O_{𝔭} be a fixed set of representatives of a 𝔽_{p}-basis of O_{𝔭}/𝔭. If (p−1) does not divide e or h_{2} is an isomorphism, then the elements

1 + ω_{i}π^{ν} where ν ∈ F_{e}, 1 ≤ i ≤ f

are a basis of the group of principal units 1 + 𝔭.

Theorem 9 (Generators of (1 + 𝔭)). Assume that (p − 1) | e and h_{2} is not an isomorphism. Choose e_{0} and μ_{0} such that p does not divide e_{0} and such that e = p^{μ_{0}−1}(p − 1)e_{0}. Let ω_{1}, …, ω_{f} ∈ O_{𝔭} be a fixed set of representatives of a 𝔽_{p}-basis of O_{𝔭}/𝔭 subject to ω_{1}^{p^{μ_{0}}} − εω_{1}^{p^{μ_{0}−1}} ≡ 0 mod 𝔭. Choose ω_{∗} ∈ O_{𝔭} such that x^{p} −εx ≡ ω_{∗} mod 𝔭 has no solution. Then the group of principal units 1+𝔭 is generated by

1 + ω_{∗}π^{p^{μ_{0}} e_{0}} and 1 + ω_{i}π^{ν} where ν ∈ F_{e}, 1 ≤ i ≤ f.

3.2 Computing a bound for the exponent of C̃ℓ

Let F be a number field and ℓ a prime number. Let C̃ℓ = C̃ℓ_{F} ≅ Ĩd/P̃r be the ℓ-group of logarithmic divisor classes. Let 𝔭_{1}, …, 𝔭_{s} be the ℓ-adic places of F.

We describe an algorithm which returns an upper bound ℓ^{m} of the exponent of C̃ℓ (see proposition 2). We denote by

• C̃ℓ(ℓ) the ℓ group of logarithmic divisor classes of degree zero:

C̃ℓ(ℓ) := {[𝔞] ∈ C̃ℓ | 𝔞 = ∑_{i=1}^{s} a_{i}𝔭_{i} with deg_{F} (𝔞) = 0}

• Cℓ′ the ℓ-group of the ℓ-ideal classes, i.e., the ℓ-part of Cℓ/⟨[𝔭_{1}], …, [𝔭_{s}]⟩.

Remark 10. If (ℓ) = 𝔭^{e} where 𝔭 is a prime ideal of O_{K} then the group C̃ℓ(ℓ) is trivial.

Lemma 11 ([DS]). Let

θ : C̃ℓ ⟶ Cℓ′, ∑_{𝔭} m_{𝔭}𝔭 ⟼ ∏_{𝔭∤ℓ} 𝔭^{(1/λ_{𝔭})m_{𝔭}}.

The sequence

0 ⟶ C̃ℓ(ℓ) ⟶ C̃ℓ ⟶^{θ} Cℓ′ ⟶ Coker θ ⟶ 0

is exact.

Proof. Recall that, if 𝔭 ∤ ℓ, ṽ_{𝔭} = λ_{𝔭}v_{𝔭}. Denote by 𝔭_{1}, …, 𝔭_{s} the ℓ-adic places of F. Let

𝔞̃ = ∑_{𝔮} a_{𝔮}𝔮 = d̃iv(α) = ∑_{𝔭} ṽ_{𝔭}(α)𝔭 = ∑_{𝔭∤ℓ} λ_{𝔭}v_{𝔭}(α)𝔭 + ∑_{i=1}^{s} ṽ_{𝔭_{i}}(α)𝔭_{i}

be a principal logarithmic divisor. A representative of the image of 𝔞̃ under θ in terms of ideals is of the form

𝔞 = ∏_{q∤(ℓ)} 𝔮^{v_{𝔮}(α)} = (αO_{K}) × ∏_{i=1}^{s} 𝔭_{i}^{−v_{p_{i}} (α)}.

This shows that the homomorphism θ is well defined. It follows immediately that Ker θ = C̃ℓ(ℓ).

Lemma 12. Set ℓ^{m′} = exp Cℓ′ and ℓm̃ = exp C̃ℓ(ℓ). Then

ℓ^{m′+m̃}𝔞 ≡ 0 mod P̃ℓ for all 𝔞 ∈ D̃ℓ.

