A new Algorithm for the Computation of logarithmic `-Class Groups of
Number Fields∗
FranciscoDiaz y Diaz, Jean-Fran¸coisJaulent, Sebastian Pauli, MichaelPohst, FlorenceSoriano-Gafiuk
Abstract. We present an algorithm for the computation of logarithmic`-class groups of number fields. Our principal motivation is the effective determination of the`-rank of the wild kernel in theK-theory of number fields.
R´esum´e. Nous d´eveloppons un nouvel algorithme pour calculer le`-groupeC`eF des classes logarithmiques d’un corps de nombresF. Il permet en particulier de d´eterminer effectivement le`-rang du noyau sauvageWK2(F) du groupeK2(F).
1 Introduction
A new invariant of number fields, called group of logarithmic classes, was intro- duced 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 theK-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 algorith- mic and original study of the wild kernel. A first algorithm for the computation of the group of logarithmic classes of a number fieldF 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` be a prime number. If a number fieldF contains the 2`-th roots of unity then the wild kernel ofF and its logarithmic`-class group have the same
`-rank. If F does not contain the 2`-th roots of unity the arithmetic of the logarithmic classes still yields the`-rank of the the wild kernel. More precisely:
• If`is odd [JS1, So] we considerF0 :=F(ζ`), whereζ` is the `-th root of unity, and use classic techniques from the theory of semi-simple algebras.
• If`= 2 [JS2] we introduce a new group, which we call the `-group of of the positive divisor classes and which can be constructed from the`-group of logarithmic classes.
In the present article we consider the general situation whereF is a number field which does not necessarily contain the 2`-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, letJF be the `-adified group of id`elesof F, i.e. the restricted product
JF =
res
Y
p
Rp
of the`-adic compactificationsRp = lim
←− Fp×/Fp×`
n
of the multiplicative groups of the completions of F at eachp. For each finite place p the subgroup Uep of Rp of the cyclotomic norms (that is to say the elements ofRp which are norms at any finite step of the local cyclotomicZ`-extensionFpc/Fp) will be calledthe group of logarithmic units ofFp. The product
UeF = Y
p
Uep
is denominated thegroup of idelic logarithmic units; it happens to be the kernel of thelogarithmic valuations
evp |x 7→ −Log`(NFp/Qp(x)) degFp ,
defined on the Rp and Z`-valued. These are obtained by taking the Iwasawa logarithm of the norm of xin the local extensionFp/Qp with a normalization factor degF pwhose precise definition is given in the next subsection.
The quotient D`F = JF/UeF is the `-group of logarithmic divisors of F; via the logarithmic valuationsevp, it may be identified with the freeZ`-module generated by the prime ideals ofF
D`F = JF/UeF = ⊕p Z`p.
Thedegreeof a logarithmic divisor d=P
pnppis then defined by degF(X
p
npp) = X
p
npdegF p,
inducing aZ`-valuedZ`-linear map on the class group of logarithmic divisors.
The logarithmic divisors of degree zero form a subgroup ofD`F denoted by D`fF = {d∈ D`F|degF d = 0}.
The image of the mapdivfF defined via the set of logarithmic valuations from the principal id`ele subgroup
RF = Z`⊗ZF×
of JF to D`fF is a subgroup denoted by P`fF, which will be referred to as the subgroup ofprincipal logarithmic divisors. The quotient
fC`F = D`fF/P`fF
is, by definition, the`-group of logarithmic classesofF. And the kernel EeF = RF∩UeF
of the morphismdivfF fromRF in D`fF is the group of globallogarithmic units.
2.2 Logarithmic ramification and `-adic-degrees.
Next we review the basic notions of the logarithmic ramification, which mimic, as a rule, the classical ones.
LetL/F be any finite extension of number fields. Letpbe a prime number.
Denote byQbcp the cyclotomicZb-extension ofQp, that is to say the compositum of all cyclotomicZq-extension of Qp on all prime numbersq. Letp be a prime ofF above (p) andPa prime ofLabovep. The logarithmic ramification (resp.
inertia) index ee(LP/Fp) (resp. fe(LP/Fp)) is defined to be the relative degree ee(LP/Fp) = [LP:LP∩QbcpFp] (resp. fe(LP/Fp) = [LP∩QbcpFp:Fp]).
