MATHEMATICS OF COMPUTATION
Volume 67, Number 223, July 1998, Pages 1225–1245
S 0025-5718(98)00939-9
MINUS CLASS GROUPS OF THE FIELDS
OF THE l-TH ROOTS OF UNITY
RENÉ SCHOOF
Abstract. We show that for any prime number l > 2 the minus class group
of the field of the l-th roots of unity Qp (ζl ) admits a finite free resolution
b
of length 1 as a module over the ring Z[G]/(1
+ ι). Here ι denotes complex
conjugation in G = Gal(Qp (ζl )/Qp ) ∼
= (Z/lZ)∗ . Moreover, for the primes
l ≤ 509 we show that the minus class group is cyclic as a module over this
ring. For these primes we also determine the structure of the minus class
group.
Introduction
Let l be an odd prime and let ζl denote a primitive l-th root of unity. In this
paper we study the cyclotomic fields Q(ζl ) and the class groups Cll of their rings of
integers Z[ζl ]. The class group Cll splits in a natural way into two parts: the natural
map from the class group Cll+ of the ring of integers of the subfield Q(ζl + ζl−1 ) to
Cll is injective [24, p.40]. Its cokernel, the minus class group of Q(ζl ), is denoted
by Cll− . There is an exact sequence
0 −→ Cll+ −→ Cll −→ Cll− −→ 0.
About the groups Cll+ little is known. For small primes l they are trivial [23].
See [3], [21] for a numerical study of these groups. In this paper we consider the
other groups, the minus class groups Cll− , which are easier to handle. There is, first
of all, an explicit and easily computable formula for their cardinalities h−
l . See [24,
p.42]:
Y 1
h−
− B1,χ ,
l = 2l
2
χ odd
where the product runs over the characters χ : (Z/lZ)∗ −→ C∗ which are odd, i.e.
which satisfy χ(−1) = −1. The numbers B1,χ are generalized Bernoulli numbers;
they are defined in section 1.
Around 1850, E. E. Kummer [9], [10] used this formula to compute the minus
class numbers h−
l for the primes l < 100. These calculations were extended by
D. H. Lehmer and J. M. Masley [15] in 1978 to the primes l ≤ 509. The numbers
−
h−
l grow very rapidly with l. For instance, h491 already has 138 decimal digits.
−
The class number hl alone does, of course, not determine the structure of the
−
−
group Cll− . If h−
l is squarefree, the group Cll is cyclic, but in general hl has
Received by the editor March 28, 1994 and, in revised form, December 2, 1996.
1991 Mathematics Subject Classification. Primary 11R18, 11R29, 11R34.
Key words and phrases. Cyclotomic fields, class groups, cohomology of groups.
c
1998
American Mathematical Society
1225
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1226
RENÉ SCHOOF
multiple factors. It is a natural problem to try and determine the structure of the
minus class groups. Kummer [12] addressed this problem in 1853. He showed, for
instance, that for l = 29 the minus class group is isomorphic to Z/2Z×Z/2Z×Z/2Z.
He claimed moreover that the minus class group of Q(ζ31 ) is cyclic of order 9. Only
in 1870 he gave a rigorous proof of this fact [11]. It involves a lenghty calculation
−
is cyclic of order 72 · 79241 is
in the field Q(ζ31 ). His claim that the group Cl71
correct, but has, as far as I know, never been justified previously [6].
In this paper we study the structure of the minus class groups Cll− as Galois
modules. Since complex conjugation ι acts as −1 on Cll− , it is natural to study Cll−
b
b denotes the profinite ring lim Z/nZ
as a module over the ring Z[G]/(1
+ ι) where Z
←
and G = Gal(Q(ζl )/Q) ∼
= (Z/lZ)∗ . We prove the following:
Theorem I. Let l be an odd prime.
b
Z[G]/(1
+ ι)-modules
Then there exist an exact sequence of
0 −→ L −→ L −→ Cll− −→ 0
Θ
b
where L is free of finite rank over Z[G]/(1
+ ι).
Theorem I is an immediate consequence of Theorems 2.2(i) and 3.2(i). For small
l we can be more precise:
Theorem II. For l ≤ 509 one can take L of rank 1 in Theorem I. In other words,
b
b
the minus class group is isomorphic to Z[G]/(1
+ ι, Θ) as a Z[G]/(1
+ ι)-module.
Moreover, for Θ one can take the modified Stickelberger element introduced in section 1.
Theorem II is proved in section 4. In the course of the proof we determine
completely the structure of the minus class groups Cll− as abelian groups for l ≤ 509.
−
, which we show to be isomorphic to a product of
As an example we mention Cl491
six cyclic groups:
Z/2Z × Z/2Z × Z/2Z × Z/982Z × Z/10802Z × Z/18680189262665824155664817/
/205804054998786681161963704417938182602575815795883211941228272982586/
/25221939971178506931727800584004906Z.
Theorem II probably holds for several other primes l, but is definitely not true in
general. It does, for instance, not hold for l = 3299. This follows from the fact
b
cyclic over Z[G]/(1
+ ι)
that, when l ≡ 3 (mod 4), the minus class group Cll− is √
if and only if the class group of the √
quadratic subfield Q( −l) ⊂ Q(ζl ) is a cyclic
group. Since the class group of Q( −3299) is isomorphic to Z/3Z × Z/9Z, the
−
b
is not cyclic as a Z[G]/(1
+ ι)-module [13, p.80].
group Cl3299
Finally, we single out a particularly simple consequence of our results. Roughly
speaking, it says that for prime divisors p of l − 1, the p-part of Cll− is cyclic
whenever it is small.
Theorem III. Let l and p be odd primes and let M denote the p-part of the minus
class group of Q(ζl ). If #M divides (l − 1)2 , then M is a cyclic group.
Theorem III is proved in section 2. Applying it with l = 31, p = 3 and l =
71, p = 7 respectively we obtain a proof of Kummer’s claims. The condition that
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MINUS CLASS GROUPS OF THE FIELDS OF THE l-TH ROOTS OF UNITY
1227
#M divide (l − 1)2 cannot be relaxed further: in section 4 we show that the 5-part
of the minus class group of Q(ζ101 ) is isomorphic to Z/125Z × Z/25Z.
Our method is, in some sense, a finite version of Iwasawa theory. It is closely
related to V. A. Kolyvagin’s work [7]. In order to obtain information about the
structure of a certain χ-eigenspace of the p-part of a minus class group, we “deform”
the Dirichlet character χ and study the extension L corresponding to χψ, where
ψ is some character of p-power order. The generalized Bernoulli numbers B1,χψ
contain information about the χ-eigenspace of the class group of this extension.
This information is obtained by viewing the field L as a “truncated” Zp -extension
and by studying the χ-part of the minus class group of L by mimicking techniques
from Iwasawa theory. The main results are Theorem III and the two criteria for
cyclicity, Theorems 2.3 and 3.3.
The main difficulty in extending Theorem II to primes l > 509 is the size of
the class numbers. For larger l one is bound to encounter composite numbers that
cannot be factored within reasonable time. Sooner or later one will also encounter
χ-parts that are not cyclic Galois modules. In these cases the methods of this paper
do not apply.
The paper is organized as follows. In section 1 we briefly recall some well known
facts concerning Z[G]-modules when G is a finite abelian group. In this section we
also discuss some elementary properties of Stickelberger elements and generalized
Bernoulli numbers. Even though there are similarities between the structure of the
odd and even parts of the minus class groups, the differences are sufficiently big
to merit separate treatment. In section 2 we consider the p-parts of minus class
groups for odd primes p. In section 3 we do the same for p = 2. Finally, in section
4, we present the numerical results and prove Theorem II.
We need to know the complete prime decomposition of the class numbers h−
l
for l ≤ 509. In the appendix a table of the prime factorizations of these numbers
is given. This table is complete and supersedes the one computed by Lehmer and
Masley [15]. The present table contains also the factorizations of the unfactored
composite numbers in their table. I thank Arjen Lenstra, Peter Montgomery, Bob
Silverman and Herman te Riele for computing the unknown prime factors, François
Morain for several primality proofs and Pietro Cornacchia for catching an error in
Table 4.4.
1. Preliminaries
In this section we recall some elementary facts concerning modules over group
rings Z[G] when G is a finite abelian group. In addition we recall some basic
properties of Stickelberger elements and generalized Bernoulli numbers.
Let G be a finite abelian group. For a G-module M , we denote by M G the
subgroup of G-invariant elements of M . Now fix a prime p and let
G∼
= π × ∆,
where π is the p-part of G and ∆ is the maximal subgroup of G of order prime to
p. We write the group ring Zp [G] as Zp [∆][π]. By the orthogonality relations there
is an isomorphism of rings
Y
Oχ .
