A new Algorithm for the Computation of logarithmic l-Class Groups of Number Fields

0.23 MB·English

by  Francisco Diaz Y Diaz

A new Algorithm for the Computation
ℓ
of logarithmic -Class Groups of
8
0
Number Fields
0 ∗
2

n
Francisco Diaz y Diaz, Jean-Franc¸ois Jaulent, Sebastian Pauli,
a
J Michael Pohst, Florence Soriano-Gafiuk

7

]
T
N Abstract. Wepresentanalgorithmforthecomputationoflogarithmicℓ-classgroups
ofnumberfields. Ourprincipalmotivationistheeffectivedeterminationoftheℓ-rank
h. of thewild kernelin the K-theory of numberfields.
t
a R´esum´e. Nous d´eveloppons un nouvel algorithme pour calculer le ℓ-groupe CeℓF des
m
classeslogarithmiquesd’uncorpsdenombresF. Ilpermetenparticulierded´eterminer
[ effectivement le ℓ-rangdu noyausauvage WK2(F) du groupeK2(F).

1
v
1 Introduction
1
2
9 A new invariantofnumber fields,calledgroupoflogarithmicclasses,was intro-
0 ducedbyJ.-F.Jaulentin1994[J1]. Theinterestinthearithmeticoflogarithmic
.
1 classes is because of its applicability in K-Theory. Indeed this new group of
0 classesisrevealedtobemysteriouslyrelatedtothewildkernelintheK-Theory
8 fornumberfields. Thenewapproachtothewildkernelissoattractivesincethe
0
arithmetic of logarithmic classes is very efficient. Thus it provides an algorith-
:
v micandoriginalstudyofthe wildkernel. Afirstalgorithmforthecomputation
i ofthegroupoflogarithmicclassesofanumberfieldF wasdevelopedbyF.Diaz
X
y Diaz and F. Soriano in 1999 [DS]. We present a new much better performing
r
a 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 kernelof F and its logarithmicℓ-classgrouphave the same
ℓ-rank. If F does not contain the 2ℓ-th roots of unity the arithmetic of the
logarithmic classesstill yields the ℓ-rank of the the wild kernel. More precisely:
If ℓ is odd [JS1, So] we consider F′ := F(ζ ), where ζ is the ℓ-th root of
ℓ ℓ
•
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
•
thepositivedivisorclassesandwhichcanbeconstructedfromtheℓ-group
of logarithmic classes.
Inthe presentarticlewe considerthe generalsituationwhereF is a number
field which does not necessarily contain the 2ℓ-th roots of unity.
∗Experimental. Math. 14(2005), 67–76.
1
2 The theoretical background
This section is devoted to the introduction of the main notions of logarithmic
arithmetic. We alsoreviewthe factsthatareofinterestforourpurpose. 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 be the ℓ-adified group of id`eles of F, i.e. the
F
J
restricted product
res
=
F p
J R
Yp
oftheℓ-adiccompactifications = lim F×/F×ℓn ofthemultiplicativegroups
p p p
R
of the completions of F at each p. F←or−each finite place p the subgroup of
p
U
of the cyclotomic norms (that is to say the elements of whichare norms
p p
R R e
at any finite step of the localcyclotomic Z -extensionFc/F ) will be called the
ℓ p p
group of logarithmic units of F . The product
p
=
F p
U U
Yp
e e
isdenominatedthegroup of idelic logarithmic units;ithappenstobethe kernel
of the logarithmic valuations
Log (N (x))
v x ℓ Fp/Qp ,
p
| 7→ − deg p
F
e
defined on the and Z -valued. These are obtained by taking the Iwasawa
p ℓ
R
logarithm of the norm of x in the local extension F /Q with a normalization
p p
factor deg p whose precise definition is given in the next subsection.
F
The quotient ℓ = / is the ℓ-group of logarithmic divisors of F;
F F F
D J U
via the logarithmic valuations v , it may be identified with the free Z -module
ep ℓ
generated by the prime ideals of F
e
ℓ = / = Z p.