As a consequence,L/F is logarithmically unramified atP, i.e. ee(LP/Fp) = 1, if and only ifLP is contained in the cyclotomic extension ofFp. Moreover, for any q 6=pthe classical and the logarithmic indexes have the sameq-part (see theorem 5). Hence they are equal as soon asp-[Fp:Qp].
As usual, in the special caseL/F =K/Q, the absolute logarithmic indexes of a finite place p of K over the prime pare just denoted byeep andfep. With these notations, the`-adic degree ofp is defined by the formula:
degKp=fepdeg`p with deg`p=
Log`p forp6=`;
` forp=`6= 2;
4 forp=`= 2.
The extension and norm maps between groups of divisors, denoted byιL/F and NL/F respectively, have their logarithmic counterparts, eιL/F and NeL/F respectively. To be more explicit,eιL/F is defined on every finite placepofF by
eιL/F(p) =P
P|peeLP/FpP, whileNeL/F is defined on allP lying abovep by
NeL/F(P) =feLP/Fp p .
These applications are compatible with the usual extension and norm maps defined betweenRL andRF, in the sense that they sit inside the commutative diagrams:
RL −−−−→divfL D`fL −−−−→degL Z` RL −−−−→divfL D`fL −−−−→degL Z`
y
NL/F
yNeL/F
and x
e
ιL/F
x
e
ιL/F
x
[L:F]
RF divfF
−−−−→ D`fF degF
−−−−→ Z` RF divfF
−−−−→ D`fF degF
−−−−→ Z`.
WhenL/F is a Galois extension with Galois group Gal(L/F), one deduces from the very definitions the unsurprising and obvious properties:
NeL/F◦eιL/F = [L:F] and eιL/F◦NeL/F =P
σ∈Gal(L/F)σ.
2.3 Ideal theoretic description of logarithmic classes.
By the weak density theorem every class in JF/UeFRF may be represented by an idele with trivial components at the`-adic places, that is to say that every class in D`F/P`F comes from a`-divisord=P
p-` αp p.
The canonical map fromRF toD`F mapsa∈ RF todivfF(a) =P
pvep(a)p.
Now for each finite placep -`, the quotienteep/ep =fp/fep of the classical and logarithmical indexes associated topis a`-adic unit (theorem 5), sayλp(which is 1 for almost all p), and one has the identity:
evp=λpvp
between the logarithmic and the classical valuations. So every`-divisordcomes from a`-idealaby the formula:
a=Q
p-`pαp 7→dF(a) =P
p-`λp αp p.
This gives the following ideal theoretic description of logarithmic classes:
Definition & Proposition 1. Let IdF ={a =Q
p-`pαp} be the group of `- ideals, IdfF ={a∈ IdF|degFdF(a) = 0} be the subgroup of `-ideals of degree 0 and PfrF ={Q
p-`pvp(a)|evp(a) = 0∀p | `} the subgroup of principal `-ideals generated by principal ideles a having logarithmic valuations 0 at every `-adic places. Then one has:
fC`F 'IdfF/PfrF
Proof. As explained above, the surjectivity follows from the weak approximation theorem. So let us consider the kernel of the canonical mapφF : IdfF 7→fC`F. Clearly we have: kerφF = {a ∈ IdfF | ∃a ∈ RF dF(a) = divfF(a)}. The condition dF(a) = divfF(a) with a ∈ IfdF implies evp(a) = 0 ∀p | `; and then gives (a)∈PrfF as expected.
The generalized Gross conjectures (for the fieldF and the prime`) asserts that the logarithmic class groupfC`F is finite (cf. [J1]). This conjecture, which is a consequence of thep-adic Schanuel conjecture was only proved in the abelian case and a few others (cf. [FG, J5]). Nevertheless, since fC`F is aZ`-module of finite type (by the`-adic class field theory), the Gross conjecture just claims the existence of an integermsuch that`mkills the logarithmic class group. As in numerical situations it is rather easy to compute such an exponentm(when the classical invariants of the number field are known), this give rise to a more suitable description offC`F in order to carry on numerical computations:
Proposition 2. Assume the integer m to be large enough such that the loga- rithmic class group fC`F is annihilated by`m. Thus introduce the group:
Idf(`
m)
F ={a∈ IdF|degFdF(a)∈`mdegFD`F}=IdfF Id`Fm Thus, denotingPfr(`
m)
F =PfrF Idf`
m
F , one has: fC`F 'Idf(`
m) F /Pfr(`
m)
F .
