Computation of 2-groups of positive classes of exceptional number fields

0.23 MB·English

by  Jean-François Jaulent

Computation of 2-groups of positive
classes of exceptional number fields∗
Jean-Franc¸ois Jaulent, Sebastian Pauli,
Michael E. Pohst & Florence Soriano–Gafiuk
8
0
0
2

n Abstract. We present an algorithm for computing the 2-group Cℓpos of the posi-
F
a tive divisor classes in case the number field F has exceptional dyadic places. As an
J
application, we compute the2-rank of thewild kernel WK2(F) in K2(F).
9
R´esum´e. Nous d´eveloppons un algorithme pour d´eterminer le 2-groupe CℓFpos des
] classes positives dans le cas ou` le corps de nombres consid´er´e F poss`ede des places
T
pairesexceptionelles. Celadonneenparticulierle2-rangdunoyausauvageWK2(F).
N
.
h 1 Introduction
t
a
m The logarithmicℓ-classgroupCℓ whasintroducedin[10]byJ.-F.Jaulentwho
F
[ used it to study the ℓ-part WK2(F) of the wild kernel in number fields: if F

contains a primitive 2ℓt-th rooftof unity (t>0), there is a naturalisomorphism
1
v µ ⊗ Cℓ ≃WK (F)/WK (F)ℓt,
ℓt Z F 2 2
7
so the ℓ-rank of WK (F) coincides with the ℓ-rank of the logarithmic group
6 2
f
3 CℓF. An algorithm for computing CℓF for Galois extensions F was developed
1 in [4] and later generalized and improved for arbitrary number fields in [3].
1. f Incasetheprimeℓisodd,theassfumptionµℓ ⊂F maybeeasilypassedifone
0 considers the cyclotomic extension F(µ ) and gets back to F via the so-called
ℓ
8 transfer(see[12],[15]and[17]). Howeverforℓ=2theconnectionbetweensym-
0
bols and logarithmic classes is more intricate: in the non-exceptional situation
:
v (i.e. when the cyclotomic Z -extension Fc contains the fourth root of unity i)
2
i
X the 2-rank of WK2(F) still coincides with the 2-rank of CℓF. Even more if the
number field F has no exceptional dyadic place (i.e. if one has i ∈ Fc for any
r q
a q|2), the same result holds if one replace the ordinary lofgarithmic class group
Cℓ byanarrowversionCℓres. The algorithmicaspectofthis istreatedin[11].
F F
Last in [13] the authors pass the difficulty in the remaining case by intro-
fducinganew2-classgroufpsCℓpos,the2-group of positive divisor classes, which
F
satisfies the rank identity: rk Cℓpos =rk WK (F).
2 F 2 2
In this paper we develop an algorithm for computing both Cℓpos and Cℓpos
F F
in case the number field F does contain exceptional dyadic places.
Weconcludewithseveralexamples. Combiningouralgorithmwiththefwork
ofBelabasandGangl[1]onthecomputationofthetamekernelofK weobtain
2
the complete structure of the wild kernel in some cases.
∗Versiondetravaildu03decembre2007
1
2 Positive divisor classes of degree zero
2.1 The group of logarithmic divisor classes of degree zero
Throughoutthis paper the prime number ℓ equals 2 and we let i be a primitive
fourth root of unity. Let F be a number field of degree n = r+2c. According
to [9], for every place p of F there exists a 2-adic valuation v which is related
p
to the wild 2-symbol in case the cyclotomic Z -extension of F contains i. The
2 p
degreedegpofpisa2-adicintegersuchthattheimageoftheemapLog||p isthe