Proof. It follows from the exact sequence in lemma 11 that for all 𝔞 ∈ D̃ℓ the congruence ℓ^{m′}θ(𝔞) ≡ 1 holds in Cℓ′. Thus ℓ^{m′}𝔞 ∈ Ker θ = C̃ℓ(ℓ) and ℓ^{m′+m̃}𝔞 ≡ 0 mod P̃ℓ.

Lemma 12 suggests setting the precision for the computation of C̃ℓ to m := m′ + m̃ ℓ-adic digits. If the ideal class group Cℓ is known we can easily compute m′. In order to find m̃ we compute a matrix of relations for C̃ℓ(ℓ).

Lemma 13 (Generators and Relations of C̃ℓ(ℓ)). Let 𝔭_{1}, …, 𝔭_{s} be the ℓadic places of F. Assume that s > 1. Reorder the 𝔭_{i} such that v_{ℓ}(deg(𝔭_{1})) = min_{1≤i≤s} v_{ℓ}(deg(𝔭_{i})). Let γ_{1}, …, γ_{r} be a basis of the ℓ-units of F. Then the group C̃ℓ(ℓ) is given by the generators [𝔤_{i}] := [𝔭_{i} − \frac{deg(𝔭_{i})}{deg(𝔭_{1})}𝔭_{1}] (i = 2, …, s) with relations ∑_{i=2}^{s} ṽ_{𝔭_{i}}(γ_{j})[𝔤_{i}] = [0].

Proof. We consider a logarithmic divisor 𝔞 = ∑_{i=1}^{s} a_{i}𝔭_{i} of degree zero over F that is constructed from the ℓ-adic places. By the choice of 𝔭_{1} and as deg(𝔞) = deg(∑_{i=1}^{s} a_{i}𝔭_{i}) = ∑_{i=1}^{s} a_{i} deg(𝔭_{i}) = 0 the coefficient a_{1} is given by the other s − 1 coefficients. Thus the [𝔤_{i}] generate C̃ℓ(ℓ).

The relations between the classes of C̃ℓ(ℓ) are of the form ∑_{i=2}^{s} b_{i}[𝔤_{i}] = [0]. That is there exists β ∈ R_{F} such that

∑_{i=2}^{s} b_{i}𝔤_{i} = ∑_{j=1}^{s} a_{j}𝔭_{j} = d̃iv(β),

with v_{𝔮}(β) = 0 for all 𝔮 ∤ (ℓ). Thus β is an element of the group of ℓ-units {α ∈ R_{F} | v_{𝔮}(α) = 0} of R_{F}. Hence we obtain the relations given above.

A version of this lemma for the case that F is Galois can be found in [DS].

Algorithm 14 (Precision). Input: a number field F and a prime number ℓ, the ℓ-adic places 𝔭_{1}, …, 𝔭_{s}

of F, and a basis γ_{1}, …, γ_{r} of the ℓ-units of F

Output: an upper bound for the exponent of C̃ℓ • Set ℓ^{m′} ← exp Cℓ′, set m ← max{m′, 4}.

• If s = 1 then return ℓ^{m′}. [Remark 10]

• Repeat

• Set m ← m + 2

• Set [Lemma 13]

A ← ( \begin{matrix}ṽ_{𝔭_{2}}(γ_{1}) & … & ṽ_{𝔭_{s}}(γ_{1}) \\ ⋮ & ... & ⋮ \\ ṽ_{𝔭_{2}}(γ_{r}) & … & ṽ_{𝔭_{s}}(γ_{r}) \end{matrix} ) mod ℓ^{m}.

• Let H be the Hermite normal form of A modulo ℓ^{m}.

• Until rank(H) = s − 1.

• Let S = (S_{i,j})_{i,j} be the Smith normal form of A modulo ℓ^{m}.

• Set m̃ ← max_{1≤i≤s−1}(v_{ℓ}(S_{i,i})), return ℓ^{m′+m̃}.

Remark 15. Algorithm 14 does not terminate in general if Gross’ conjecture is false.