F F F p ℓ
D J U ⊕
e
The degree of a logarithmic divisor d= n p is then defined by
p p
P
deg ( n p) = n deg p,
F p p F
Xp Xp
inducing a Z -valued Z -linear map on the class group of logarithmic divisors.
ℓ ℓ
The logarithmic divisors of degree zero form a subgroup of ℓ denoted by
F
D
ℓ = d ℓ deg d = 0 .
D F { ∈D F | F }
TheimageofthemafpdivF definedviathesetoflogarithmicvaluationsfrom
the principal id`ele subgroup
f
= Z F×
F ℓ Z
R ⊗
2
of to ℓ is a subgroup denoted by ℓ , which will be referred to as the
F F F
J D P
subgroup of principal logarithmic divisors. The quotient
f f
ℓ = ℓ / ℓ
F F F
C D P
f f f
is, by definition, the ℓ-group of logarithmic classes of F. And the kernel
=
F F F
E R ∩U
of the morphism div from in ℓ is the groupof globallogarithmic units.
F F e F e
R D
f f
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 extensionof number fields. Let p be a prime number.
Denote by Qc the cyclotomic Z-extensionof Q , that is to say the compositum
p p
of all cyclotomic Z -extension of Q on all prime numbers q. Let p be a prime
b q b p
ofF above(p)andPa primeofL abovep. The logarithmicramification(resp.
inertia) index e(L /F ) (resp. f(L /F )) is defined to be the relative degree
P p P p
e(L /F )=e [L :L QcFe] (resp. f(L /F )=[L QcF :F ]).
P p P P∩ p p P p P∩ p p p
As aeconsequence, L/F is lobgarithmically uneramified at P, i.e.be(L /F ) = 1,
P p
if and only if L is contained in the cyclotomic extension of F . Moreover, for
P p
any q = p the classical and the logarithmic indexes have the saeme q-part (see
6
theorem 5). Hence they are equal as soon as p∤[F :Q ].
p 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 e and f . With
p p
these notations, the ℓ-adic degree of p is defined by the formula:
e e
Log p for p=ℓ;
ℓ 6
deg p=f deg p with deg p=ℓ for p=ℓ=2;
K p ℓ ℓ  6
4 for p=ℓ=2.
e

The extension and norm maps between groups of divisors, denoted by ι
L/F
and N respectively, have their logarithmic counterparts, ι and N
L/F L/F L/F
respectively. Tobe moreexplicit, ι is definedoneveryfinite placepof F by
L/F e e
ι (p)= e P ,
L/F e P|p LP/Fp
P
while NL/F is defined on aell P lying aboveep by
e NL/F(P)=fLP/Fp p .
These applications are compatible with the usual extension and norm maps
e e
defined between and , in the sense that they sit inside the commutative
L F
R R
diagrams:
dfivL ℓ degL Z dfivL ℓ degL Z
L L ℓ L L ℓ
R −−−−→ D −−−−→ R −−−−→ D −−−−→
NL/F fNeL/F and eιL/F feιL/F [L:F]
  (cid:13) x x x
y dfivF yℓ degF Z(cid:13)(cid:13)  dfivF ℓ degF Z.
F F ℓ F F ℓ
R −−−−→ D −−−−→ R −−−−→ D −−−−→
f f
3
WhenL/F isaGaloisextensionwithGaloisgroupGal(L/F),onededucesfrom
the very definitions the unsurprising and obvious properties:
N ι =[L:F] and ι N = σ .
L/F ◦ L/F L/F ◦ L/F σ∈Gal(L/F)
P
e e e e
2.3 Ideal theoretic description of logarithmic classes.
By the weak density theorem every class in / may be represented by
F F F
J U R
an idele with trivial components at the ℓ-adic places, that is to say that every
e
class in ℓ / ℓ comes from a ℓ-divisor d= α p.
D F P F p∤ℓ p
The canonicalmap from to ℓ mapsPa to div (a)= v (a)p.