Z -module deg(p)Z (see [10]). (By Log we mean the usual 2-adic logarithm.)
2 2
The construction of the 2-adic logarithmic valuations v yields
p
∀α∈R :=Z ⊗ F× : v (α)deg(p) = 0, (1)
F 2 Z p e
p∈XPlF0
e
where Pl0 denotes the set of finite places of the number field F. Setting
F
div(α) := v (α)p
p
p∈XPlF0
we obtain by Z -linearity: f e
2
deg(div(α)) = 0. (2)
We define the 2-group of logarithmic divisors of degree 0 as the kernel of the
f
degree map deg in the direct sum Dℓ = Z p:
F p∈Pl0 2
F
P
Dℓ := a p∈Dℓ | a deg(p)=0 ;
F p∈Pl0 p F p∈Pl0 p
F F
n o
and the subgroup of prPincipal logarithmic diPvisors as the image of the logarith-
f
mical map div:
Pℓ := {div(α)|α∈R } .
F F
f
Because of (2) Pℓ is clearly a subgroup of Dℓ . More ever by the so-called
F F
f f
extended Gross conjecture, the factorgroup
f f
Cℓ := Dℓ /Pℓ
F F F
is a finite 2-group, the 2-group of logarithmic divisor classes. So, under this
f f f
conjecture, Cℓ is just the torsion subgroup of the group
F
Cℓ :=Dℓ /Pℓ
f F F F
of logarithmic classes (without any asumption of degree).
f
Remark 1. Let F+ be the set of all totally positive elements of F× (i.e. the
subgroup F+ :={x∈F×|x >0 for all real p}. For
p
Pℓ+ := {div(α)|α∈R+ :=Z ⊗ F+}
F F 2 Z
the factor group
f f
Cℓres := Dℓ /Pℓ+ (resp. Cℓres := Dℓ /Pℓ+)
F F F F F F
is the 2-group of narrow logarithmic divisor classes ofthe number field F (resp.
f f f f
the 2-group of narrow logarithmic divisor classes of degree 0) introducedin [16]
and computed in [11].
2
2.2 Signs and places
For a field F we denote by Fc, (respectively Fc[i]) the cyclotomic Z -extension
2
(resp. the maximal cyclotomic pro-2-extension) of F.
We adopt the notations and definitions in this section from [13].
Definition 1 (signed places). Let F be a number field. We say that a non-
complex place p of F is signed if and only if F does not contains the fourth
p
root i. These are the places which do not decompose in the extension F[i]/F.
Wesaythatpislogarithmically signedifandonlyifthecyclotomicZ -extension
2
Fcdoesnotcontaini. ThesearetheplaceswhichdonotdecomposeinFc[i]/Fc.
p
Definition 2 (sets of signed places). By PS, respectively PLS, we denote
the sets of signed, respectively logarithmically signed, places:
PS := {p|i6∈F } ,
p
PLS := {p|i6∈Fc} .
p
A finite place p ∈ PLS is called exceptional. The set of exceptional places is
denoted by PE. Exceptional places are even (i.e. finite places dividing 2).
These sets satisfy the following inclusions:
P⊂PLS =PE∪PR⊂Pl(2)∪Pl(∞)
where Pl(2), Pl(∞), PR denote the sets of even, infinite and real places of F,
respectively. From this the finiteness of PLS is obvious.
We recall the canonical decomposition Q× =2Z×(1+4Z )×h−1i and we
2 2
denote by ǫ the projection from Q× onto h−1i.
2
Definition 3 (sign function). For all places p we define a sign function via
1 for pcomplex
sign(x) for preal
sg : F× →h−1i : x7→ .
p p  ǫǫ((NNKp−p/νQp2(x()x))Np−νp(x)) ffoorr pp6||22∞
These sign functions satisfy the product formula:
∀x∈F× sg(x)=1.
p∈YPlF
In addition we have:
Proposition 1. The places p of F satisfy the following properties:
(i) if p∈PLS then (sg ,v ) is surjective;
p p
(ii) if p∈PS\PLS then esgp( )=(−1)vep( ) and vp is surjective;
(iii) if p6∈PS then sg (F×)=1 and v is surjective.
p p p e
Remark 2. The logarithmic valuation v is surjective in all three cases. Part
p
e
2 of the preceding result is often used for testing p∈PLS.
e
3
2.3 The group of positive divisor classes
For the introduction of that group we modify several notations from [13] in
order to make them suitable for actual computations.
Since PLS is finite we can fix the order of the logarithmically signed places,
say PLS = {p ,··· ,p }, with PE = {p ,··· ,p } and PR = {p ,··· ,p }.
1 m 1 e e+1 m
Accordingly we define vectors e=(e ,··· ,e )∈{±1}m.
1 m
For each divisor a= a p, we form pairs (a,e) and put
p∈Pl0 p
F
P m
sg(a,e) := (−1)ap × e (3)
i
p∈PYS\PLS Yi=1
LetDℓ (PE):= a∈Dℓ a= a p betheZ -submoduleofDℓ gen-
F F p∈PE p 2 F
erated by the exnceptional (cid:12)dyadiPc places. Aond let DℓPFE be the factor group
(cid:12)
Dℓ /Dℓ (PE). Thus the group of positive divisors is the Z -module:
F F 2
Dℓpos := (a,e)∈DℓPE ×{±1}m sg(a,e)=1 (4)
F F
n (cid:12) o
For α∈RF :=Z2⊗ZF×, let div′(α) denotes the i(cid:12)(cid:12)mage of div(α) in DℓPFE and
sg(α) the vector of signs (sg (α),...,sg (α)) in {±1}m. Then
p1 pm
f f
Pℓpos := (div′(α),sg(α))∈DℓPE ×{±1}m α∈R (5)
F F F
n (cid:12) o
is obviouslyfa submodulefof Dℓpos which is called the pri(cid:12)ncipal submodule.
F (cid:12)
Definition 4 (positive divisor classes). With the notations above:
(i) The group of positive logarithmic divisor classes is the factor group
Cℓpos = Dℓpos/Pℓpos .
F F F
(ii) The subgroup of positive logarithmic divifsor classes of degree zero is the
kernel Cℓpos of the degree map deg in Cℓpos:
F F
f CℓFpos :={(a,e)+PℓFpos | deg(a)∈deg(DℓF(PE))}.
Remark 3. ThefgroupCℓpos isinfifnitewheneverthenumberfieldF hasnoex-
F
ceptionalplaces,sinceinthiscasedeg(Cℓpos)isisomorphictoZ . Thefiniteness
F 2
ofCℓpos incasePE 6=∅followsfromthe so-calledgeneralizedGrossconjecture.
F
For the computation of Cℓpos we need to introduce primitive divisors.
F
Definition 5. A divisor b of F is called a primitive divisor if deg(b) generates
f
the Z -module deg(Dℓ )=4[F ∩Qc :Q]Z .
2 F 2
We close this section by presenting a method for exhibiting such a divisor:
Let q ,··· ,q be all dyadic primes; and p ,··· ,p be a finite set of non-
1 s 1 s
dyadic primes which generates the 2-group of 2-ideal-classes Cℓ′ (i.e. the quo-
F
tient of the usual 2-class group by the subgroup generated by ideals above 2).
Then every p ∈ {q ,··· ,q ,p ,··· ,p } with minimal 2-valuation ν (degp)
1 s 1 t 2
is primitive.
4
2.4 Galois interpretations and applications to K-theory
Let Flc be the locally cyclototomic 2-extension of F (i.e. the maximal abelian
pro-2-extensionofF whichiscompletelysplitateveryplaceoverthecyclotomic
Z -extensionFc. Thenbyℓ-adicclassfieldtheory(cf. [9]),onehasthefollowing
2
interpretations of the logarithmic class groups:
Gal(Flc/F)≃Cℓ and Gal(Flc/Fc)≃Cℓ .
F F
Remark 4. Letusassumei∈/ Fc. Thuswemaylistthefollowingspecialcases:
f
(i) In case PLS =∅, the group Cℓpos ≃Z ⊕Cℓpos of positive divisor classes
F 2 F
hasindex2inthegroupCℓ ≃Z ⊕Cℓ oflogarithmicclassesofarbitrary
F 2 F
degree; as a consequence its torsion subgrfoup Cℓpos has index 2 in the
F
finite group Cℓ of logarithmic classfes of degree 0 yet computed in [3].
F
f
(ii) In case PE = ∅, the group Cℓpos ≃ Z ⊕Cℓpos has index 2 in the group
f F 2 F
Cℓres ≃Z ⊕Cℓres of narrow logarithmic classes of arbitrary degree; and
F 2 F
its torsionsubgroupCℓpos has index 2 in tfhe finite groupCℓres of narrow
F F
logarithmic clfasses of degree 0 introduced in [16] and computed in [11].
f f
Definition 6. We adopt the following conventions from [6, 7, 13, 14]:
(i) F is exceptional whenever one has i∈/ Fc (i.e. [Fc[i]:Fc]=2);
(ii) F is logarithmically signed whenever one has i∈/ Flc (i.e. PLS 6=∅);
(iii) F is primitive whenever one at least between the exceptional places does
not split in (the first step of the cyclotomic Z -extension) Fc/F.
2
The following theorem is a consequence of the results in [6, 7, 9, 10, 13, 14]:
Theorem 1. Let WK (F) (resp. K∞(F) :=∩ K2n(F)) be be the 2-part of
2 2 n≥1 2
the wild kernel (resp. the 2-subgroup of infinite height elements) in K (F).
2
(i) In case i∈Flc (i.e. in case PLS =∅), we have both:
rk WK (F) = rk Cℓ = rk Cℓres.
2 2 2 F 2 F
(ii) In case i∈/ Flc but F has no exceptionfal places (if.e. PE =∅), we have:
rk WK (F) = rk Cℓres.
2 2 2 F
(iii) In case PE 6=∅, then we have f
rk WK (F) = rk Cℓpos.
2 2 2 F
And in this last situation there are two subcases:
(a) If F is primitive, i.e. if the set PE of exceptional dyadic places
contains a primitive place, we have:
K∞(F) = WK (F) .
2 2
(b) If F is imprimitive and K∞(F)=⊕n Z/2niZ, we get:
2 i=1
i. WK2(F)=Z/2n1+1Z⊕(⊕ni=2Z/2niZ) if rk2(CℓFpos)=rk2(CℓFpos);
ii. WK2(F)=Z/2Z⊕(⊕ni=1Z/2niZ) if rk2(CℓFpos)<rk2(CℓFpos).
f
f
5
3 Computation of positive divisor classes
We assume in the following that the set PE of exceptional places is not empty.
3.1 Computation of exceptional units
Classicallythe groupoflogarithmicunits is the kernelin R ofthe logarithmic
F
valuations (see [9]):
E ={x∈R |∀p:v (x)=0}
F F p
In order to compute poseitive divisor classes ien case PE is not empty, we ought
to introduce a new group of units:
Definition7. Wedefinethegroupoflogarithmic exceptionalunitsasthekernel
of the non-exceptional logarihtmic valuations:
Eexc ={x∈R |∀p∈/ PE :v (x)=0} (6)
F F p
We just know thaet is a subgroup of the 2-greoup of 2-units EF′ = Z2⊗EF′ .
If we assume that there are exactly s places in F containing 2 we have, say:
E′ = µ ×hε ,··· ,ε i
F F 1 r+c−1+s
For the calculationofEexc we use the sameprecision η as for our2-adicapprox-
F
imations usedinthe courseofthe calculationof Cℓ . Thenweobtaina system
F
of generators of Eexc byecomputing the nullspace of the matrix
F
f
e | 2η ··· 0
B = v (ε ) | · ··· ·
 pi j 
| 0 ··· 2η
 e 
with r+c−1+s+e columns and e rows,where e is the cardinality of PE and
the precision η is determined as explained in [3].
We assume that the nullspace is generated by the columns of the matrix
C
 