3.3 Computing C̃ℓ

We use the ideal theoretic description from section 2.3 for the computation of C̃ℓ ≅ Ĩd/P̃r. In the previous section we have seen how we can compute a bound for the exponent of C̃ℓ. It is clear that this bound also gives a lower bound for the precision in our calculations.

Theorem 16 (Generators of C̃ℓ). Let 𝔞_{1}, …, 𝔞_{t} be a basis of the ideal classgroup of F with gcd(𝔞_{i}, ℓ) = 1 for all 1 ≤ i ≤ t. Denote by 𝔭_{1}, …, 𝔭_{s} the ℓ-adic places of F. Let α_{1}, …, α_{s} be elements of R_{F} with ṽ_{𝔭_{i}}(α_{j}) = δ_{i,j} (i, j = 1, …, s) and gcd((α_{i}), ℓ) = 1 for all 1 ≤ i ≤ s. Set 𝔞_{t+i} := (α_{i}) for 1 ≤ i ≤ s. For an ideal 𝔞 of F denote by \overline{𝔞} the projection of 𝔞 from ⨁_{𝔭} 𝔭^{ℤ_{ℓ}} to ⨁_{𝔭∤(ℓ)} 𝔭^{ℤ_{ℓ}}. We distinguish two cases:

I. If deg_{ℓ}(𝔞_{i}) = 0 for all 1 ≤ i ≤ t + s then set 𝔟_{i} := 𝔞_{i}. The group C̃ℓ_{F} is generated by \overline{𝔟}_{1}, …, \overline{𝔟}_{t+s}.

II. Otherwise let 1 ≤ j ≤ t+s such that v_{ℓ}(deg_{ℓ}(𝔞_{j})) = min_{1≤i≤t+s} v_{ℓ}(deg_{ℓ}(𝔞_{i})). Set 𝔟_{i} := 𝔞_{i}/𝔞_{j}^{d} with d ≡ \frac{deg_{ℓ}(𝔞_{i})}{deg_{ℓ}(𝔞_{j})} mod ℓ^{m} where ℓ^{m} > exp(C̃ℓ). The group

C̃ℓ_{F} is generated by \overline{𝔟}_{1}, …, \overline{𝔟}_{j−1}, \overline{𝔟}_{j+1}, …, \overline{𝔟}_{t+s}.

Proof. Let 𝔞 ∈ Ĩd. There exist γ ∈ R_{F} and a_{1}, …, a_{t} ∈ ℤ_{ℓ} such that 𝔞 = ∏_{i=1}^{t} 𝔞_{i}^{a_{i}} · (γ). Set g_{i} := ṽ_{𝔭_{i}}(γ) for 1 ≤ i ≤ s. Now

𝔞 = ∏_{i=1}^{s} 𝔞_{i}^{a_{i}} · ((γ) · ∏_{j=1}^{s}(α_{i})^{−g_{i}}) · (∏_{j=1}^{s}(α_{i})^{g_{i}}).

By the definition of Id (Definition and Proposition 1) we have

𝔞 = \overline{𝔞} = ∏_{i=1}^{t} \overline{𝔞}_{i}^{a_{i}} · (\overline{(γ) · ∏_{j=1}^{s}(α_{j})^{−g_{j}}}) · (∏_{j=1}^{s} \overline{(α_{j})^{g_{j}}})

As ṽ_{𝔭_{i}}((γ) ∏_{j=1}^{s}(α_{j})^{−g_{j}}) = 0 for i = 1, …, s we obtain

𝔞 ≡ ∏_{i=1}^{t} 𝔞_{i}^{a_{i}} · (∏_{j=1}^{s} \overline{(α_{j})}^{g_{j}}) mod P̃r.

