Class Number Formula for Dihedral Extensions Luca Caputo, Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio

Class number formula for dihedral extensions Luca Caputo, Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio

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Luca Caputo, Filippo Alberto Edoardo Nuccio Mortarino Majno Di Capriglio. Class number formula for dihedral extensions. Glasgow Mathematical Journal, Cambridge University Press (CUP), In press. hal-01728713

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LUCA CAPUTO AND FILIPPO A. E. NUCCIO MORTARINO MAJNO DI CAPRIGLIO

Abstract. We give an algebraic proof of a class number formula for dihedral extensions of number ﬁelds of degree 2q, where q is any odd integer. Our formula expresses the ratio of class numbers as a ratio of orders of cohomology groups of units and recovers similar formulas which have appeared in the literature as special cases. As a corollary of our main result we obtain explicit bounds on the (ﬁnitely many) possible values which can occur as ratio of class numbers in dihedral extensions. Such bounds are obtained by arithmetic means, without resorting to deep integral representation theory.

1. Introduction

Let k be a number field and let M/k be a finite Galois extension. For a subgroup H ⊆ Gal(M/k), let χH denote the rational character of the permutation representation defined by the Gal(M/k)-set Gal(M/k)/H. If M/k is non-cyclic, then there exists a relation X nH · χH = 0 H where nH ∈ Z are not all 0 and H runs through the subgroups of Gal(M/k). Using the formalism of Artin L-functions and the formula for the residue at 1 of the Dedekind zeta-function, Brauer showed in [Bra51] that the above relation translates into the following formula

Y nH Y nH (1.1) hM H = (wM H RM H ) H H where, for a number field E, hE, RE, wE denote the class number, the regulator and the order of the group of roots of unity of E, respectively. Although the regulator of a number field is in general an irrational number, the right-hand side of the above equality is clearly rational, being equal to the left-hand side. As a matter of fact, Brauer himself proved that when k = Q the regulator can be translated into an algebraic invariant of the Galois structure of the unit group. It is therefore natural to look for an algebraic proof of Brauer formula, independent of any analytical result. Brauer also showed that the left-hand side of (1.1) can take only finitely many values as M ranges over all Galois extension of Q with Galois group isomorphic to a fixed finite group. It thus becomes interesting to look for an explicit description of a finite set of integers (as small as possible) containing all the values which are attainable by the class number ratio, as well as for generalizations beyond the case k = Q. √ √ Historically, these questions were investigated for the first time in the case where M = Q( d, −d) (with d > 1 a square-free integer). In this case Dirichlet showed, by analytical means, that h 1 M = Q h √ h √ 2 ( d) Q( −d) Q √ where Q is 1 or 2 depending on whether a certain unit of M has norm 1 or not in Q( −1) (see[Dir42]). Hilbert later gave an algebraic proof of the above equality (see [Hil94] and [Lem94] for a more general result). Another interesting particular case is that of dihedral extensions, which is the one we focus on in this paper. This setting has already been considered by other authors: notably, Walter computed explicitly in [Wal77]

Date: March 11, 2018. 2010 Mathematics Subject Classification. 11R29, 11R20, 11R34. 1 CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 2 the regulator term, building upon Brauer’s previous work and obtained a refined class number formula in the dihedral case1 over an arbitrary base field k, still relying on analytical results about Artin L-functions. Bartel provided a different representation-theoretic intepretation of Walter’s result in [Bar12] for dihedral extensions of order 2q of arbitrary number fields, where q is any odd integer. Our main result is the following theorem whose proof is purely algebraic and does not rely on any argument involving L-functions. Theorem 3.13. Let L/k be a Galois extension of number fields whose Galois group D = Gal(L/k) is dihedral of order 2q with q odd. Let Σ ⊂ D be a subgroup of order 2 and G ⊂ D be the subgroup of order q. Set K = LD and F = LG. Then 2 0 × 1 0 × −1 × hLh bh D, O ⊗ Z bh (D, O ) bh (Σ, O ) k = L 2 = L L . 2 × 1 × × hF h −1 −1 0 K bh D, OL ⊗ Z 2 bh (D, OL ) bh (Σ, OL )