RF D F ∈RF F p p
Now for each finite place p ∤ ℓ, the quotient e /e = f /f of the claPssical and
p p p pf e
logarithmicalindexesassociatedtopisaℓ-adicunit(theorem5),sayλ (which
is 1 for almost all p), and one has the identitey: e p
v =λ v
p p p
betweenthelogarithmicandtheclassicalvaluations. Soeveryℓ-divisordcomes
e
from a ℓ-ideal a by the formula:
a= p∤ℓpαp 7→dF(a)= p∤ℓλp αp p .
This gives the followiQng ideal theoretic dePscription of logarithmic classes:
Definition & Proposition 1. Let IdF = {a = p∤ℓpαp} be the group of ℓ-
ideals, d = a d deg d (a) = 0 be the sQubgroup of ℓ-ideals of degree
I F { ∈ I F| F F }
0 and PfrF = { p∤ℓpvp(a)|vp(a) = 0 ∀p | ℓ} the subgroup of principal ℓ-ideals
generated by priQncipal ideles a having logarithmic valuations 0 at every ℓ-adic
places.fThen one has: e
ℓ d / r
F F F
C ≃I P
Proof. Asexplainedabove,thesfurjectifvityfofllowsfromtheweakapproximation
theorem. So let us consider the kernel of the canonical map φ : d ℓ .
F F F
I 7→ C
Clearly we have: kerφ = a d a d (a) = div (a) . The
F { ∈ I F | ∃ ∈ RF F Ff } f
condition d (a) = div (a) with a d implies v (a) = 0 p ℓ; and then
F F f∈ I F p ∀f|
gives (a) r as expected.
∈P F f f e
The genefralized Gross conjectures (for the field F and the prime ℓ) asserts
thatthelogarithmicclassgroup ℓ isfinite(cf. [J1]). Thisconjecture,whichis
F
C
a consequence of the p-adic Schanuel conjecture was only provedin the abelian
f
case and a few others (cf. [FG, J5]). Nevertheless, since ℓ is a Z -module
F ℓ
C
of finite type (by the ℓ-adic class field theory), the Gross conjecture just claims
f
the existence of an integer m such that ℓm kills the logarithmicclass 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 ℓ in order to carry on numerical computations:
F
C
Proposition 2. Assumfe the integer m to be large enough such that the loga-
rithmic class group ℓ is annihilated by ℓm. Thus introduce the group:
F
C
d(ℓm) = fa d deg d (a) ℓmdeg ℓ = d dℓm
I F { ∈I F| F F ∈ F D F} I F I F
(ℓm) ℓm (ℓm) (ℓm)
Thus, denfoting r = r d , one has: ℓ d f/ r .
P F P F I F C F ≃I F P F
f f f f f f
4
ℓm
Proof. The hypothesis gives d r and by a straightforwardcalculation:
I F ⊂P F
d(ℓm)/ r(ℓm) = d fdℓm/ fr dℓm d /( d r dℓm)
I F P F I FI P FI ≃I F I F ∩P FI
ℓm
f f fd / r fd = d /fr fℓ . f
≃I F P FI F I F P F ≃C F
f f f f f f
Remark3. Alowerboundformwhichwillberequiredforasufficientprecision
of the p-adic calculations will be given after lemma 12.
3 The algorithms
ThroughoutthissectionafiniteabeliangroupGispresentedbyacolumnvector
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 of M. We note that for every
a G there is a v Zm satisfying a = vTg. If g ,...,g is a basis of G,
1 m
∈ ∈
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 vTg is
an abbreviation for gv1 gvm.
1 ··· 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 (α)=0 for all p∤s
p
∈ |
(cid:8) (cid:9)
the s-units of F.
For this section let F be a fixed number field. We denote the ideal class
group of F by ℓ= ℓ . We also write ℓ for ℓ , ℓ for ℓ , and so on.