B′ =  − − − 
 
 
 
 D 
 
 
 
 
where C has r+c−1+s and D exactly e rows. It suffices to consider C. Each
column (n ,··· ,n )tr of C corresponds to a unit
1 r+c−1+s
r+c−1+s
εni ∈EexcR2η
i F F
i=1
Y
e
6
so that we can choose
r+c−1+s
ε:= εni
i
i=1
Y
asanapproximationforanexceeptionalunit. Thisprocedureyieldsk ≥r+c+e
exceptional units, say: ε ,··· ,ε . By the so-called generalized conjecture of
1 k
Grosswewouldhaveexactlyr+c+esuchunits. Soweassumeinthefollowing
that the procedure does give k = r + c + e (otherwise we would refute the
e e
conjecture). Hence, from now on we may assume that we have determined
exactly r+c+e generators ε ,··· ,ε of Eexc, and we write:
1 r+c+1 F
EeFxc =e h−1i×ehε1,··· ,eεr+c−1+ei
Definition 8. The kerenel of the canoneical maep RF → DℓFpos is the subgroup
of positive logarithmic units:
Epos = {ε∈Eexc |∀p∈PLS sg (ε)=+1}
F F p
The subgroupeEFpos hasefinitee index in the group EeeFxc of exceptional units.
3.2 The algoerithm for computing Cℓpose
F
We assumePE 6=∅ andthatthe logarithmic2-classgroupCℓ is isomorphicto
F
the direct sum
Cℓ ∼= ⊕ν Z/2niZ f
F i=1
subject to 1≤n ≤···≤n . Leta (1≤i≤ν) be fixedrepresentativesofthe
1 ν i
f
ν generating divisor classes. Then any divisor a of Dℓ can be written as
F
ν
a = a a +λb+div(α)
i i
i=1
X
f
with suitable integers a ∈ Z , a primitive divisor b, λ = deg(a) and an appro-
i 2 deg(b)
priate element α of R . With each divisor a we associate a vector
F i
e := (sg(a ,1),1,··· ,1)∈{±1}m ,
i i
where m again denotes the number of divisors in PLS. Clearly, that represen-
tation then satisfies sg(a ,e ) = 1, hence the element (a ,e ) belongs to Dℓpos.
i i i i F
Setting e =(sg(b,1),1,··· ,1) as above and writing
b
ν
e′ := sg(α)× eai ×e×eλ
i b
i=1
Y
for abbreviation any element (a,e) of Dℓpos can then be written in the form
F
ν ν
(a,e) = a a +λb+div(α),e′× eai ×sg(α)×eλ
i i i b
!
i=1 i=1
X Y
ν f
= a (a ,e )+λ(b,e )+(0,e′)+(div(α),sg(α)) .
i i i b
i=1
X
f
7
The multiplications are carried out coordinatewise. The vector e′ is therefore
contained in the Z -module generated by g ∈ Zm (1 ≤ i ≤ m) with g =
2 i 1
(1,··· ,1), whereas g has first and i-th coordinate -1, all other coordinates 1
i
for i>1.
As a consequence, the set
{(a ,e )|1≤j ≤ν}∪{(0,g )|2≤i≤m}∪{(b,e}
j j i
contains a system of generators of Cℓpos ( note that (0,g ) is trivial in Cℓpos).
F 1 F
We still need to expose the relations among those. But the latter are easy to
characterize. We must have
ν m
a (a ,e )+ b (0,g )+λ(b,e ) ≡ 0 mod Pℓpos ,
j j j i i b F
j=1 i=2
X X
ν m f
a (a ,e )+ b (0,g )+λ(b,e ) = (div(α),sg(α))+ (d p,1)
j j j i i b p
j=1 i=2 p∈PE
X X X
f
with indeterminates a ,b ,d from Z . Considering the two components sepa-
j i p 2
rately, we obtain the conditions
ν
a a +λb ≡ d p mod Pℓ (7)
j j p F
j=1 p∈PE
X X
f
and
ν m
eaj × gbi ×eλ = sg(α) . (8)
j i b
j=1 i=2
Y Y
Let us recall that we have already ordered PLS so that exactly the first e ele-
ments p ,··· ,p belong to PE. Then the first one of the conditions above is
1 e
tantamount to
ν e
degp
a a ≡ d p − ib mod Pℓ .
j j pi i degb F
j=1 i=1 (cid:18) (cid:19)
X X
f
The divisors
degp
p − ib
i degb
on the right-hand side can again be expressed by the a . For 1≤i≤e we let
j
ν
degp
div(α ) + p − ib = c a .
i i degb ij j
j=1
X
f
The calculation of the α ,c is described in [15].
i ij
Consequently, the coefficient vectors (a ,··· ,a ,λ) can be chosen as Z -
1 ν 2
linear combinations of the rows of the following matrix A∈Z(ν+e)×(ν+1):
2
8
2n1 0 ··· 0 0 | 0
0 2n2 ··· 0 0 | 0
 