× i × Here OL is the group of units of L and, for a subgroup H ⊆ D and an integer i, bh (H, OL ) denotes the order × of the i-th Tate cohomology group of H with values in OL . Similar algebraic proofs have appeared in the literature although, as far as we know, only for special cases. Halter-Koch in [HK77] and Moser in [Mos79] analysed the case of a dihedral extensions of Q of degree 2p, where p is an odd prime. Their result expresses the ratio of class numbers in terms of a unit index, proving that in this setting it holds

hL × × × × −r 2 2 = OL : OK OK0 OF p hF hK where K0 is a Galois conjugate of K in L and r = 1 if F is imaginary and 2 otherwise. Their result was generalized by Jaulent in [Jau81, Proposition 12], who gave an algebraic proof of a similar formula for dihedral extensions of degree 2ps when the ground field k is Q or an imaginary quadratic field and s ≥ 1 is any integer. Jaulent actually obtained his result as a particular case of a formula for Galois extensions over an arbitrary ground field with Galois group of the form Z/ps o T where T is an abelian group of exponent dividing p − 1. In such general formula the ratio of class numbers is expressed in terms of orders of cohomology groups of units, as in our Theorem 3.13. This approach was not developed further and in fact Jaulent’s paper is not cited in subsequent works in the literature. The study of class number formulas for dihedral extensions from an algebraic viewpoint was taken up more than 20 years later by Lemmermeyer. In [Lem05], he generalized Halter-Koch’s formula to arbitrary ground fields, always under the assumption that the degree is 2p and under some ramification condition. The quoted works also provide an explicit description of the finitely many possible values which may occur as class number ratio, clearly only under their working assumptions. These values are obtained using the explicit classification of the indecomposable integral representations of D, in order to find an exhaustive list of the possible indexes of units which can appear as regulators ratio. This classification is not available for dihedral groups of arbitrary order 2q, as discussed in the introduction of [HR63], for instance. Combining our formula with the knowledge of the Herbrand quotient of the units for the cyclic subextension L/F , we find a finite set of integers containing all the attainable values of the ratio of class numbers, without resorting to any integral representation theoretic results. This indicates that the traditional way of expressing the class number ratio as a unit index, although suggestive, is less effective than our cohomological approach. Our main results in this direction are the following (for a prime number `, v` denotes the `-adic valuation). Corollary 3.14. For every prime `, the following bounds hold 2 hLhk −av`(q) ≤ v` 2 ≤ bv`(q) hF hK

1Walter’s result actually hodls for more general Frobenius groups. CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 3

× × where a = rkZ(OF )+βF (q)+1 and b = rkZ(Ok )+βk(q). In particular, if F is totally real of degree [F : Q] = 2d, then 2 hLhk −2dv`(q) ≤ v` 2 ≤ (d − 1)v`(q). hF hK We refer the reader to Section 3 for the definition of βk(q), βF (q) ∈ {0, 1}. In case F is assumed to be a CM field and k = F + is its totally real subfield, we can strengthen our bounds as follows.