F F F
C C C C D D
f f f f
3.1 Computing deg (p) and v (·)
F p
We describe how invariants of the logarithmic objects can be computed. Some
e
of the tools presented here also applied directly in the computation of the lo-
garithmic 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× = pZ F× (1+2pZ )
∈ p ∼ × p × p
denote by a the projection of a to (1+2pZ ). Let F be the completion of F
p p
h i
with respect to p. For α F define
∈
Log N (α)
h (α):= ph Fp/Qp i.
p
[F :Q ] deg p
p p · p
The p-part of the logarithmic ramification index e is [h (F×) : Z ]. For all
p p p p
primes q with q =p the q-part of e is the q-part of the ramification index e of
p p
p. 6 e
e
5
For a proof see [J1].
In section 2.2 we have seen that the degree deg (p) of a place p can be
F
computed as deg (p)=f deg p. From section 2.3 we know that e f = e f .
F p ℓ p p p p
We have
e Log (N (x)) e e
v (x)= ℓ Fp/Qp .
p
− deg (p)
F
Thus we can concentrateeon the computation of e for which we need the com-
p
pletion F of F at p and generators of the unit group F×.
p p
The Round Four Algorithm was originally ceonceived 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
completionsofnumberfields;andthirdly,itcanbeusedfordeterminingintegral
bases of maximal orders.
Let Φ(x) be a monic, squarefree polynomial over Z . The algorithm for
p
factoring polynomials over local fields as described in [Pa] returns
a factorization Φ(x) = Φ (x) Φ (x) of Φ(x) into irreducible factors
1 s
• ·····
Φ (x) (1 i s) over Z ,
i p
≤ ≤
the inertia degrees e and ramification indexes f of the extensions of Q
i i p
•
given by the Φ (x) (1 i s), and
i
≤ ≤
two element certificates (Γ (x),Π (x)) with Γ (x),Π (x) F[x] such that
i i i i
• ∈
v Π (α ) = 1/e and [F (Γ (α )) : F ] = f where α is a root of Φ (x)
i i i i p i i p i i i
in(cid:0)F[x]/(Φ(cid:1)i(x)), vi is an extension of the exponential valuation vp of Qp
to Q [x]/(Φ (x)) with v =v .
p i i|Qp p
Thefactorizationalgorithmin[FPR]returnsthecertificatescombinedinone
polynomial for each irreducible factor. The data returned by these algorithms
can be applied in severalways.
An integral basis of the extension of Q generated by a root α of Φ (x)
p i i
•
is given by the elements Γ (α )hΠ (α )j with 0 h f and 0 j e .
i i i i i i
≤ ≤ ≤ ≤
The localintegralbasescanbe combinedto a globalintegralbasisfor the
extension of Q generated by Φ(x).
For the computationof v we need to compute the normof an element in
p
•
the completions of F. The completions of F are given by the irreducible
factors of the generatingepolynomial of F over Q.
Lemma 6 (Ideal Factorization). Let Φ(x) Z [x] be irreducible over Q. Let
p
∈
Φ (x),...,Φ (x) Z [x] betheirreducible factors of Φ(x)with twoelement cer-
1 s p
∈
tificates (Γ (x),Π (x)). Denote by e the ramification indexes of the extensions
i i i
of Q given by the Φ (x) (1 i s). The Chinese Remainder Theorem gives
p i
≤ ≤
polynomials Θ (x),...,Θ (x) Q [x] with
1 s p
∈
Θ (x) Π (x)modΦ (x)
i i i
≡
Θ (x) 1mod Φ (x).
i ≡ j6=i j
Q
Let L:=Q(α) where α is a root of Φ(x) in C. Then
(p)= p,Θ (α) e1 p,Θ (α) es
1 s
·····
(cid:0) (cid:1) (cid:0) (cid:1)
is a factorization of (p) into prime ideals.
6
In order to compute [h (F×):Z ] it is sufficient to compute the image of a
p p p
set of generators of F×. Algorithms for this task were recently developed with
p
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)
Letp be the prime idealoverthe prime number p in . Lete be the ram-
p p
O
ification index and f the inertia degree of p. We define the set of fundamental
p
levels
:= ν 0<ν < pep,p∤ν
Fe | p−1
(cid:8) (cid:9)
and let ε × such that p= πeε. Furthermore we define the map
p
∈O −
h :a+p ap εa+p.