Thus all elements of C̃ℓ can be represented by \overline{𝔞}_{1}, …, \overline{𝔞}_{t}, \overline{𝔞}_{t+1} = \overline{(α_{1})}, …, \overline{𝔞}_{t+s} = \overline{(α_{s})}. For the two cases we obtain:

I. It follows immediately that \overline{𝔟}_{1}, …, \overline{𝔟}_{t+s} are generators of C̃ℓ.

II. If we have \overline{𝔞} ≡ \overline{𝔞}_{1}^{a_{1}} · · · · · \overline{𝔞}_{t+s}^{a_{t+s}} mod P̃r for an ideal 𝔞 ∈ Ĩd then 0 = deg(\overline{𝔞}) = ∑_{i=1}^{t+s} a_{i} deg_{ℓ}(\overline{𝔞}_{i}), thus −a_{j} = ∑_{i≠j}^{s} a_{i} deg_{ℓ}(\overline{𝔞}_{i})/ deg_{ℓ}(\overline{𝔞}_{j}). Hence \overline{𝔟}_{1}, …, \overline{𝔟}_{j−1}, \overline{𝔟}_{j+1}, …, \overline{𝔟}_{t+s} are generators of C̃ℓ.

We continue to use the notation from theorem 16. Set Cℓ′ := Cℓ/⟨𝔭_{1}, …, 𝔭_{s}⟩.

Remark 17. The definition of Cℓ′ in this section and the previous section, where we considered the ℓ-part of Cℓ/⟨𝔭_{1}, …, 𝔭_{s}⟩, differ. The definition we chose here makes the description of the algorithm easier. In the algorithm we make sure that only the ℓ-part of the group appears in the result by computing the ℓ-adic Hermite normal form of the relation matrix.

The relations between the generators \overline{𝔞}_{1}, …, \overline{𝔞}_{t} of the group Cℓ′ are of the form ∏_{i=1}^{t} \overline{𝔞}_{i}^{a_{i}} = \overline{(α)} with α ∈ R_{F}. There exist integers c_{1}, …, c_{n} such that \overline{(α)} ≡ ∏_{i=1}^{s} \overline{(α_{i})}^{c_{i}} mod P̃r. This yields the relation ∏_{i=1}^{t} \overline{𝔞}_{i}^{a_{i}} ≡ ∏_{i=1}^{s} \overline{(α_{i})}^{c_{i}} mod P̃r in C̃ℓ. We can derive all relations involving the generators \overline{𝔞}_{i} +P̃r from their relations as generators of the group Cℓ′ in this way.

The other relations between the generators of C̃ℓ are obtained as follows. A relation between the generators \overline{α}_{i} is of the form ∏_{i=1}^{s} \overline{(α_{i})}^{v_{i}} ≡ (1) mod Pr or equivalently ∏_{i=1}^{s}(α_{i})^{v_{i}} · ∏_{i=1}^{s} 𝔭_{i}^{w_{i}} = (α) for some α ∈ R_{F}. The last equality is fulfilled if and only if ∏_{i=1}^{s} 𝔭_{i}^{w_{i}} is principal, i.e., if ∏_{i=1}^{s} 𝔭_{i}^{w_{i}} is an (ℓ)-unit. Assume that ∏_{i=1}^{s} 𝔭_{i}^{w_{i}} = (γ) for some γ ∈ R_{F}. As ṽ_{𝔭_{j}}(α) = 0 for all \overline{(α)} ∈ P̃r and 𝔭_{j} | (ℓ) the equation ṽ_{𝔭_{j}}(∏_{i=1}^{s} α_{i}^{v_{i}} · γ) = 0 must hold. By the definition of the β_{i} we obtain v_{i} = −ṽ_{𝔭_{i}}(γ) for 1 ≤ i ≤ s.

Corollary 18 (Relations of C̃ℓ). Let ((\overline{𝔞}_{1}, …, \overline{𝔞}_{t}), (a_{i,j})_{i,j∈\{1,...,t\}}) be a basis and a relation matrix of Cℓ′ := Cℓ⟨𝔭_{1}, …, 𝔭_{s}⟩. Let 𝔞_{t+1} = (α_{1}), …, 𝔞_{t+s} = (α_{s}) be as above. For each 1 ≤ k ≤ t we find c_{k,2}, …, c_{k,s} such that ∏_{i=1}^{t} \overline{𝔟}_{i}^{a_{k,i}} = ∏_{i=2}^{s} \overline{(α_{i})}^{c_{k,i}}. Let γ_{1}, …, γ_{r} be a basis of the (ℓ)-units of R_{F}. Set v_{i,j} := ṽ_{𝔭_{j}}(γ_{i}) (1 ≤ i ≤ r, 2 ≤ j ≤ s). Set

M := ( \begin{matrix}b_{1,1} & … & b_{1,t} & −c_{1,2} & … & −c_{1,s} \\ ⋮ & ... & ⋮ & ⋮ & ... & ⋮ \\ b_{t,1} & … & b_{t,t} & −c_{t,2} & … & −c_{t,s} \\ 0 & … & 0 & v_{1,2} & … & v_{1,s} \\ ⋮ & ... & ⋮ & ⋮ & ... & ⋮ \\ 0 & … & 0 & v_{r,2} & … & v_{r,s} \end{matrix} ) .