Zp [∆] ∼
=
χ
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1228
RENÉ SCHOOF
∗
Here χ runs over the characters χ : ∆ −→ Qp up to Gal(Qp /Qp )-conjugacy. The
rings Oχ are unramified extensions of Zp generated by the values of χ. They are
Zp [∆]-algebras via the rule σ · x = χ(σ)x for x ∈ Oχ and σ ∈ ∆. The ring
isomorphism is given by mapping σ ∈ ∆ to χ(σ) in each component Oχ . The
residue field of Oχ is Fp (ζd ) where d is the order of χ.
∗
Definition. Let M be a Zp [G]-module and let χ : ∆ −→ Qp be a character.
Equivalently, χ is a character of G of order prime to p. The χ-eigenspace M (χ) or
χ-part of M is defined by
M (χ) = M ⊗Zp [∆] Oχ .
We have a decomposition into eigenspaces of M :
Y
M (χ),
M∼
=
χ
∗
where χ runs over the characters χ : ∆ −→ Qp up to Gal(Qp /Qp )-conjugacy. Each
eigenspace M (χ) is a module over the local ring Oχ [π]. The residue field of this
ring is equal to the residue field of Oχ which is Fp (ζd ), where d is the order of χ.
We frequently use the following properties of the Tate cohomology groups [2]. Let
M be a G-module and let P ⊂ π. The natural action of P on the Tate cohomology
b q (P, M ) is trivial, but ∆ acts, in general, in a non-trivial way. Note that
groups H
b q (P, M ) are Zp [∆]-modules, because they are killed by #P .
the groups H
Lemma 1.1. Let p be a prime and let G be a finite abelian group. Let π and ∆ be
as above and let P be a subgroup of π.
b q (P, M ∆ ) ∼
b q (P, M )∆ for all
(i) For every Z[G]-module M we have that H
= H
q ∈ Z.
∗
(ii) For every Zp [G]-module M and every character χ : ∆ −→ Qp we have that
b q (P, M )(χ)
b q (P, M (χ)) ∼
H
=H
for all q ∈ Z.
Proof. (i) Since the actions of ∆ and P commute, the inclusion i : M ∆ ,→ M and
the ∆-norm map N : M → M ∆ are P -morphisms. The maps i · N and N · i induce
b q (P, M ∆ ) respectively. Since #∆ and
b q (P, M )∆ and H
multiplication by #∆ on H
#P are coprime, multiplication by #∆ is an isomorphism and (i) follows.
(ii) Since the actions of ∆ and P commute, the eigenspaces M (χ) are P -modules.
∗
Taking the sum over the characters χ : ∆ −→ Qp , up to Gal(Qp /Qp )-conjugacy,
b q (P, M (χ)) −→ H
b q (P, M )(χ), we obtain precisely the map
of the natural maps H
L
L bq
q
b
χ H (P, M (χ)) −→ H (P, M ) induced by the isomorphism
χ M (χ) −→ M .
This proves (ii).
The remainder of this section is devoted to properties of Stickelberger elements
and generalized Bernoulli numbers. Let f 6≡ 2 (mod 4) be a conductor and let
G = (Z/f Z)∗ . The Stickelberger element θf of conductor f is given by
f
X
1
a
−
[a]−1 ∈ Q[G].
θf =
f
2
a=1
gcd(a,f )=1
For any prime number p we write G = π × ∆ as above. We have Qp [G] ∼
=
L
∗
K
[π]
where
the
sum
runs
over
the
characters
χ
:
∆
−→
Q
up
to
Gal(Q
/Q
χ
p
p
p )χ
conjugacy and Kχ is the quotient field of Oχ . We denote the algebra homomorphism
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MINUS CLASS GROUPS OF THE FIELDS OF THE l-TH ROOTS OF UNITY
1229
Qp [G] −→ Kχ [π] induced by χ again by χ. For every character χ 6= ω, the image
∗
1
1
∗
2 χ(θf ) of 2 θf in Kχ [π] is an element of the subring Oχ [π]. Here ω : (Z/pZ) −→ Qp
denotes the Teichmüller character. It is the character that gives the action of
Gal(Q/Q) on the group µp of p-th roots of unity. Note that ω = 1 when p = 2. For
odd p the element 12 θf annihilates the χ-part of the p-part of the ideal class group
of Q(ζf ). This is Stickelberger’s Theorem [24, Chpt.6]. For p = 2, C. Greither [4]
has shown the same when π is cyclic and the conductor f is odd.
For any character ϕ of G of conductor f , the generalized Bernoulli number B1,ϕ
is simply the value of the algebra homomorphism Qp [G] −→ Qp induced by ϕ
evaluated on the Stickelberger element:
B1,ϕ = ϕ(θf ) =
f
X
a=1
gcd(a,f )=1
a 1
−
f
2
ϕ(a)−1 ∈ Qp .
Finally we assume that f = l is prime, so that G = (Z/lZ)∗ and we introduce the
b
modified Stickelberger element Θ ∈ Z[G]/(1
+ ι) that occurs in Theorem II. We
Q
∼
b
have that Z[G]/(1 + ι) = p Zp [G]/(1 + ι). Moreover, each factor Zp [G]/(1 + ι) is
Q
isomorphic to χ Oχ [πp ], where the χ run over all odd characters of order prime
to p when p is odd and all characters of odd order when p = 2 respectively. Here πp
denotes the p-part of G. Therefore it suffices to describe the various components
χ(Θ) of Θ: if p = l and χ = ω or if p = 2 and χ = 1, we let χ(Θ) = 1. In all other
cases χ(Θ) = 12 χ(θl ).
b
The modified Stickelberger element Θ ∈ Z[G](1
+ ι) annihilates Cll− . The order
b
of Z[G](1
+ ι, Θ) is equal to the minus class number h−
l .
2. Odd primes p
In this section we study the p-parts of the minus class groups of complex abelian
number fields for odd primes p. We show that certain eigenspaces of these groups
are cohomologically trivial Galois modules. This puts restraints on their structure.
We derive an easily applicable criterion for these eigenspaces to be cyclic Galois
modules.
In this section p 6= 2 is a prime. We fix a complex abelian number K field with
G = Gal(K/Q). Let π denote the p-part of G and F = K π its fixed field. We
∗
fix an odd character χ : G −→ Qp of order prime to p, which is not equal to the
−
(χ) = ClK (χ). Therefore
Teichmüller character ω . Since p 6= 2, we have that ClK
we work, in this section, with the class group ClK itself rather than the minus class
−
group ClK
.
Theorem 2.1. Let P ⊂ G be a subgroup of π with fixed field E = K P . Suppose
that that for all primes r that are ramified in E ⊂ K we have that χ(r) 6= 1. Then
(i) the eigenspace ClK (χ) is a cohomologically trivial Oχ [P ]-module;
(ii) the natural map ClE (χ) −→ ClK (χ)P is bijective and the norm map ClK (χ)
−→ ClE (χ) is surjective.
b q (P, ClK (χ)) = 0 for all q ∈ Z. Let OK denote
Proof. (i) It suffices to show that H
the ring of integers of K, let CK denote the idèle class group of K and let UK denote
the group of unit idèles, i.e. the group of K-idèles that have trivial valuation at all
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1230
RENÉ SCHOOF
finite primes. We have the exact sequence of G-modules [2]
∗
0 −→ OK
−→ UK −→ CK −→ ClK −→ 0.
We show that the χ-parts of the Tate P -cohomology groups of these modules are
∗
we have the following exact sequence [24, p.39]
all zero. For the unit group OK
∗
∗
0 −→ {1, −1} −→ µK × OK
+ −→ OK −→ Q −→ 0.
Here OK + is the ring of integers of the maximal real subfield K + of K and µK
denotes the group of roots of unity in K. The group Q has order at most 2.
∗
Complex conjugation acts trivially on {1, −1}, on Q and on OK
+ . Since χ is an
q
∗
q
∼
b
b
odd character, we have, by Lemma 1.1, that H (P, OK )(χ) = H (K, µK )(χ) for all
q ∈ Z. Since χ is not the Teichmüller character, the χ-part of µK is zero so that,
b q (P, O∗ )(χ) = 0 for all q ∈ Z.
by Lemma 1.1, H
K
b q (P, CK ) ∼
By global class field theory there are natural isomorphisms H
=
b q−2 (P, Z) for all q ∈ Z. Since G acts trivially on Z, it follows from Lemma 1.1
H
b q (P, CK )(χ) = 0 for all q ∈ Z.
that H
We use local class field theory to compute the cohomology of UK . See also [20].