2
7−→ −
Theorem 8 (Basisof(1+p)). Let ω ,...,ω be a fixed set of representa-
1 f p
∈O
tives of a F -basis of /p. If (p 1)does not divide e or h is an isomorphism,
p p 2
O −
then the elements
1+ω πν where ν ,1 i f
i e
∈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 is not an
2
− |
isomorphism. Choose e and µ such that p does not divide e and such that
0 0 0
e = pµ0−1(p 1)e0. Let ω1,...,ωf p be a fixed set of representatives of a
F -basis of −/p subject to ωpµ0 ε∈ωpOµ0−1 0modp. Choose ω such
p Op 1 − 1 ≡ ∗ ∈ Op
that xp εx ω modp has no solution. Then the group of principal units1+p
∗
− ≡
is generated by
1+ω πpµ0e0 and 1+ω πν where ν , 1 i f.
∗ i e
∈F ≤ ≤
3.2 Computing a bound for the exponent of Cℓ
Let F be a number field and ℓ a prime number. Let Cℓ=CℓF ∼=fId/Pr be the
ℓ-group of logarithmic divisor classes. Let p ,...,p be the ℓ-adic places of F.
1 s f f f f
Wedescribeanalgorithmwhichreturnsanupperboundℓm oftheexponent
of ℓ (see proposition 2). We denote by
C
f ℓ(ℓ) the ℓ group of logarithmic divisor classes of degree zero:
• C
f ℓ(ℓ):= [a] ℓ a= s a p with deg (a)=0
C ∈C | i=1 i i F
(cid:8) P (cid:9)
ℓ′ the ℓ-gfroup of the ℓ-idfeal classes, i.e., the ℓ-part of ℓ/[p ],...,[p ] .
1 s
• C C h i
Remark 10. If (ℓ) = pe where p is a prime ideal of then the group ℓ(ℓ)
K
O C
is trivial.
f
Lemma 11 ([DS]). Let
θ :Cℓ−→Cℓ′, pmpp7−→ p∤ℓp(1/λp)mp.
P Q
The sequence f
0 ℓ(ℓ) ℓ θ ℓ′ Cokerθ 0
−→C −→C −→C −→ −→
is exact. f f
7
Proof. Recall that, if p∤ℓ, v =λ v . Denote by p ,...,p the ℓ-adic places of
p p p 1 s
F. Let
e
s
a= a q=div(α)= v (α)p= λ v (α)p+ v (α)p
q p p p pi i
Xq Xp Xp∤ℓ Xi=1
e f e e
be a principal logarithmic divisor. A representative of the image of a under θ
in terms of ideals is of the form
e
s
a= qvq(α) =(α ) p−vpi(α).
OK × i
qY∤(ℓ) iY=1
This shows that the homomorphism θ is well defined. It follows immediately
that Kerθ = ℓ(ℓ).
C
Lemma 12.fSet ℓm′ =exp ℓ′ and ℓme =exp ℓ(ℓ). Then
C C
ℓm′+mea 0mod ℓ forfall a ℓ.
≡ P ∈D
Proof. It follows from the exact sequefnce in lemmaf11 that for all a ℓ
the congruence ℓm′θ(a) 1 holds in ℓ′. Thus ℓm′a Kerθ = ℓ(ℓ)∈aDnd
ℓm′+mea 0mod ℓ. ≡ C ∈ C f
≡ P f
Lemma 12 sugfgests setting the precision for the computation of ℓ to m:=
C
m′+mℓ-adicdigits. Ifthe idealclassgroup ℓisknownwecaneasilycompute
C f
m′. In order to find m we compute a matrix of relations for ℓ(ℓ).