For the two cases we obtain:

I. ((\overline{𝔟}_{1}, \overline{𝔟}_{t+s}), M) are generators and relations of C̃ℓ.

II. Let j be chosen as in Theorem 16. Denote by N the matrix obtained by removing the j-th column from M. Then ((\overline{𝔟}_{1}, …, \overline{𝔟}_{j−1}, \overline{𝔟}_{j+1}, …, \overline{𝔟}_{t+s}), N) are generators and relations of C̃ℓ.

Now we only need to find the elements α_{1}, …, α_{s} with ṽ_{𝔭_{i}}(α_{j}) = δ_{i,j}. Let η_{i,1}, …, η_{i,r_{i}} be a system of generators of O_{𝔭}^{×} for 1 ≤ i ≤ s. Let

M := ( \begin{matrix}ṽ_{𝔭_{1}}(η_{1,1}) & … & v_{𝔭_{s}}(η_{1,1}) \\ ⋮ & ... & ⋮ \\ ṽ_{𝔭_{1}}(η_{1,r_{1}}) & … & v_{𝔭_{s}}(η_{1,r_{1}}) \\ ⋮ & ⋮ & ⋮ \\ ṽ_{𝔭_{1}}(η_{s,1}) & … & v_{𝔭_{s}}(η_{s,1}) \\ ⋮ & ... & ⋮ \\ ṽ_{𝔭_{1}}(η_{s,r_{s}}) & … & v_{𝔭_{s}}(η_{s,r_{s}}) \end{matrix} ) .

Let S = LMR be the ℓ-adic Smith normal form of M with transformation matrices L and R. Application of the left transformation matrix L to the generators η_{1,1}, …, η_{s,r_{s}} yields elements α_{1}, …, α_{s} with the desired properties.

Algorithm 19 (Logarithmic Classgroup).

Input: a number field F and a prime number ℓ

Output: generators g and and a relation matrix H for C̃l_{F}

• Determine a bound ℓ^{m} for the exponent of C̃l_{F} and use it as the precision for the rest of the algorithm. [Algorithm 14]

• Compute generators 𝔞_{1}, …, 𝔞_{t} of Cℓ′ = Cℓ/⟨𝔭_{1}, …, 𝔭_{s}⟩, where 𝔭_{1}, …, 𝔭_{s} are the ideals of F over ℓ.

• Determine 𝔞_{t+1} = (α_{1}), …, 𝔞_{t+s} = (α_{s}) with ṽ_{𝔭_{i}}(α_{j}) = δ_{i,j}.

• Compute generators g := (\overline{𝔟}_{1}, …, \overline{𝔟}_{t+s})^{T} with deg(𝔟_{i}) = 0 [Theorem 16] from 𝔞_{1}, …, 𝔞_{t+s}.

• Compute a relation matrix M between the generators g. [Corollary 18]

• In case II. remove the j-th column from M and the j-th generator from g.

• Compute the ℓ-adic Hermite normal form H of M.

• Return (g, H).

4 Examples

All methods presented here have been implemented in the computer algebra system Magma [C^{+}].

We recomputed the logarithmic class groups from [DS, section 6] with our new algorithm. Our results differ in one example. For the field F = ℚ(i, \sqrt{1173}) and ℓ = 2 we obtain C̃ℓ_{F} ≅ C_{2} × C_{2} × C_{2} instead of C̃ℓ_{F} ≅ C_{2} × C_{2} × C_{2} × C_{2}. As F contains the 4th roots of unity, the 2-rank of the wild kernel of F is 3.