By Shapiro’s lemma we have
M
MM
∗
∗
b q (Pr , Ow
b q (Pr , Ow
b q (P, UK ) ∼
)=
)
H
H
H
=
v
r
v|r
where v runs over the prime ideals of E and r runs over ordinary prime numbers.
The ring Ow is the ring of integers of the completion Kw of K at a prime w of
K over v. We have Qr ⊂ Ev ⊂ Kw with Galois groups Gr = Gal(Kw /Qr ),
Pr = Gal(Kw /Ev ) and Hr = Gal(Ev /Qr ). Since G is abelian, the decomposition
b q (Pr , O∗ ) vanishes when v
groups Pr and Hr only depend on the prime r. Since H
w
is unramified in K, it suffices to consider only primes r that are ramified in E ⊂ K.
For each prime ideal v of F dividing a ramified prime r, there is an exact sequence
of Gr -modules
∗
∗
0 −→ Ow
−→ Kw
−→ Z −→ 0.
Consider the long exact sequence of Tate Pr -cohomology groups. By Lemma 1.1,
b q (Pr , Z). By local class field
the group Hr acts trivially on the cohomology groups H
q
∗ ∼ b q−2
b
(Pr , Z) for all q ∈ Z, so
theory there are natural isomorphisms H (Pr , Kw ) = H
b q (Pr , K ∗ ). Let ∆r denote the maximal
that Hr also acts trivially on the groups H
w
subgroup of Hr of order prime to p. Then ∆r and Pr have coprime orders, so
that the long cohomology sequence remains exact when we take ∆r -invariants. It
b q (Pr , O∗ ) is ∆r -invariant. Therefore ∆r acts trivially on the sum
follows that H
w
L
q
∗
b
(P
,
O
).
Since
χ(r) 6= 1 for all ramified primes r, we see that ∆r 6⊂ ker(χ).
H
r
w
v|r
L
∗
b q (Pr , Ow
) is zero.
H
This implies that the χ-part of
v|r
b q (G, UK )(χ) = 0 for all q ∈ Z. Combining all this and using
It follows that H
b q (P, ClK (χ)) = 0 for all q ∈ Z. This
Lemma 1.1 one more time, we deduce that H
proves (i).
−→ ClE /N (ClK ) is sur(ii) It is easy to see that the natural map CE /N (CK ) −→
b 0 (P, CK ) ∼
b −2 (P, Z) has trivial
jective. Since χ 6= 1, the group CE /N (CK ) = H
=H
χ-part, and it follows that the norm map N : ClK (χ) −→
−→ ClE (χ) is surjective.
Notice that in order to prove surjectivity of this norm map we have not really used
the condition on χ, but merely the fact that χ is not trivial.
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MINUS CLASS GROUPS OF THE FIELDS OF THE l-TH ROOTS OF UNITY
1231
∗
The P -cohomology groups of each module in the exact sequence 0 −→ OK
−→
∗
UK −→ CK −→ ClK −→ 0 have trivial χ-parts. Since the natural maps OE
→
∗ P
P
P
OK , UE → UK and CE → CK are all isomorphisms, so is ClE (χ) → ClK (χ)P .
This proves (ii).
Theorem 2.2. If for all primes r that are ramified in F ⊂ K we have that χ(r) 6=
1, then
(i) there is an exact sequence of Oχ [π]-modules
Θ
0 −→ Oχ [π]d −→ Oχ [π]d −→ ClK (χ) −→ 0
where d is the Oχ -rank of ClF (χ);
(ii) we have
Y
#ClK (χ) = #Oχ /( B1,χ−1 ψ )
ψ
∗
where ψ runs over all characters ψ : π −→ Qp .
−→
Proof. By Nakayama’s lemma there is a surjective Oχ [π] morphism Oχ [π]d −→
ClK (χ). By Theorem 2.1, the class group ClK (χ) and hence the kernel of this map
are cohomologically trivial. Now one copies the proof of [2, p.113, Thm.8] with Z
replaced by the discrete valuation ring Oχ . It follows that the kernel is a projective
Oχ [π]-module. Since Oχ [π] is local, the kernel is therefore free. It has rank d since
it is of finite index in Oχ [π]d . This proves (i).
Part (ii) is a generalization of the Theorem of B. Mazur and A. Wiles [7], [16],
[17], [18]. By D. Solomon’s Theorem [22, p.472], we have for every subgroup P ⊂ π
with cyclic quotient π/P ,
Y
B1,χ−1 ψ ).
#ClK P (χ)[NP 0 /NP ] = #Oχ /(
ker ψ=P
Here the ψ run over the characters of G for which ker ψ = P . Here P 0 denotes
the unique subgroup of P
π containing P
P as a subgroup of index p and NP and NP 0
denote the norm maps σ∈P σ and σ∈P 0 σ respectively. In the exceptional case
P
[NP 0 /NP ] we
P = π the group P 0 is not defined and we simply put NP 0 = 0. By ClK
denote the kernel of the relative norm map NP 0 /NP from the class group ClK P (χ)
to itself.
Q
Put Sχ = P NP Oχ [π]/NP 0 Oχ [π]. Here P runs over the subgroups of π with
cyclic quotient π/P . The natural map
g : Oχ [π] −→ Sχ
becomes an isomorphism when we take the tensor product with the quotient field
Kχ of Oχ . Therefore g is injective and has finite cokernel.
All modules occurring in the exact sequence of part (i) areQcohomologically trivial. Therefore it remains exact when we apply the functor P NP (−)/NP 0 (−) to
it. We obtain the following diagram with exact rows.
0 −→
Oχ[π]d
g
y
0 −→
Sχd
Θ
−→
−→
Oχ[π]d
g
y
Sχd
−→
ClK(χ)
−→ 0

y
Q
−→
P NP ClK (χ)/NP 0 ClK (χ) −→ 0
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1232
RENÉ SCHOOF
Theorem 2.1(i) and (ii) and an application of the snake lemma then gives that
Y
Y
#ClK (χ) =
#(NP ClK (χ)/NP 0 ClK (χ)) =
#(ClK P (χ)[NP 0 /NP ])
P
P
and the result follows from Solomon’s Theorem.
It is not difficult to express the order of ClK (χ) in terms of the matrix Θ of
Theorem 2.1(i). One has [1, III, sect.9, Prop.6]
Y
#ClK (χ) = #Oχ /( ψ(det(Θ))).
ψ
Here ψ runs over the characters of π, and ψ(det(Θ)) indicates the value of the
natural extension of ψ to an algebra homomorphism Oχ [π] −→ Qp on det(Θ) ∈
Oχ [π].
Next we deduce a sufficient condition for the eigenspace ClK (χ) to be a cyclic
Oχ [π]-module.
Theorem 2.3. Suppose that for all primes r that are ramified in F ⊂ K we have
that χ(r) 6= 1. If one of the following conditions holds:
– B1,χ−1 = pu for some unit u ∈ Oχ∗ ;
∗
– there exists a character ϕ : Gal(Q/Q) −→ Qp of order pk > 1 such that
B1,χ−1 ϕ = (1 − ζpk )u for some unit u in Oχ [ζpk ],
then there is an isomorphism of Oχ [π]-modules
ClK (χ) ∼
= Oχ [π]/(θχ ).
In particular, ClK (χ) is a cyclic Oχ [π]-module.
Proof. We first show that ClF (χ) is a cyclic Oχ -module. If B1,χ−1 = pu for some
unit u ∈ Oχ∗ , it follows from Theorem 2.2(ii) that #ClF (χ) is equal to the order of
the residue field Oχ /(p). Therefore ClF (χ) is cyclic over Oχ .
ker ϕ
F and let P = Gal(E/F ). Then P is cyclic and
In the other case, let E = Q
we let F ⊂ E 0 ⊂ E be the unique subfield of E of index p. Since ϕ 6= 1, it follows
from Theorem 2.1(ii) that the norm map NE/E 0 : ClE (χ) −→ ClE 0 (χ) is surjective.
To compute the order of the kernel of NE/E 0 , we observe that
Norm(B1,χ−1 ϕ ) = Norm(1 − ζpk ) = p
(here the Norm is the Qp (ζpk )/Qp -norm). By Solomon’s Theorem [22, Thm. II,
1], we conclude that ClE (χ)[NE/E 0 ] has the same order as the residue field Oχ /(p)
of Rχ . Therefore so does ClE (χ)/(NE/E 0 ). By Nakayama’s lemma, ClE (χ) is
therefore cyclic over the group ring Oχ [P ]. It follows that ClF (χ) is cyclic over Oχ
in this case as well.