e C
Lemma 13 (Generaetors and Relations of ℓ(ℓ)). Let p1,..f.,ps be the ℓ-adic
C
places of F. Assume that s > 1. Reorder the p such that v (deg(p )) =
f i ℓ 1
min v (deg(p )). Let γ ,...,γ be a basis of the ℓ-units of F. Then the
1≤i≤s ℓ i 1 r
group ℓ(ℓ) is given by the generators [g ]:= p deg(pi)p (i=2,...,s) with
C i i− deg(p1) 1
relations s v (γ )[g ]=[0]. (cid:2) (cid:3)
f i=2 pi j i
Proof. WPe conseider a logarithmic divisor a = si=1aipi of degree zero over F
that is constructed from the ℓ-adic places. By Pthe choice of p1 and as deg(a)=
deg( s a p ) = s a deg(p ) = 0 the coefficient a is given by the other
i=1 i i i=1 i i 1
s 1Pcoefficients. TPhus the [g ] generate ℓ(ℓ).
i
− C
The relations between the classes of ℓ(ℓ) are of the form s b [g ]= [0].
Cf i=2 i i
That is there exists β ∈RF such that f P
s s
b g = a p =div(β),
i i j j
Xi=2 Xj=1
f
with v (β) = 0 for all q ∤ (ℓ). Thus β is an element of the group of ℓ-units
q
α v (α)=0 of . Hence we obtain the relations given above.
F q F
{ ∈R | } R
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 ,...,p
1 s
of F, and a basis γ ,...,γ of the ℓ-units of F
1 r
Output: an upper bound for the exponent of ℓ
C
f
8
Set ℓm′ exp ℓ′, set m max m′,4 .
• ← C ← { }
If s=1 then return ℓm′. [Remark 10]
•
Repeat
•
Set m m+2
• ←
Set [Lemma 13]
•
v (γ ) ... v (γ )
p2 1 ps 1
A← e ... ... e ... modℓm.
 v (γ ) ... v (γ ) 
 p2 r ps r 
Let H be the Hermiteenormal form oef A modulo ℓm.
•
Until rank(H)=s 1.
• −
Let S =(S ) be the Smith normal form of A modulo ℓm.
i,j i,j
•
Set m max v (S ) , return ℓm′+me.
1≤i≤s−1 ℓ i,i
• ←
(cid:0) (cid:1)
e
Remark 15. Algorithm 14 does not terminate in general if Gross’ conjecture
is false.
3.3 Computing Cℓ
We use the ideal theoretifc description from section 2.3 for the computation of
ℓ= d/ r. Intheprevioussectionwehaveseenhowwecancomputeabound
C ∼I P
for the exponent of ℓ. It is clear that this bound also gives a lower bound for
f f f C
the precision in our calculations.
f
Theorem16(Generatorsof ℓ). Leta ,...,a beabasisoftheidealclassgroup
1 t
C
of F with gcd(a ,ℓ)=1 for all 1 i t. Denote by p ,...,p the ℓ-adic places
i f ≤ ≤ 1 s
of F. Let α ,...,α be elements of with v (α )=δ (i,j =1,...,s) and
1 s RF pi j i,j
gcd((α ),ℓ)=1 for all 1 i s. Set a :=(α ) for 1 i s. For an ideal a
i t+i i
of F denote by a the proj≤ectio≤n of a from epZℓ to ≤ p≤Zℓ. We distinguish
p p∤(ℓ)
two cases: L L
I. If deg (a ) = 0 for all 1 i t+s then set b := a . The group ℓ is
ℓ i ≤ ≤ i i C F
generated by b ,...,b .
1 t+s f
II. Otherwiselet1 j t+ssuchthatv (deg (a ))=min v (deg (a )).
≤ ≤ ℓ ℓ j 1≤i≤t+s ℓ ℓ i
Setb :=a /ad with d degℓ(ai) mod ℓm where ℓm >exp( ℓ). The group
i i j ≡ degℓ(aj) C
ℓF is generated by b1,...,bj−1,bj+1,...,bt+s. f
C
f
Proof. Let a d. There exist γ and a ,...,a Z such that a =
F 1 t ℓ
t aai (γ).∈SeIt g :=v (γ) for 1∈ Ri s. Now ∈
i=1 i · f i pi ≤ ≤
Q
a= si=1eaaii · (γ)· sj=1(αi)−gi · sj=1(αi)gi .