The table contains examples of logarithmic ℓ-class groups C̃ℓ of selected number fields F together with their class groups Cℓ, Galois groups Gal, and the factorization of the ideals (ℓ). χ_{α}(x) denotes the minimal polynomial of α and i denotes a root of x^{2} + 1. The class groups are presented as a list of the orders of their cyclic factors, Cℓ′ = Cℓ/⟨𝔭_{1}, …, 𝔭_{s}⟩, and ℓ^{m} is the bound for the exponent of C̃ℓ as returned by algorithm 14.

The logarithmic 2-class group of ℚ(i, \sqrt{78}) is an example for the fact that the cokernel of θ in the exact sequence in lemma 11 is not trivial in general. Indeed one can show [DuS] that for F = ℚ(i, \sqrt{d}) with d ≠ 2 and d squarefree

Coker(θ) ≅ \{ \begin{matrix}C_{2} & if d ≡ ±2 mod 16, \\ C_{1} & otherwise. \end{matrix}

F  Cℓ  Gal  ℓ (ℓ)  Cℓ′  ℓ^{m} C̃ℓ
ℚ(\sqrt{−521951})  [1024]  S(2)  2 𝔭_{1}𝔭_{2}  [4]  8 [2,4]
ℚ(i, \sqrt{11})  [1]  E(4)  5 𝔭_{1} ⋯ 𝔭_{4} [1]  5 [5]
ℚ(i, \sqrt{78})  [2,2]  E(4)  2 𝔭_{1}^{4}  [2]  2 [1]
ℚ(i, \sqrt{455})  [2,2,10]  E(4)  2 𝔭_{1}^{2}𝔭_{2}^{2}  [2,2]  512 [2,512]
ℚ(i, \sqrt{1173})  [2,2,6]  E(4)  2 𝔭_{1}^{2}  [2,2,2]  2 [2,2,2]
ℚ(i, \sqrt{1227})  [4,4]  E(4) 613 𝔭_{1} ⋯ 𝔭_{4} [4,4]  613 [613]
ℚ(α)  [14]  D(4) 2 𝔭_{1}^{2}𝔭_{2}^{2}  [1]  1 [1]
χ_{α}(x) = x^{4} + 13x^{2} − 12x + 52  3 𝔭_{1}^{2}𝔭_{2}^{2}  [1]  3 [3]
7 𝔭_{1}  [14]  7 [7]
ℚ(\sqrt{1234577}, \sqrt{−3}) [273]  E(4)  2 𝔭_{1}𝔭_{2}  [273]  4 [4,4]
3 𝔭_{1}^{2}  [273]  3 [3]
13 𝔭_{1}𝔭_{2}  [273] 169 [13,13]
ℚ(ζ_{3}, \sqrt{303})  [14]  E(4)  2 𝔭_{1}^{2}  [14]  2 [2]
3 𝔭_{1}^{2}𝔭_{2}^{2}  [1]  9 [9]
7 𝔭_{1} ⋯ 𝔭_{4} [1]  1 [1]
ℚ(β)  [2,6,6]  S(5)  2 𝔭_{1}𝔭_{2}  [2,2,6]  2 [2,2,2]
χ_{β}(x)=x^{5}+2x^{4}+18x^{3}+34x^{2}+17x+3^{10} 3 𝔭_{1} ⋯ 𝔭_{4} [6]  3 [3]
ℚ(ζ_{5}, \sqrt{5029})  [15,150] [2,4]  2 𝔭_{1}𝔭_{2}  [3,150]  4 [2,2]
3 𝔭_{1}𝔭_{2}  [15,150] 3 [3,3]
5 𝔭_{1}𝔭_{2}  [3,150] 25 [5,25]
ℚ(i, \sqrt{11}, \sqrt{−499}) [3,105]  E(8)  5 𝔭_{1} ⋯ 𝔭_{8} [3]  25 [5,5,25]
ℚ(i, \sqrt{11}, γ)  [2,2,2,6] S(3)× 2 𝔭_{1}^{2}  [2,2,2,6] 2 [2,2,2,2]
χ_{γ}(x) = x^{3} + 3x^{2} + 2x + 125 E(4)  3 𝔭_{1}𝔭_{2}  [2,2,2,6] 9 [3,3]
5 𝔭_{1} ⋯ 𝔭_{12} [2]  5 [5,5]

References

[C^{+}] J.J. Canon et al., The computer algebra system Magma, University of Sydney

    (2003), http://magma.maths.usyd.edu.au/magma/.