· · ··· · · | ·
 · · ··· · · | · 
 
 0 0 ··· 2nν−1 0 | 0 
 
A =  0 0 ··· 0 2nν | 0 
 
 −− −− −−− −− −− −−− 
 
 | deg(p1) 
 deg(b) 
 . 
 c | .. 
 ij 
 | deg(pe) 
 deg(b) 
 
Eachrow(a ,··· ,a ,λ) ofA correspondsto alinear combinationsatisfying
1 ν
ν
a a +λb ≡ div(α) mod Dℓ (PE) . (9)
j j F
j=1
X
f
Condition (8) gives
m ν
gbi = sg(α)× eaj ×eλ . (10)
i j b
i=2 j=1
Y Y
Obviously, the family (g ) is free over F implying that the exponents
i 2≤i≤m 2
b are uniquely defined. Consequently, if the k-th coordinate of the product
i
sg(α)× ν eaj×eλis−1wemusthaveb =1,otherwiseb =0for2≤k ≤m.
j=1 j b k k
(Wenotethattheproductoverallcoordinatesisalways1.) Therefore,wedenote
Q
by b ,··· ,b the exponents of the relation belonging to the j-th column of
2,j m,j
A for j =1,··· ,ν+e.
Unfortunately,theelementsαareonlygivenuptoexceptionalunits. Hence,
we must additionally consider the signs of the exceptional units of F. For
Eexc = h−1i×hε ,··· ,ε i (11)
F 1 r+c−1+e
we put:
e me e
sg(ε )= gbi,j+v+e . (12)
j i
i=1
Y
Using the notations of (11)eand (12) the rows of the following matrix A′ ∈
Z(ν+e+r+c)×(ν+m) generate all relations for the (a ,e ), (b,e ), (0,g ).
2 j j b i
| b2,1 ··· bm,1
0 1
| · ··· ·
B A | · ··· · C
B C
B | · ··· · C
BB | b2,ν+e ··· bm,ν+e CC
A′ = BB − −−− − | − −−− − CC .
BBB | b2,ν+e+1 ··· bm,ν+e+1 CCC
B | · ··· · C
B C
B O | · ··· · C
B C
B | · ··· · C
B C
@ | b2,ν+e+r+c ··· bm,ν+e+r+c A
9
3.3 The algorithm for computing Cℓpos
F
We assume that PE = {p ,··· ,p } 6= ∅ is ordered by increasing 2-valuations
1 e
v (degp ); that the group Cℓpos of positive diviseor classes is isomorphic to the
2 i F
direct sum
Cℓpos ∼= ⊕w Z/2miZ;
F i=1
andthatweknowafullsetofrepresentatives(b ,f ) (1≤i≤w)forallclasses.
i i
Then each (b,f)∈Dℓpos satisfies deg(b)∈deg(Dℓ (PE)) and
F F
w
f
b ≡ b b mod(Dℓ (PE)+Pℓ ) .
i i F F
i=1
X
f
Obviously, we obtain
w
0 ≡ deg(b) ≡ b deg(b ) mod deg(Dℓ (PE)) .
i i F
i=1
X
We reorder the b if necessary so that
i
v (deg(b )) ≤ v (deg(b )) (2≤i≤w)
2 1 2 i
is fulfilled. We put
t := max(min({v (deg(p))|p∈Dℓ (PE)})−v (deg(b )),0)
2 F 2 1
= max(v (deg(p ))−v (deg(b ),0)
2 1 2 1
and
w
deg(b )
i
δ := b + b .
1 deg(b ) i
1
i=2
X
Then we get:
w
deg(b )
b ≡ b b − i b +δb mod (Dℓ (PE)+Pℓ )
i i deg(b ) 1 1 F F
i=2 (cid:18) 1 (cid:19)
X
f
and so
degb≡0≡ b ×0+δdegb moddegDℓ (PE).
i 1 F
From this it is immediateXthat a full set of representatives of the elements of
Cℓpos is given by
F
f b − deg(bi)b ,f ×f−deg(bi)/deg(b1) for 2≤i≤w
i deg(b ) 1 i 1
(cid:18) 1 (cid:19)
and
degb
(b′ :=2tb −2t 1p ,f2t) .
1 1 degp 1 1
1
Let us denote the class of (c,f) in Cℓpos by [c,f].
F
f
10