Q (cid:0) Q (cid:1) (cid:0)Q (cid:1)
By the definition of d (Definition and Proposition1) we have
I
a=a= ti=1aaii · (γ)· sj=1(αj)−gj · sj=1(αj)gj
Q (cid:0) Q (cid:1) (cid:0)Q (cid:1)
9
As vpi (γ) sj=1(αj)−gj =0 for i=1,...,s we obtain
(cid:0) Q (cid:1)
e a t aai s (α )gj mod r.
≡ i=1 i · j=1 j P
Q (cid:0)Q (cid:1)
f
Thusallelementsof ℓcanberepresentedbya ,...,a ,a =(α ),...,a =
1 t t+1 1 t+s
C
(α ). For the two cases we obtain:
s f
I. It follows immediately that b ,...,b are generators of ℓ.
1 t+s
C
II. If we have a aa1 aat+s mod r for an ideal a f d then 0 =
≡ 1 · ··· · t+s P ∈ I
deg(a) = t+sa deg (a ), thus a = s a deg (a )/deg (a ). Hence
i=1 i ℓ i − j f i6=j i ℓ i fℓ j
b ,...,b P,b ,...,b are generatorPs of ℓ.
1 j−1 j+1 t+s
C
f
Wecontinuetousethenotationfromtheorem16. Set ℓ′ := ℓ/ p ,...,p .
1 s
C C h i
Remark17. Thedefinitionof ℓ′inthissectionandtheprevioussection,where
C
we considered the ℓ-part of ℓ/ p ,...,p , differ. The definition we chose here
1 s
C h i
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 a ,...,a of the group ℓ′ are of the
1 t
form t aai = (α) with α . There exist integers c ,...,Cc such that
(α) Qi=s1 (iα )ci mod r. Th∈isRyiFeldstherelation t aai 1 s n(α )ci mod
≡ i=1 i P i=1 i ≡ i=1 i
r in Qℓ. WecanderiveallrelationsinvolvingthegQeneratorsaQ+ r fromtheir
P C f i P
relations as generators of the group ℓ′ in this way.
f f C f
The other relations between the generatorsof ℓ are obtained as follows. A
relation between the generators α is of the formC s (α )vi (1)mod r or
iesqufuivlfiallelendtlyifQansi=d1o(αnliy)viif·Qssi=1ppwwiiiii=s p(αri)ncfoiprasl,omi.eefQ.,αii=f∈1 RsiF.pTw≡iheislaasnt(eℓq)Pu-uanliitty.
i=1 i i=1 i
Assume that s pwi =Q(γ) for some γ . As v (Qα) = 0 for all (α) r
tahnedβpijw|e(ℓo)btthaQeinie=qv1iua=iti−onvpveip(jγ(cid:0))Qfosi=r11α≤viii·≤γ∈(cid:1)sR=. F0 mustephjold. By the definiti∈onPfof
Corollary 18 (Relationes of Cℓ). Let (a ,...,a ),(a ) be a basis
1 t i,j i,j∈{1,...,t}
baendasaarbeolavtei.onFmorateraicxho1f Cℓ′k:=fCℓthwp1e,.fi.(cid:0)n.d,pcsi. ,L.e.t.,act+1 =su(cαh1t)h,a..t.,(cid:1)att+sb=ak(,iαs=)
s (α )ck,i. Letγ ,...,≤γ be≤a basis of thek(,2ℓ)-unitsk,osf . Set vQi=:=1 vi (γ )
i=2 i 1 r RF i,j pj i
(Q1 i r,2 j s). Set
≤ ≤ ≤ ≤ e
b ... b c ... c
1,1 1,t 1,2 1,s
− −
 .. .. .. .. .. .. 
. . . . . .
M := bt,1 ... bt,t −ct,2 ... −ct,s .
 0 ... 0 v ... v 
 1,2 1,s 
 .. .. .. .. .. .. 
 . . . . . . 
 
 0 ... 0 v ... v 
 r,2 r,s 
For the two cases we obtain:
10