Proof. The hypothesis givesIdf`
m
F ⊂PfrF and by a straightforward calculation:
Idf(`
m) F /Pfr(`
m)
F =IdfFId`m/PfrFId`m 'IdfF/(fIdF∩PfrFId`m) 'IdfF/PrfFIdf`
m
F =IdfF/PfrF 'fC`F.
Remark 3. A lower bound formwhich will be required for a sufficient precision of thep-adic calculations will be given after lemma 12.
3 The algorithms
Throughout this section a finite abelian groupGis presented by a column vector g ∈Gm, whose entries form a system of generators for G, and by a matrix of relations M ∈Zn×m of rank m, such that vTg = 0 for v ∈ Zm if and only if vT is an integral linear combination of the rows ofM. We note that for every a ∈ G there is a v ∈ Zm satisfying a = vTg. If g1, . . . , gm is a basis of G, M is usually a diagonal matrix. Algorithms for calculations with finite abelian groups can be found in [C2]. IfGis a multiplicative abelian group, thenvTg is an abbreviation for gv11· · ·gmvm.
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× |vp(α) = 0 for allp-s thes-units ofF.
For this section let F be a fixed number field. We denote the ideal class group of F byC`=C`F. We also writefC` forfC`F,D`f forD`fF, and so on.
3.1 Computing degF(p) and vep(·)
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 lo- garithmic class group.
Definition & Proposition 5. Let pbe a prime number. Let F be a number field. Let p be a prime ideal ofF overp. Fora∈Q×p ∼=pZ×F×p ×(1 + 2pZp) denote by haithe projection ofa to(1 + 2pZp). LetFp be the completion of F with respect top. Forα∈F define
hp(α) := LogphNFp/Qp(α)i [Fp:Qp]·degpp.
The p-part of the logarithmic ramification index eep is [hp(Fp×) : Zp]. For all primesqwithq6=ptheq-part ofeep is theq-part of the ramification index ep of p.
For a proof see [J1].
In section 2.2 we have seen that the degree degF(p) of a place p can be computed as degF(p) =fepdeg`p. From section 2.3 we know thateepfep=epfp. We have
evp(x) =−Log`(NFp/Qp(x)) degF(p) .
Thus we can concentrate on the computation of eepfor which we need the com- pletionFp ofF at pand generators of the unit groupFp×.
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 fac- toring 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 Zp. The algorithm for factoring polynomials over local fields as described in [Pa] returns
• a factorization Φ(x) = Φ1(x)· · · · ·Φs(x) of Φ(x) into irreducible factors Φi(x) (1≤i≤s) overZp,
• the inertia degreesei and ramification indexes fi of the extensions ofQp
given by the Φi(x) (1≤i≤s), and
• two element certificates (Γi(x),Πi(x)) with Γi(x),Πi(x)∈F[x] such that vi Πi(αi)
= 1/ei and [Fp(Γi(αi)) :Fp] =fi where αi is a root of Φi(x) in F[x]/(Φi(x)), vi is an extension of the exponential valuationvp ofQp
toQp[x]/(Φi(x)) withvi|Qp=vp.
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 ofQp generated by a rootαi of Φi(x) is given by the elements Γi(αi)hΠi(αi)j with 0≤h≤fi and 0≤j≤ei. The local integral bases can be combined to a global integral basis for the extension ofQgenerated by Φ(x).
• For the computation ofevp we need to compute the norm of an element in the completions ofF. The completions of F are given by the irreducible factors of the generating polynomial ofF overQ.