To complete the proof, we observe that, by Theorem 2.1, ClK (χ) is cohomologically trivial and the π-norm map induces an Oχ -isomorphism between ClF (χ)
and ClK (χ) modulo the augmentation ideal of Oχ [π]. It follows from Nakayama’s
lemma that ClK (χ) is cyclic over Oχ [π]. By Stickelberger’s theorem there is there−→ ClK (χ), which is an isomorphism because both
fore a surjection Oχ [π]/(θχ ) −→
groups have the same order by Theorem 2.2. This proves Theorem 2.3.
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MINUS CLASS GROUPS OF THE FIELDS OF THE l-TH ROOTS OF UNITY
1233
In the case the p-group π is cyclic of order pe , say, we can be a little bit more
explicit. We have the usual isomorphism of local rings, familiar in Iwasawa theory
e
Oχ [π] ∼
= Oχ [T ]/((1 + T )p − 1),
where 1 + T corresponds to some generator of π. The maximal ideal of this local
i
ring is (T, p). For i ≥ 0, we let ωi (T ) = (1 + T )p − 1.
By the Weierstrass Preparation theorem [24], every non-zero f (T ) ∈ Oχ [[T ]]/
e
((1 + T )p − 1) is the residue class of a polynomial of the form pµ u(T )h(T ) where µ
is a non-negative integer, u(T ) a unit and h(T ) = T λ + aλ−1 T λ−1 + . . . + a1 T + a0
is a Weierstrass polynomial of degree λ < pe . This means that ai ≡ 0 (mod p) for
i = 0, 1, . . . , λ − 1.
Proposition 2.4. Suppose that for all primes r that are ramified in F ⊂ K we
have that χ(r) 6= 1. Suppose that the Galois group π is cyclic of order pe and that
ClF (χ) is a cyclic Oχ -module. If for some character ψ of π of order p, for some
λ < p − 1 and for some unit u ∈ Oχ [ζp ], we have that B1,χ−1 ψ = (1 − ζp )λ u, then
ClK (χ) ∼
= (Oχ /(pe ))λ−1 × Oχ /(pe B1,χ−1 )
as an Oχ -module.
Proof. We write Oχ [π] = Oχ [T ]/(ωe (T )) as above. Since ClF (χ) is a cyclic Oχ module, it follows from Theorem 2.1 that the eigenspace ClK (χ) is a cohomologically trivial cyclic Oχ [π]-module. Therefore ClK (χ) ∼
= Oχ [π]/(pµ f (T )) for
∼
some Weierstrass polynomial f (T ). Since ClF (χ) = Oχ [π]/(T ) ∼
= Oχ /(pµ f (0)),
µ
we have that p f (0) = B1,χ−1 , up to a p-adic unit. Similarly, for the subfield
F ⊂ E ⊂ K of degree p over F we have that ClE ∼
= Oχ [T ]/(f (T ), ω1(T )). Applying Solomon’s Theorem [22, Thm. II, 1], we find that, up to a p-adic unit,
f (1 − ζp ) = B1,χ−1 ψ = (1 − ζp )λ .
Since λ < p − 1, this implies µ = 0 and deg(f ) = λ. Since Oχ [T ]/(f (T ), ωe(T ))
is cohomologically trivial, we have the following exact sequence
T
0 −→ Oχ [T ]/(f (T ), ωe (T )/T ) −→ Oχ [T ]/(f (T ), ωe(T )) −→ Oχ /(f (0)) −→ 0.
We analyze the ideal (f (T ), ωe (T )/T ). Consider for 0 ≤ i < e the quotient
i
i
ωi+1 (T )
= (1 + T )p (p−1) + . . . + (1 + T )p + 1.
ωi (T )
Since λ < p − 1 we have that T p−1 ≡ T pg(T ) (mod f (T )) for some polynomial
g(T ) ∈ Oχ [T ]. This implies that ωi+1 /ωi = p + pT h(T ) for some h(T ) ∈ Oχ [T ].
Therefore
e−1
Y ωi+1
ωe (T )
=
≡ pe · u(T ) (mod f (T ))
T
ω
i
i=0
where u(T ) is some unit in Oχ [T ]/(ωe (T )). This shows that the ideals (f (T ),
ωe (T )/T ) and (f (T ), pe ) are equal and that there is an isomorphism of Oχ -modules
Oχ [T ]/(f (T ), ωe(T )/T ) ∼
= (Oχ /pe Oχ )λ .
To complete the proof, we observe that f (0) ∈ Oχ [T ]/(f (T ), ωe (T )) is the image of
f (T ) − f (0)
∈ Oχ [T ]/(f (T ), ωe(T )/T ) = Oχ [T ]/(f (T ), pe ),
T
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1234
RENÉ SCHOOF
under the multiplication by T map. Since f is monic, this implies that f (0) has
order pe . Therefore 1 ∈ Oχ [T ]/(ωe (T ), f (T )) has, up to p-adic unit, order f (0)pe .
This completes the proof
The following simple result often suffices to determine the structure of the p-part
of the minus class group of Q(ζl ) when p divides l − 1. Note that the proof does
not rely on the theorems of Mazur-Wiles, Kolyvagin or Solomon.
Theorem III. Let l and p be odd primes and let M be the p-part of the minus
class group of Q(ζl ). If #M divides (l − 1)2 , then M is a cyclic group.
Proof. Let π denote the p-part of G = Gal(Q(ζl )/Q); it is a cyclic group of order pe .
Let F be the fixed field of π, let χ be a character of G of order prime to p and let
M (χ) be the corresponding eigenspace of M . We assume that M (χ) 6= 0. Since
the condition of Theorem 2.1 is satisfied for K = Q(ζl ), there is an exact sequence
Θ
0 −→ Oχ [π]d −→ Oχ [π]d −→ M (χ) −→ 0,
where d is the Oχ -rank of ClF (χ). Let q = pa denote the number of elements in
the residue field of Oχ . We write det(Θ) = pµ u(T )f (T ) ∈ Oχ [π] ∼
= Oχ [T ]/(ωe (T ))
λ
λ−1
+
a
T
+
.
.
.
+
a
for some Weierstrass polynomial f (T
)
=
T
λ−1
1 T + a0 and some
Q
unit u(T ). Then #M (χ) = #Oχ /( ζ pe =1 pµ f (ζ − 1)), so that
e
#M (χ) ≥ q µp
+min(λ,p−1)e+1
and hence
2e ≥ a(µpe + min(λ, p − 1)e + 1).
Since 2e < pe + 1, we have µ = 0. Since M (χ) 6= 0, this implies that λ > 0.
Moreover, since a · min(λ, p − 1) < 2, we have that λ = 1 and a = 1 so that
Oχ = Zp . This shows that, up to a unit, f (T ) = det(Θ) = T − β for some
−→ Cll (χ) is
β ∈ pZp . Since d is the Oχ -rank of ClF (χ), any surjection Oχ [π]d −→
an isomorphism modulo the maximal ideal m of the local ring Oχ [π]. This implies
that all entries of the matrix Θ are contained in m so that det(Θ) ∈ md .
e
It follows that d = 1, so that M (χ) ∼
= Zp /pe βZp
= Zp [T ]/((1 + T )p − 1, T − β) ∼
is a cyclic group. We conclude the proof by observing that #M (χ) ≥ pe+1 , so that
only one eigenspace M (χ) is non-trivial and hence M = M (χ).
3. The 2-part
In this section we study the 2-part of the minus class group of a complex abelian
number field K. We show that certain eigenspaces of the 2-part are cohomologically
trivial Galois modules. This has consequences for their structure. Finally we prove
a criterion for cyclicity of these eigenspaces as Galois modules.
Let G = Gal(K/Q), let ι ∈ G denote complex conjugation and let K + denote
the fixed field of ι. We have inclusions of idèle class groups CK + ⊂ CK and of idèle
unit groups UK + ⊂ UK . There is a natural map ClK + −→ ClK . We define
−
UK
= UK /UK + ,
−
CK
= CK /CK + ,
−
ClK
= ClK /im ClK + ,
−
µ−
K = µK ∩ U K .
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MINUS CLASS GROUPS OF THE FIELDS OF THE l-TH ROOTS OF UNITY
1235
−
1−ι
−
Note that UK
is isomorphic to the submodule UK
of UK . The intersection µK ∩UK
is taken inside UK .
∗
A diagram chase involving the exact sequence 0 −→ OK
−→ UK −→ CK −→
+
ClK −→ 0 and the analogous sequence for K shows that there is an exact sequence [19]
−
−
−
0 −→ µ−
K −→ UK −→ CK −→ ClK −→ 0.