Corollary 3.17. Assume that F is a CM field of degree [F : Q] = 2d and that k is its totally real subfield. × q × × q ∞ Let s be the order of the quotient (OL ) ∩ OF /(OF ) and let t be the order of µF (q )/µF (q). Then, for every prime `, 2 hLhk −v`(q) − v`(s) − v`(t) ≤ v` 2 ≤ (d − 1)v`(q). hF hK When q = p is prime and k = Q, we can further improve Corollary 3.14 and prove in Corollary 3.19 that the ratio of class numbers verifies hL 1 1 2 ∈ 1, , 2 . hF hK p p This answers the MathOverflow question [Bar] of Bartel, who asks for a proof of this fact that does not rely on the analytical class number formula, nor on any integral representation theory. Our Corollary 3.14 actually shows that a similar result holds without the assumption that k = Q or that q = p be prime. We now briefly describe the structure of the paper. Section2 contains all the technical algebraic lemmas used in the rest of the paper. The proof of Theorem 3.13, which is divided in two parts, is contained in Section 3. The proof of the 2-part follows a strategy of Walter (see [Wal77]) by combining an elementary result about abelian groups with a D-action and a simple arithmetic piece of information. On the other hand, the proof for the odd components heavily relies on cohomological class field theory and is partly inspired by the proof of Chevalley’s ambiguous class number formula. Since the extension L/k is non-abelian we cannot apply class field theory directly: however, since we are restricted to odd parts, the action of ∆ = Gal(F/k) on the cohomology of G = Gal(L/F ) can be described very explicitly as discussed in Section2. We conclude the article with Section 4 where we translate our cohomological formula in an explicit unit index (see Corollary 4.2). Acknowledgment. This paper has a long history. We started working on it in 2007, during a summer school in Iwasawa theory held at McMaster University, following a suggestion of Ralph Greenberg. At the time our main focus was on what we called “fake Zp-extensions” and the formula for dihedral extensions was simply a tool which we obtained by translating the right-hand side of (1.1) into a unit index essentially using linear algebra. We posted a version of our work (quite different from the present one) on arXiv in 2009 (see [CN09]) and it became part of the second author’s PhD thesis [Nuc09]. At that time we were confident that our techniques would easily generalize to cover other classes of extensions, so we decided to wait. Then basically nothing happened until the autumn of 2016 when the second author came across a question [Bar] posted by Alex Bartel on MathOverflow. This motivated us to revise and improve our work, and to split the original version in two articles. This is the first of the two papers, concentrating on the class number formula for finite dihedral extensions, while in the second [CN], which is still in progress, we focus on fake Zp-extensions. We are grateful to Henri Johnston for encouraging us to take up this revision task.

2. Cohomological preliminaries This section contains all the technical algebraic tools we need in the rest of the paper. It contains no arithmetic results and could be used as black-box in a ﬁrst reading. We ﬁx an odd positive integer q ≥ 3 and we let D denote the dihedral group of order 2q. We choose generators σ, ρ ∈ D such that σ2 = ρq = 1, σρ = ρ−1σ. CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 4

We set Σ := hσi,Σ0 := hσρi and G := hρi. We say that an abelian group B is uniquely 2-divisible when multiplication by 2 is an automorphism of B. As we shall see below, the D-Tate cohomology group Hb i(D,B) of a uniquely 2-divisible D-module B is the subgroup of Hb i(G, B) on which ∆ acts trivially. We find a similar cohomological description for the subgroup Hb i(G, B) on which ∆ acts as −1: the precise statement is given in Proposition 2.1. H Notation. Given a group H and an H-module B, we denote by B (resp. BH ) the maximal submodule (resp. the H maximal quotient) of B on which H acts trivially. We will sometimes refer to B (resp. BH ) as the invariants P (resp. coinvariants) of B. Moreover, let NH = h∈H h ∈ Z[H] be the norm, B[NH ] the kernel of multiplication by NH and IH the augmentation ideal of Z[H] defined as IH = Ker(Z[H] −→ Z) where the morphism is induced by h 7→ 1. Given i ∈ Z, we can attach to every H-module B the ith Tate cohomology group Hb i(H,B) of H with values i in B. Similarly, for i ≥ 0, H (H,B) (resp. Hi(H,B)) is the ith cohomology (resp. homology) group of H with values in B. For standard properties of these groups (which will be implicitly used without specific mention) the reader is referred to [CE56, Chapter XII]. If B0 is another H-module and f : B → B0 is a homomorphism of abelian groups, we say that f is H- equivariant (resp. H-antiequivariant) if f(hb) = hf(b) (resp. f(hb) = −hf(b)) for every h ∈ H and b ∈ B. Suppose now that H = {1, h} has order 2. An object B of the category of uniquely 2-divisible H-modules admits a functorial decomposition B = B+ ⊕ B− where we denote by B+ (resp. B−) the maximal submodule 1+h 1−h on which H acts trivially (resp. as −1). Such decomposition is obtained by writing b = 2 b + 2 b for b ∈ B. Then H + − (2.1) B = B = B/B = BH . Observe that, if B is a uniquely 2-divisible D-module, the same holds for BH for every subgroup H ⊆ D. Indeed, if b ∈ BH and c ∈ B is such that b = 2c, we have 2c = b = hb = 2hc for every h ∈ H. Since B is uniquely 2-divisible, this implies hc = c, i. e. c ∈ BH which means that BH is 2-divisible (and therefore uniquely 2-divisible being a submodule of B). In particular, for every i ∈ Z, the map ·2 Hb i(∆,BG) −→ Hb i(∆,BG) is an automorphism and therefore (2.2) Hb i(∆,BG) = 0 because these groups are killed by 2 = |∆|. In a similar way we can prove that if B is uniquely 2-divisible, the same holds for IH B. Indeed, we can assume that H is cyclic since IH B is generated by IhhiB for h ∈ H. Now, if h generates H, the kernel of multiplication by (1 − h) is BH which is uniquely 2-divisible. From the exact sequence