Lemma 6 (Ideal Factorization). LetΦ(x)∈Zp[x]be irreducible overQ. Let Φ1(x), . . . ,Φs(x)∈Zp[x]be the irreducible factors ofΦ(x)with two element cer- tificates(Γi(x),Πi(x)). Denote by ei the ramification indexes of the extensions of Qp given by the Φi(x) (1≤i ≤s). The Chinese Remainder Theorem gives polynomials Θ1(x), . . . ,Θs(x)∈Qp[x]with
Θi(x) ≡ Πi(x) mod Φi(x) Θi(x) ≡ 1 modQ
j6=iΦj(x).
Let L:=Q(α)where αis a root ofΦ(x)inC. Then (p) = p,Θ1(α)e1
· · · · · p,Θs(α)es
is a factorization of(p) into prime ideals.
In order to compute [hp(Fp×) :Zp] it is sufficient to compute the image of a set of generators of Fp×. 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.
Fp×∼=πZ×(Op/p)××(1 +p)
Letpbe the prime ideal over the prime numberpinOp. Letep be the ram- ification index andfp the inertia degree ofp. We define the set of fundamental levels
Fe:=
ν |0< ν < p−1pep, p-ν
and letε∈ O×p such that p=−πeε. Furthermore we define the map h2:a+p7−→ap−εa+p.
Theorem 8 (Basis of(1+p)). Letω1, . . . , ωf ∈ Opbe a fixed set of representa- tives of aFp-basis ofOp/p. If(p−1)does not divideeorh2 is an isomorphism, then the elements
1 +ωiπν whereν ∈ Fe,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 andh2 is not an isomorphism. Choosee0andµ0such thatpdoes not dividee0 and such that e =pµ0−1(p−1)e0. Let ω1, . . . , ωf ∈ Op be a fixed set of representatives of a Fp-basis of Op/p subject to ω1pµ0 −εωp1µ0−1 ≡0 modp. Choose ω∗ ∈ Op such thatxp−εx≡ω∗modphas no solution. Then the group of principal units1 +p is generated by
1 +ω∗πpµ0e0 and1 +ωiπν where ν∈ Fe,1≤i≤f.
3.2 Computing a bound for the exponent of C`f
LetF be a number field and`a prime number. LetfC`=fC`F ∼=Ifd/Prf be the
`-group of logarithmic divisor classes. Let p1, . . . ,ps be the`-adic places ofF.
We describe an algorithm which returns an upper bound`mof the exponent offC`(see proposition 2). We denote by
• fC`(`) the`group of logarithmic divisor classes of degree zero:
fC`(`) :=
[a]∈fC`|a=Ps
i=1aipi with degF(a) = 0
• C`0 the`-group of the`-ideal classes,i.e., the`-part ofC`/h[p1], . . . ,[ps]i.
Remark 10. If (`) =pe wherep is a prime ideal of OK then the groupfC`(`) is trivial.
Lemma 11 ([DS]). Let
θ:fC`−→ C`0, P
pmpp7−→Q
p-`p(1/λp)mp. The sequence
0−→fC`(`)−→fC`−→ C`θ 0 −→Cokerθ−→0 is exact.
Proof. Recall that, ifp-`,evp=λpvp. Denote byp1, . . . ,psthe`-adic places of F. Let
ea=X
q
aqq=div(α) =f X
p
evp(α)p=X
p-`
λpvp(α)p+
s
X
i=1
evpi(α)pi
be a principal logarithmic divisor. A representative of the image of ea under θ in terms of ideals is of the form
a= Y
q-(`)
qvq(α)= (αOK)×
s
Y
i=1
p−vi pi(α).
This shows that the homomorphism θ is well defined. It follows immediately that Kerθ=fC`(`).
Lemma 12. Set`m0 = expC`0 and`me = expfC`(`). Then
`m0+mea≡0 modPf` for alla∈D`.f
Proof. It follows from the exact sequence in lemma 11 that for all a ∈ D`f the congruence `m0θ(a) ≡ 1 holds in C`0. Thus `m0a ∈ Kerθ = fC`(`) and
`m0+mea≡0 modPf`.
Lemma 12 suggests setting the precision for the computation offC`to m:=
m0+m `-adic digits. If the ideal class groupe C`is known we can easily compute m0. In order to findme we compute a matrix of relations for fC`(`).