−
It is important to use the definition of the minus class group ClK
that we give
here. Often the minus class group of an abelian number field K is defined to be
the kernel of the norm map N : ClK −→ ClK + . The present definition differs at
most in the 2-part. It has several advantages: as we will see below, it is easy to
−
; the results for the 2-part are very similar
compute the Galois cohomology of ClK
to the results for the odd parts. I don’t know how to do the calculations using the
other definition.
Another advantage over the usual definition is the following. It is easy to deduce
−
the following formula for the order of ClK
from the usual class number formula:
Y 1
2
−
=
− B1,χ .
#ClK
− #µK
2
[µK : µK ]
χ odd
This formula does not involve the unit index “QK ” of Hasse [5, Ch.20], which is,
in general, difficult to compute. This time there is the factor 2/[µK : µ−
K ], which is
either 1 or 2, but this quantity is easy to compute; it captures, in some sense, only
the easy aspects of the unit index QK and its calculation is precisely the content
of Hasse’s Satz 22 in [5].
In this secton we fix a complex abelian number field K with G = Gal(K/Q).
Let π be the 2-part of G with fixed field k = K π . We fix a non-trivial character
χ of G of odd order. We denote the fixed field of K under ι by K + . Note that
k ⊂ K +.
Theorem 3.1. Let P ⊂ π be a 2-group that does not contain ι and let E = K P .
Let E + be the fixed field of E under ι. If all primes r that ramify in E + ⊂ K satisfy
χ(r) 6= 1, then
−
(χ) is a cohomologically trivial Oχ [P ]-module;
(i) ClK
−
−
(χ) −→ ClK
(χ)P is bijective and the norm map N :
(ii) the natural map ClE
−
−
ClK (χ) −→ ClE (χ) is surjective.
Proof. Note that Gal(K/E + ) ∼
= P × {1, ι}. The proof follows the pattern of the
proof of Theorem 2.1.
b q (P, Cl− (χ)) = 0 for all q ∈ Z. Consider the exact
(i) It suffices to show that H
K
sequence
−
−
−
0 −→ µ−
K −→ UK −→ CK −→ ClK −→ 0.
We show that the χ-parts of the P -cohomology groups of the first three modules
b q (P, Cl− (χ)) = 0 for all q ∈ Z.
are trivial. Lemma 1.1 then implies that H
K
Since χ has odd order, it acts trivially on the 2-part of µ−
K and therefore on its
b q (P, µ− )(χ) = 0 for all q ∈ Z. By global
P -cohomology groups. This shows that H
K
b q (P, CK ) and H
b q (P, CK + ) are isomorphic to H
b q−2 (P, Z) and
class field theory H
have therefore trivial G-action and, since χ 6= 1, trivial χ-parts. It follows that
b q (P, CK − )(χ) = 0 for all q ∈ Z.
H
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1236
RENÉ SCHOOF
By local class field theory and the fact that χ(r) 6= 1 for the primes r that ramify
b q (P, UK ) and H
b q (P, UK + ) have trivial
in E ⊂ K and E + ⊂ K + we have that H
χ-parts. The proofs are similar to the proof of part (i) of Theorem 2.1.
−
−
−
−
/N (CK
) −→ ClE
/N (ClK
) is surjective. We saw already
(ii) The natural map CE
−
−
−
0
b
in the proof of part (i) that CE /N (CK ) = H (P, CK
) has trivial χ-part. Therefore
−
−
(χ) −→
−→ ClE
(χ) is surjective. Note that we only used the
the norm map N : ClK
fact that χ 6= 1 to prove this.
To prove the second statement, we consider the following diagram:
0 −→
0 −→
µ−
E

y
µ−
K
P
−→
UE−

y
−
−→ UK
P
−
CE

y
−→
−
−→ CK
P
−→
−
Cl
E

y
−
−→ ClK
P
−→ 0
−→ 0
An easy diagram chase shows that the first three vertical arrows are injective and
have cokernels with trivial χ-parts. By the proof of part (i), the P -cohomology
−
−
−
groups of each of the modules µ−
K , UK , CK and ClK have trivial χ-parts as well.
−
(χ) −→
This easily implies that the rightmost map induces an isomorphism ClE
−
ClK (χ)P as required.
Theorem 3.2. If all primes r that ramify in k ⊂ K satisfy χ(r) 6= 1, then
(i) there is an exact sequence
−
0 −→ (Oχ [π]/(1 + ι))d −→(Oχ [π]/(1 + ι))d −→ ClK
(χ) −→ 0;
Θ
(ii) If, in addition, the prime 2 is not ramified in the field K, then
Y1
−
B −1 ),
#ClK
(χ) = Oχ /(
2 1,χ ψ
ψ
where the product runs over the odd characters ψ of G of 2-power order.
Proof. Choose σ ∈ π so that hσi is a direct summand of π containing ι. Let 2e
denote the order of σ and let P be a complement of hσi in π: we have π = P × hσi.
e−1
−
(χ) is a Oχ [π]-module on which ι = σ 2
acts as −1. Therefore
The eigenspace ClK
−
(χ) is a module over the ring Oχ [P × hσi]/(1 + ι) ∼
ClK
= Oχ [ζ2e ][P ].
−
(χ) is a cohomologically trivial P -module. Let Oχ [ζ2e ][P ]d
By Theorem 3.1, ClK
−
ClK (χ) be a surjective Oχ [ζ2e ][P ]-homomorphism. The kernel is a cohomologically trivial torsion-free Oχ [ζ2e ][P ]-module. As in the proof of Theorem 2.3, we
copy the proof of [2, p.113, Thm.8] with Z replaced by the discrete valuation ring
Oχ [ζ2e ]. It follows that the kernel is projective and hence free over the local ring
Oχ [ζ2e ][P ]. Since the quotient is finite, the kernel has rank d. This proves (i).
(ii) We proceed with induction with respect to the order of π. Since 2 is unramified we may apply C. Greither’s Theorem [4, p.453, Thms. A and B] and we see
that the result holds when π is cyclic. Suppose π is not cyclic. Writing π = hσi × P
as in part (i), the group P is not trivial. Let τ ∈ P be an element of order 2.
The fixed fields K τ and K τ ι of τ and τ ι are both complex abelian number fields
containing k. The set of odd characters of G is the disjoint union of the sets of odd
characters of Gal(K τ /Q) and Gal(K τ ι /Q).
By induction, the result holds for the fields K τ and K τ ι . By Theorem 3.1(i),
−
(χ) is cohomologically trivial, both as a {1, τ }-module and as a {1, τ ι}M = ClK
module. Moreover, by part (ii) of that theorem, (1 + τ )M and (1 + τ ι)M are
isomorphic to the χ-part of the 2-part of the minus class group of K τ and K τ ι
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MINUS CLASS GROUPS OF THE FIELDS OF THE l-TH ROOTS OF UNITY
1237
respectively. Since ι acts as −1 on M , it follows from the cohomological triviality
of M that #M = #(1 + τ )M · #(1 − τ )M = #(1 + τ )M · #(1 + τ ι)M . This
proves (ii).
−
Finally we prove a sufficient condition for the eigenspace ClK
(χ) to be a cyclic
Oχ [π]/(1 + ι)-module.
Theorem 3.3. Suppose that all primes r that ramify in k ⊂ K satisfy χ(r) =
6 1.
If there exists an odd character ϕ of odd conductor and of order 2k for which each
of the following two conditions hold:
– 12 B1,χ−1 ϕ = (1 − ζ2k )u for some unit u ∈ Oχ [ζ2e ]∗ ,
– χ(r) 6= 1 for all primes r dividing the conductor of ϕ,
−
(χ) is a cyclic Oχ [π]/(1 + ι)-module.
then ClK
Proof. Let kϕ denote the composite field kQker ϕ and let Kϕ denote KQker ϕ . Both
fields kϕ ⊂ Kϕ are complex. Put π 0 = Gal(Kϕ /k) and P = Gal(Kϕ /kϕ ). We have
that ι 6∈ P .
