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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 ´ SCHOOF RENE 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

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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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#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, Fran¸cois 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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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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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 Teichm¨ uller 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 Teichm¨ uller 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 id`ele class group of K and let UK denote the group of unit id`eles, i.e. the group of K-id`eles that have trivial valuation at all

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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 Teichm¨ uller 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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∗ 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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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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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

1234

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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 id`ele class groups CK + ⊂ CK and of id`ele 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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− 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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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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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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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.

MINUS CLASS GROUPS OF THE FIELDS OF THE l-TH ROOTS OF UNITY

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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 Teichm¨ uller eigenspace Cll− (ω) is always trivial. Therefore we only have contributions for the

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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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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.

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 ´ SCHOOF RENE

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

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 ´ SCHOOF RENE

MINUS CLASS GROUPS OF THE FIELDS OF THE l-TH ROOTS OF UNITY

1245

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[13] [14] [15]

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[21] [22] [23] [24]

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` di Roma “Tor Vergata”, I-00133 Rome, Dipartimento di Matematica, 2a Universita Italy E-mail address: [email protected]