Lemma 13 (Generators and Relations of fC`(`)). Let p1, . . . ,ps be the `- adic places of F. Assume that s >1. Reorder the pi such that v`(deg(p1)) = min1≤i≤sv`(deg(pi)). Let γ1, . . . , γr be a basis of the `-units of F. Then the groupfC`(`)is given by the generators[gi] :=
pi−deg(pdeg(pi)
1)p1
(i= 2, . . . , s)with relationsPs
i=2vepi(γj)[gi] = [0].
Proof. We consider a logarithmic divisor a =Ps
i=1aipi of degree zero over F that is constructed from the`-adic places. By the choice ofp1 and as deg(a) = deg(Ps
i=1aipi) = Ps
i=1aideg(pi) = 0 the coefficient a1 is given by the other s−1 coefficients. Thus the [gi] generatefC`(`).
The relations between the classes offC`(`) are of the formPs
i=2bi[gi] = [0].
That is there existsβ∈ RF such that
s
X
i=2
bigi=
s
X
j=1
ajpj=div(βf ),
with vq(β) = 0 for all q - (`). Thus β is an element of the group of `-units {α∈ RF |vq(α) = 0} ofRF. Hence we obtain the relations given above.
A version of this lemma for the case thatF is Galois can be found in [DS].
Algorithm 14 (Precision).
Input: a number fieldF and a prime number`, the`-adic placesp1, . . . ,ps
ofF, and a basisγ1, . . . , γrof the`-units of F Output: an upper bound for the exponent offC`
• Set`m0 ←expC`0, set m←max{m0,4}.
• Ifs= 1 then return`m0. [Remark 10]
• Repeat
• Setm←m+ 2
• Set [Lemma 13]
A←
evp2(γ1) . . . evps(γ1) ... . .. ... evp2(γr) . . . evps(γr)
mod`m.
• LetH be the Hermite normal form ofAmodulo`m.
• Until rank(H) =s−1.
• LetS= (Si,j)i,j be the Smith normal form ofAmodulo`m.
• Setme ←max1≤i≤s−1 v`(Si,i)
, return`m0+me.
Remark 15. Algorithm 14 does not terminate in general if Gross’ conjecture is false.
3.3 Computing C`f
We use the ideal theoretic description from section 2.3 for the computation of fC`∼=Id/f Pr. In the previous section we have seen how we can compute a boundf for the exponent offC`. It is clear that this bound also gives a lower bound for the precision in our calculations.
Theorem 16 (Generators of fC`). Leta1, . . . ,at be a basis of the ideal class- group of F withgcd(ai, `) = 1for all 1≤i≤t. Denote byp1, . . . ,ps the`-adic places ofF. Letα1, . . . , αsbe elements ofRF withevpi(αj) =δi,j(i, j= 1, . . . , s) and gcd((αi), `) = 1 for all 1 ≤ i ≤ s. Set at+i := (αi) for 1 ≤ i ≤ s. For an ideal a of F denote by a the projection ofa fromL
ppZ` toL
p-(`)pZ`. We distinguish two cases:
I. If deg`(ai) = 0 for all 1 ≤i ≤t+sthen set bi :=ai. The groupfC`F is generated byb1, . . . ,bt+s.
II. Otherwise let1≤j≤t+ssuch thatv`(deg`(aj)) = min1≤i≤t+sv`(deg`(ai)).
Setbi:=ai/adj withd≡ degdeg`(ai)
`(aj) mod`mwhere`m>exp(fC`). The group fC`F is generated by b1, . . . ,bj−1,bj+1, . . . ,bt+s.
Proof. Let a ∈ Id. There existf γ ∈ RF and a1, . . . , at ∈ Z` such that a = Qt
i=1aaii·(γ). Setgi:=evpi(γ) for 1≤i≤s. Now a=Qs
i=1aaii· (γ)·Qs
j=1(αi)−gi
· Qs
j=1(αi)gi .
By the definition ofId(Definition and Proposition 1) we have a=a=Qt
i=1aaii· (γ)·Qs
j=1(αj)−gj
· Qs
j=1(αj)gj