Since 2 is not ramified, it follows from Greither’s Theorem that the order of
Clk−ϕ (χ) is equal to the order of Oχ /(Norm( 12 B1,χ−1 ϕ )). Here the Norm is the
Oχ [ζ2k ]/Oχ -Norm. Since Norm( 12 B1,χ−1 ϕ ) = Norm(1 − ζ2k ) = 2, we see that
the order of Clk−ϕ (χ) is equal to the order of the residue field of Oχ . Therefore
Clk−ϕ (χ) is a cyclic Galois module. By Theorem 3.1, applied to E = kϕ ⊂ Kϕ , the
−
eigenspace ClK
(χ) is a cohomologically trivial P -module and the P -norm map inϕ
−
duces an isomorphism between Clk−ϕ (χ) and ClK
(χ) modulo the P -augmentation
ϕ
−
ideal. Therefore another application of Nakayama’s Lemma implies that ClK
(χ)
ϕ
0
is a cyclic Oχ [P ]-module and hence a cyclic Oχ [π ]/(1 + ι)-module. Therefore its
−
(χ) is a cyclic Oχ [π]/(1 + ι)-module, as required.
quotient ClK
If the group π is cyclic, then Oχ [π]/(1 + ι) ∼
= Oχ [ζ2e ] where #π = 2e . Since
the ring Oχ [ζ2e ] is a discrete valuation ring, the structure of finite modules over
Oχ [π]/(1 + ι) is particularly simple.
−
Proposition 3.4. Suppose that π is cyclic and that ClK
(χ) is cyclic over Oχ [π].
−
ft
f
If #ClK (χ) = 2 , where 2 is the order of the residue field Oχ /(2), then there is
an isomorphism of Oχ [ζ2e ]-modules
−
(χ) ∼
ClK
= Oχ [ζ2e ]/((1 − ζ2e )t )
and there is an isomorphism of abelian groups
−
ClK
(χ) ∼
= (Z/2r Z)f (2
e−1
−s)
× (Z/2r+1 Z)f s
where r, s ∈ Z are determined by t = r2e−1 + s and 0 ≤ s < 2e−1 .
Proof. This follows from the fact that Oχ [ζ2e ] is a discrete valuation ring with
uniformizing element 1 − ζ2e .
4. Tables
In this section we present the proof of Theorem II. An essential ingredient is the
table of class numbers h−
l given in the appendix. We briefly explain the notation.
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1238
RENÉ SCHOOF
Table 4.1
l
233
269
317
337
359
379
383
389
397
401
409
p14 · p29
p16 · p31
p25 · p49
p13 · p15 · p15
p13 · p30 · p45
p22 · p24
p19 · p24 · p46
p24 · p60
p8 · p26 · p27
p16 · p18 · p31
p12 · p52
PM
PM
HtR
PM,
PM,
BS
PM,
AL
PM,
PM,
PM
PM
HtR
HtR
BS
PM
l
419
433
439
449
463
467
479
487
499
503
509
p16 · p30 · p49
p14 · p34
p11 · p21 · p23 · p24
p18 · p84
p18 · p21 · p25
p19 · p49 · p55
p20 · p27 · p70
p30 · p49
p15 · p18 · p47
p12 · p14 · p112
p16 · p28 · p101
PM,
PM
PM,
PM
PM,
PM,
PM,
HtR
PM,
PM,
PM,
HtR
PM, PM
BS
AL
AL
PM
PM
AL
Let l be an odd prime. We have l − 1 = 2e · m with m odd. For every divisor d
of l − 1 which itself is divisible by 2e we define
Y
1
h−
− B1,χ
l (d) =
2
ord(χ)=d
where the product runs over the characters χ : (Z/lZ)∗ −→ C∗ of order d; except
when d = l − 1, in which case we multiply this product by l, and when d = 2e , in
which case we multiply it by 2. In the rare occasion when l − 1 is equal to 2e , the
only possible value for d is l − 1 = 2e and we put
Y
1
h−
− B1,χ .
l (d) = 2l
2
ord(χ)=d
This last case occurs only when l is a Fermat prime i.e., when l = 3, 5, 17, 257,
65537 or has more than 2 500 000 decimal digits.
The numbers h−
l (d) are listed in the appendix. They are rational integers [5],
[24] and they are related to the minus class number h−
l by
Y
−
h−
h−
l = #Cll =
l (d).
2e |d|l−1
In [15] D. H. Lehmer and J. M. Masley presented a table with the numbers h−
l (d)
for l ≤ 509. Of most of these numbers the complete prime factorization was given,
but their table contains 22 unfactored composite numbers. These were factored by
Peter Montgomery (PM), Bob Silverman (BS), Herman te Riele (HtR) and Arjen
Lenstra (AL). The most laborious factorization, for l = 467, was performed by
Arjen Lenstra, who factored a 103 digit factor of h−
467 into a product of two primes
of 49 and 55 digits respectively. We list the various contributions in Table 4.1. By
pn we denote a prime factor of n decimal digits. The order in which the initials are
given corresponds to the order of the prime factors. In order to prove Theorem II
and, at the same time, determine the structure of Cll− as an abelian group, we
study the table of numbers h−
l (d) of the appendix. Clearly, if a prime p divides the
exactly
once,
the p-part of Cll− is cyclic as a group and hence as
class number h−
l
a Galois module. This happens for most large prime divisors. All other cases are
listed below. Tables 4.2, 4.3 and 4.4 contain the prime pairs (p, l) with l ≤ 509 for
which p2 divides h−
l . We discuss each table in some detail.
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MINUS CLASS GROUPS OF THE FIELDS OF THE l-TH ROOTS OF UNITY
1239
The class group Cll− is a product of its p-parts and each p-part is a product of
eigenspaces Cll (χ). The minus class group Cll− is a cyclic Galois module if and
only if for each prime p, each eigenspace Cll− (χ) is cyclic over the local ring Oχ [π],
where π is the p-part of G = Gal(Q(ζl )/Q).
Table 4.2. Primes p not dividing l − 1
l
41
131
139
p
11
3
47
277
149
151
157
211
3
11
157
281
227
241
277
281
293
313
337
353
379
2939
47
47
11
41
3
37
17
353
379
397
401
409
419
443
457
467
479
487
23
41
5
3
3
5
467
5
7
491
37
3
11
491
d
40
26
46
46
138
4
30
156
14
70
226
16
276
40
40
4
24
16
352
42
378
132
80
24
2
26
24
466
2
2
6
18
2
10
98
490
f
2
3
1
1
1
2
2
1
1
1
1
2
2
2
1
2
2
1
1
1
1
2
2
2
1
3
2
1
1
1
1
1
1
1
1
1
hl (d)
112
33
472
277
277
32
112
1572
281
281
29393
472
472
112
412
32
372
172
3532
379
379
232
412
52
32
36
52
4672
52
7
7
372
32
113
491
4912
class group
11 × 11
3×3×3
2209
277
277
3×3
11 × 11
157 × 157
281
281
2939 × 2939 × 2939
47 × 47
47 × 47
11 × 11
1681
3×3
37 × 37
17 × 17
353 × 353
379
379
23 × 23
41 × 41
5×5
9
9×9×9
5×5
467 × 467
25
7
7
37 × 37
9
11 × 121
491
491 × 491
Thm.2.3 with r = 283
Thm.2.2
Thm.2.2
Thm.2.3 with r = 83
Thm.2.2
Thm.2.2
Thm.2.3 with r = 7
Thm.2.3 with r = 7
Thm.2.2
Thm.2.3 with r = 11
Thm.2.2
Thm.2.3 with r = 7
Thm.2.2, Thm.2.3 with r = 23
Thm.2.2
In Table 4.2 we have listed all pairs (p, l) for which p is odd and p2 divides h−
l ,
but p does not divide l − 1. In this case the p-part π of the Galois group of Q(ζl )
over Q is trivial and an eigenspace Cll (χ) is cyclic as a Galois module if and only
if it is a cyclic Oχ -module. It turns out that in all cases every Cll (χ) is cyclic as
an Oχ -module.
To explain the table, we first note that in the case l = p, the Teichmüller
eigenspace Cll− (ω) is always trivial. Therefore we only have contributions for the
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1240
RENÉ SCHOOF
Table 4.3. Odd primes p dividing l − 1
`
31
71
101
131
137
139
157
181
199
211
283
307
331
337
367
379
409
421
439
461
463
499
p
3
7
5
5
17
3
13
5
3
3
7
3
3
3
3
7
3
3
17
5
3
5
7
7
3
d
2
2
4
2
8
2
12
4
2
2
6
2
2
2
10
16
2
2
8
4
2
4
2
6
2
h0 , h1 , . . .