[C1] H. Cohen, A course in computational algebraic number theory, Springer Verlag, New York, 1993.

[C2] H. Cohen, Advanced topics in computational number theory, Springer Verlag, New York, 2000.

[DS] F. Diaz y Diaz & F. Soriano, Approche algorithmique du groupe des classes logarithmiques, J. Number Theory 76 (1999), 1–15.

[DuS] I. Dubois & F. Soriano, Un nouveau reǵulateur de type Gross, submitted Abh. Hamburg.

[FG] L.J. Federer & B.H. Gross, (with an appendix by W. Sinnot) Regulators and Iwasawa modules, Invent. Math. 62 (1981), 443–457.

[FPR] D. Ford, S. Pauli & X.-F. Roblot, A fast algorithm for polynomial factorization over ℚ_{p}, J. Théor. Nombres Bordeaux 14 (2002), 151–170.

[Gr] G. Gras, Class Field Theory, Springer Monographs in Mathematics (2003).

[Ha] H. Hasse, Number theory, Springer Verlag, Berlin, 1980.

[He] F. Hess, Zur Klassengruppenberechnung in algebraischen Zahlkörpern, Diplomarbeit, TU - Berlin, 1996,

    http://www.math.TU-Berlin.DE/~kant/publications/diplom/hess.ps.gz.

[HPP] F. Hess, S. Pauli & M.E. Pohst, Computing the Multiplicative Group of Residue Class Rings, Mathematics of Computation 72 (2003).

[J1] J.-F. Jaulent, Classes logarithmiques des corps de nombres, J. Théor. Nombres Bordeaux 6 (1994), 301–325.

[J2] J.-F. Jaulent, Théorie ℓ-adique du corps de classes, J. Théor. Nombres Bordeaux 10 (1998), 355–397.

[J3] J.-F. Jaulent, Classes logarithmiques signées des corps de nombres, J. Théor. Nombres Bordeaux 12 (2000), 455–474.

[J4] J.-F. Jaulent, Corrigendum à Classes logarithmiques signées des corps de nombres, J. Théor. Nombres Bordeaux 14 (2002), 1–5.

[J5] J.-F. Jaulent, Classes logarithmiques des corps totalement réels, Acta Arithmetica 103 (2002), 1–7.

[JS1] J.-F. Jaulent & F. Soriano, Sur le noyau sauvage des corps de nombres et le groupe des classes logarithmiques, Math. Z. 238 (2001), 335–354.

[JS2] Jean-François Jaulent & Florence Soriano-Gafiuk, 2-groupe des classes positives d’un corps de nombres et noyau sauvage de la K-thorie, submitted to J. Number Theory.

[Pa] S. Pauli, Factoring polynomials over local fields, J. Symb. Comp. 32 (2001), 533–547.

[PZ] M.E. Pohst and H. Zassenhaus, Algorithmic algebraic number theory, Cambridge University Press, 1989.

[So] F. Soriano, Sur le noyau hilbertien d’un corps de nombres, C. R. Acad. Sci. Paris, t. 330, Série I (2000), 863-866.

Francisco Diaz y Diaz Université Bordeaux I Laboratoire A2X 351 Cours de la Libération 33405 Talence Cedex, France

diaz@math.u-bordeaux.fr

Jean-François Jaulent Université Bordeaux I Institut des Mathématiques de Bordeaux 351 Cours de la Libération 33405 Talence Cedex, France

jaulent@math.u-bordeaux.fr

Florence Soriano-Gafiuk Université de Metz Département de Mathématiques Ile du Saulcy 57000 Metz, France

soriano@poncelet.univ-metz.fr

Sebastian Pauli Technische Universität Berlin Institut für Mathematik - MA 8-1 Straße des 17. Juni 136 10623 Berlin, Germany

pauli@math.tu-berlin.de

Michael E. Pohst Technische Universität Berlin Institut für Mathematik - MA 8-1 Straße des 17. Juni 136 10623 Berlin, Germany

pohst@math.tu-berlin.de