3, 3
7, 7
5, 25, 25
5, 5
17, 17
3, 3
13, 13
25, 5
9, 3, 3
3, 3
7, 7
3, 3
3, 3, 3
3, 9
81, 81
49, 49
9, 3
3, 3, 3, 3
17, 17
25, 5
3, 27
25, 25
7, 7
7, 7
3, 3
group
9
49
25 × 125
25
289
9
169
125
81
9
49
9
27
3×9
9×9×9×9
49 × 49
27
81
289
125
9×9
5 × 125
49
49
9
Prop.2.4, λ = 2
Prop.2.4, λ = 1
Thm.2.3,
Prop.2.4,
Prop.2.4,
Prop.2.4,
θ = T 2 − 15T + 3
λ=1
λ=1
λ=1
Prop.2.4, λ = 1
Thm.2.3, θ = T 2 − 3T − 3
Thm.2.3 with r = 11; Prop.2.4, λ = 2
characters χ 6= ω. Let d be a divisor of l − 1 for which p divides h−
l (d). Then for all
characters χ of order d the ring Oχ has a residue field with pf elements where f is
the order of p modulo d. If pf happens to be the exact power of p dividing h−
l (d),
then it is clear that for exactly one character χ of order d the eigenspace Cll− (χ)
is isomorphic to Oχ /(2) while all others are trivial. These cases are listed without
comment. In the remaining cases we apply the theorem of Mazur and Wiles which
is the case with trivial π of Theorem 2.2. If the precise power of p dividing h−
l (d)
is pf a and for precisely a characters χ of order d the generalized Bernoulli number
B1,χ−1 is divisible by p, then each eigenspace Cll− (χ) is either isomorphic to Oχ /(2)
or is zero. In particular, each Cll (χ) is a cyclic Galois module. This happens in
all but seven cases. In the remaining seven cases we use Theorem 2.3 and show
that each eigenspace is a cyclic Oχ module by computing an additional Bernoulli
number B1,χ−1 ϕ where ϕ is a suitable even character of order p and conductor r.
In Table 4.3 we have listed all pairs (p, l) with p 6= 2 dividing l − 1. We’ll see
2
below that in this case the class number h−
l is automatically divisible by p , so
−
that Table 4.3 actually contains all pairs (p, l) for which p divides gcd(hl , l − 1).
In order to explain the contents of the table, we fix p and l and we let pe be the
exact power of p dividing l − 1.
If d and d0 are two divisors of l − 1 that only differ by a power of p, then B1,ϕ−1 ≡
B1,ϕ0−1 modulo (1 − ζpe ) for all characters ϕ of order d and ϕ0 of order d0 . Therefore,
− 0
as Lehmer observed [14, Thm.5], either both h−
l (d) and hl (d ) are divisible by p
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MINUS CLASS GROUPS OF THE FIELDS OF THE l-TH ROOTS OF UNITY
1241
Table 4.4. p = 2
`
29
113
163
197
239
277
311
337
349
373
397
421
463
491
d
28
112
6
28
14
12
62
336
12
124
12
60
14
14
ord(χ)
7
7
3
7
7
3
31
21
3
31
3
15
7
7
2e
4
16
2
4
2
4
2
16
4
4
4
4
2
2
f
3
3
2
3
3
2
5
6
2
5
2
4
3
3
h−
l (d)
8
8
4
8
82
42
322
64
42
32
43
16
8
82
2–class group
2×2×2
2×2×2
2×2
2×2×2
4×4×4
2×2×2×2
2×2×2×2×2×2×2×2×2×2
2×2×2×2×2×2
2×2×2×2
2×2×2×2×2
4×4×2×2
2×2×2×2
2×2×2
2×2×2×2×2×2
r
3
3
7
3
or none is. For this reason we have ordered the class numbers as follows: for each
divisor d of l − 1 which is itself not divisible by p but for which h−
l (d) is divisible
i
(dp
).
By
Lehmer’s
observation,
by p, we list, for i = 0, 1, . . . , e the p-part hi of h−
l
each hi is divisible by p. We note in passing that this implies that h−
l is divisible
by p2 .
For each character χ of order d the residue field of Oχ has order pf where f is
the order of p modulo d. In all but one case either h0 = pf or h1 = pf . In the latter
case we have that, up to a unit, B1,χ−1 ψ = 1 − ζp for the characters ψ of conductor l
and order p. In either case Theorem 2.3 applies and we see that Cll (χ) is cyclic over
Oχ [π]. The only exception is l = 461 with p = 5. In this case h0 = h1 = 25. In this
case we have applied Theorem 2.3 with ϕ a character of order 5 and conductor 11.
It turns out that in this exceptional case Cll (χ) is a cyclic Oχ [π]-module as well.
In most cases we can apply Theorem III and conclude that the eigenspace is a
cyclic group. These cases are listed without comment. In the cases (l, p) = (101, 5),
(337, 7), (461, 5) and (331, 3) (the latter for d = 10) an application of Proposition
2.4 immediately gives the structure of Cll (χ). Finally, in the cases (l, p) = (439, 3)
and (331, 3) (the latter for d = 2) we have explicitly computed the Stickelberger
element θ and applied Theorem 2.3 directly.
Finally we discuss the contents of Table 4.4. Let χ be a character of (Z/lZ)∗ of
odd order. The 2-part of Cll− is a module over Oχ [π]/(1 + ι) ∼
= Oχ [ζ2e ]. Here 2e
−
is the exact power of 2 dividing l − 1. It is well known that Cll (χ) is trivial when
χ = 1. This implies that the prime p = 2 never divides h−
l with multiplicity 1.
Therefore Table 4.4 actually contains all primes l ≤ 509 for which h−
l is even.
It turns out that Cll− (χ) is in all cases a cyclic Galois module. This
Q follows from
several applications of Theorem 3.3. In all but 4 cases we have that ψ 12 B1,χ−1 ψ =
2u for some unit u ∈ Oχ . Here the product runs over the odd characters ψ of 2power order and conductor l. In this case Cll− (χ) ∼
= Oχ /(2) which is a vector
space of dimension f over F2 . Here f is the degree of F2 (ζd ) over F2 and d is the
order of χ. In the remaining cases we applied Theorem 3.3 with an odd quadratic
character ϕ of conductor r. Here r ≡ 3 (mod 4) is a prime for which χ(r) 6= 1.
The structure of Cll− (χ) then follows easily from Theorem 3.4.
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License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
71
67
61
59
53
47
43
41
37
31
29
23
17
19
13
11
l
3
5
7
d
2
4
2
6
2
10
4
12
16
2
6
18
2
22
4
28
2
6
10
30
4
12
36
8
40
2
6
14
42
2
46
4
52
2
58
4
12
20
60
2
6
22
66
2
10
14
h−
l (d)
1
1
1
1
1
1
1
1
1
1
1
1
3
1
1
23
3
3
1
1
1
1
37
1
112
1
1
1
211
5
139
1
4889
3
59 · 233
1
1
41
1861
1
1
67
12739
7
1
7
149
139
137
131
127
113
109
107
103
101
97
89
83
79
73
l
d
70
8
24
72
2
6
26
78
2
82
8
88
32
96
4
20
100
2
6
34
102
2
106
4
12
36
108
16
112
2
6
14
18
42
126
2
10
26
130
8
136
2
6
46
138
4
h−
l (d)
79241
89
1
134353
5
1
53
377911
3
279405653
113
118401449
3457
577 · 206209
5
52
52 · 101 · 601 · 18701
5
1
1021
103 · 17247691
3
743 · 9859 · 2886593
17
1
1009
9431866153
17
23 · 11853470598257
5
13
43
3079
547
883 · 626599
5
5
33 · 53
131 · 1301 · 4673706701
17
17 · 47737 · 46890540621121
3
3
472 · 277
277 · 967 · 1188961909
32
4
12
20
36
60
180
2
10
38
190
64
192
4
28
196
2
6
181
193
197
199
191
2
178
4
172
2
6
18
54
162
2
166
2
6
10
30
50
150
4
12
52
156
d
148
179
173
167
163
157
151
l
h−
l (d)
149 · 5129663383200408/
/05461
7
1
281
112
25951
1207501 · 312885301
5
13
3148601
13 · 1572 · 1093 · 1873 · 4/
/18861
1
22
181
365473
23167 · 441845817162679
11
499 · 5123189985484229/
/035947419
5
20297 · 231169 · 725717/
/29362851870621
5
1069 · 144586673923349/
/48286764635121
52
37
5 · 41
2521
61 · 1321
5488435782589277701
13
11
51263
612771091 · 3673395066/
/9733713761
192026280449
6529 · 15361 · 29761 · 91/
/969 · 10369729
5
23 · 1877
7841 · 939830268487086/
/6656225611549
32
3
Appendix
256
2
10
50
250
251
257
16
48
80
240
2
14
34
238
239
241
8
232
4
12
76
228
2
226
2
6
10
14
30
42
70
210
2
6
74
222
d
18
22
66
198
233
229
227
223
211
l
h−
l (d)
3 · 19
727
25645093
207293548177 · 31681904128/
/39
3
3·7
41
281
181
7 · 421
71 · 281 · 12251
1051 · 113981701 · 4343510221
7
43
17909933575379
11757537731851 · 342480448/
/3726447
5
29393 · 1692824021974901·
·13444015915122722869
17
13
705053 · 47824141
457 · 7753 · 41415390332169/
/2666991589
1433
233 · 79933937980769 · 13046/
/008204119903320572430489
3·5
26
511123
14136487 · 123373184789 · 2/
/2497399987891136953079
472
2359873
15601 · 126767281
13921 · 518123008737871423/
/891201
7
11
348270001
9631365977251 · 3696311145/
/67755437243663626501
257 · 20738946049 · 1022997/
/74456391196156129869818/
/3419037149697
1242
RENÉ SCHOOF
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
331 2
6
10
307 2
6
18
34
102
306
311 2
10
62
310
313 8
24
104
312
317 4
316
l
d
263 2
262
269 4
268
271 2
6
10
18
30
54
90
270
277 4
12
92
276
281 8
40
56
280
283 2
6
94
282
293 4
292
h−
l (d)
13
263 · 787 · 385927 · 418759100955678867328189444629948074260186283
13
40170973189 · 8625962877077617 · 8297860832320483544484903227261
11
1
31
37
1201
751928131
21961 · 7288651
271 · 811 · 1621 · 15391 · 20238391 · 666587726641
17
24
89977 · 1371353 · 30697273
472 · 829 · 4873333 · 1776834909244716811072486129
17
112 · 412 · 401
64523056921
3235961 · 977343139976233968569461075411406081
3
3
2064523 · 39341481709417
283 · 5484646647490654799157896194266098076673
32
293 · 38901409 · 52561753 · 354041533 · 19844792749 · 702405569982494626097/
/54079833
3
3
3 · 37
137 · 443 · 1429
307 · 10191268178209
613 · 919 · 512412441029648479897766391339165893563
19
41
210 · 9918966461
311 · 856882084088129553550988747251311805392434897275868681
233
372
65386361 · 30358065621833
155288017 · 82941207961 · 986685963782009603919680953
13
1438031130902847137607233 · 8097705990409820600574529770502809400397/
/943027841
3
32
34
397 4
12
36
389 4
388
367 2
6
122
366
373 4
12
124
372
379 2
6
14
18
42
54
126
378
383 2
382
359 2
358
349 4
12
116
348
353 32
352
d
22
30
66
110
330
337 16
48
112
336
347 2
346
l
h−
l (d)
23 · 67
4
3 · 61
17406850561
476506973241784667381
270271 · 221475181712309125848473872740271
72 · 172 · 353
238321
72 · 894469355265098929
26 · 3246769 · 3622267546801 · 110537863229809 · 225164259907777
5
347 · 1954086942666238828259012186195350500935086726556960834433397/
/220152315402574339617
5
24 · 13
421081 · 943429 · 2021708236660033
2089 · 17749 · 29247661 · 16684629796320170064136004281782850431997
6113 · 9473
3532 · 281249 · 1380611233 · 3001891553 · 394388386054183213731974638871/
/81225470103134619777
19
5862361010431 · 813287316389858595758239885873 · 58922190801687625383/
/9609863906122210269152723
32
3
733 · 268738874461290742168853881
39163 · 127480330983805586375654833118494134773442493271686377913
5
61
25 · 1117 · 6218451821 · 1699148567515153
1489 · 191953 · 124204598699794021789479401683826456140588477617076789
3
3 · 13
1499
3 · 991
379 · 547
3 · 29997973
127 · 757 · 9199 · 154412119
379 · 1087873417 · 3111358344381146608939 · 214670345683920446286163
17
300032351 · 3000702226373096449 · 290945169106342852317343 · 250644232/
/2771948099181404130620436761970705901
41
389 · 1553 · 4847366257 · 128029167243805465177973 · 1027742679263367083/
/43655333188809496622747915533012083866597
13
26
109 · 4861
MINUS CLASS GROUPS OF THE FIELDS OF THE l-TH ROOTS OF UNITY
1243
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
457 8
24
449 64
448
443 2
26
34
442
439 2
6
146
438
433 16
48
144
432
421 4
12
20
28
60
84
140
420
431 2
10
86
430
409 8
24
136
408
419 2
22
38
418
d
44
132
396
401 16
80
400
l
h−
l (d)
23910808769
232 · 132189553 · 1917436489
9901 · 14141557 · 28894150148400351045400753 · 241092554399010330726544957
64849
412 · 476056112401
401 · 462972001 · 3692494801 · 2106370412068801 · 166771329637484801 · 348925/
/0662765811145388290782801
52 · 17
73 · 1321
17 · 122181721 · 7960379881 · 29097077764969
409 · 725945254273 · 6183699722087375941883228469840272721633145678440121
32
647747
1103 · 5410099
2719452561369347 · 440305024994584776198045120721 · 38089642480704298751/
/25494615628571625716516342483
52
37
5 · 2521
29 · 39509
24 · 22064701
70309 · 46085341
409781 · 16521541 · 672896721281
421 · 39901 · 3455761 · 57979541174101 · 2655579516751331409910861
3·7
11 · 701
676649 · 2709472364809333
14621 · 7970051 · 112225988494992246639243672859450218083129490012657313/
/823968596573192207124531
842353
4727329
3457 · 3021564742348701537217
433 · 12097 · 21601 · 47521 · 1403137 · 102550753 · 96686549358769 · 64340730822/
/61985367563988399449713
3·5
33
293 · 527207 · 7171667 · 50898521 · 327151064937209
40139516617 · 607057872831881225737 · 15343765387604391577783 · 7611086694/
/50601851817037
5
36 · 79 · 157
367926037
12377 · 2099059 · 309860291076943369037303413323285158985313526398152831/
/008871913595050372353059812436688273929
500402969557121
168449 · 226736972834339969 · 772865886177933052632667046915246737827100/
/790144773744195236265619879496879953539649
41
52 · 577
509
503
499
491
487
479
467
463
461
l
d
h−
l (d)
152 1217 · 43777 · 23353152677443223648257268496337
456 63841 · 28668613681009535839148397954381101468353560199403645535773916736/
/6347873193
4
52
20 52 · 661
92 461 · 463413261346674397069
460 161461 · 3702458172193117785898149655903648058852928086226081699845637442/
/0371674719539068279993529581
2
7
6
7
14 23 · 7 · 29
22 89 · 1123
42 7 · 631 · 673
66 4423 · 33642841
154 463 · 664064207818594609257539327251
462 8779 · 604417477499456083 · 334167173856936895861 · 1451125083064477390379041
2
7
466 4672 · 7842513546558078253 · 154987811800520892460672570209646897293261969/
/1231 · 4511882445351575687067360009368178199225508063847112361
2
52
478 48757 · 62141 · 2560169 · 26756241308309805857 · 177581990178050932739148007·
·3939232521558670638697337486372397962981765904709957802472308181004309
2
7
6
7
18 372
54 919 · 2647 · 10909
162 105792786991 · 1355141213869532941
486 58321 · 105290443 · 294594702996402697646390639203 · 90058027084074393088174/
/14913576150427261734980259
2
32
10 113
14 26 · 29
70 1262296191031
98 491 · 101566319 · 2311247713517
490 4912 · 8489251 · 17841391 · 74468731 · 18022473215169065702224279183302091210/
/994749548801576948376558921841
2
3
6
3
166 167 · 8170189 · 4568950377354424102616078873671968013
498 628477 · 2498605441 · 476526575352703 · 125184090531384337 · 2313122953817705/
/5312162275545594472697442144611
2
3·7
502 15061 · 182337132259 · 67961871500791 · 142639305944396395662911180592353348/
/442031813108145092050553010609968433975432 1688566291891565574466073368/
/455407
4
13
508 1102305661663669 · 3595837345204924707130453993 · 285986765137386082677131/
/210874962327994154402550613015614414986549035966985 8574049275462019230/
/8152597
1244
RENÉ SCHOOF
MINUS CLASS GROUPS OF THE FIELDS OF THE l-TH ROOTS OF UNITY
1245
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Cassels, J.W.S and Fröhlich, A.: Algebraic Number Theory, Academic Press, London 1967.
MR 35:6500
Cornacchia, P.: Anderson’s module and ideal class groups of abelian fields, J. Number Theory,
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Greither, C.: Class groups of abelian fields, and the main conjecture, Ann. de l’Institut
Fourier, 42 (1992), 449–499. MR 93j:11071
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Dipartimento di Matematica, 2a Università di Roma “Tor Vergata”, I-00133 Rome,
Italy
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