H ·(1−h) 0 −→ B −→ B −→ IH B −→ 0 we deduce that IH B must be uniquely 2-divisible. The next proposition is the key of all our cohomological computations.

Proposition 2.1. Let B be a uniquely 2-divisible D-module. Then, for every i ∈ Z, restriction induces isomorphisms Hb i(D,B) =∼ Hb i(G, B)+. Moreover, the Tate isomorphism Hb i(G, B) =∼ Hb i+2(G, B) is ∆-antiequivariant. In particular, for every i ∈ Z, there are isomorphisms Hb i(G, B)− =∼ Hb i+2(G, B)+ =∼ Hb i+2(D,B) Hb i(G, B)+ =∼ Hb i+2(G, B)− =∼ Hb i(D,B) CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 5 of abelian groups.

Proof. We start by proving the ﬁrst isomorphisms for i ∈ Z with i 6= 0, −1, i. e. for standard cohomology and i G G homology. Since H (∆,B ) = Hi(∆,B ) = 0 by (2.2), using the Hochschild–Serre spectral sequence (see, for instance, [CE56, Chapter XVI, § 6]), we deduce that restriction induces isomorphisms Hi(D,B) =∼ Hi(G, B)∆ = i + ∼ + H (G, B) and Hi(D,B) = Hi(G, B)∆ = Hi(G, B) for i ≥ 0. We now consider the case i = 0. Restriction induces a commutative diagram with exact rows

0 0 0 / NDB / H (D,B) / Hb (D,B) / 0 (2.3) + 0 + 0 + 0 / (NGB) / H (G, B) / Hb (G, B) / 0

The bottom row is exact since NGB is uniquely 2-divisible, hence ∆-cohomologically trivial as shown in (2.2). We showed above that the central vertical arrow is an isomorphism. In particular the right vertical arrow is surjective. It is also injective since restriction followed by corestriction is multiplication by 2 which is an automorphism of Hb 0(D,B). Therefore Hb 0(G, B)+ =∼ Hb 0(D,B) −1 as claimed. The proof is similar in degree −1, by writing Hb (D,B) = Ker ND : H0(D,B) → NDB and similarly for G. Finally, we discuss the ∆-antiequivariance of Tate isomorphisms. Recall that, for any i ∈ Z, the Tate isomorphism is given by the cup product with a ﬁxed generator χ of H2(G, Z): Hb i(G, B) −→ Hb i+2(G, B) x 7−→ x ∪ χ.

i As in [CE56, Chap. X, § 7], we denote by cδ the automorphism of Hb (G, B) induced by δ ∈ ∆ ). Then cδ is multiplication by −1 on H2(G, Z): indeed, since δ acts as −1 on G, it also acts as −1 on Hom(G, Q/Z). Our claim follows by considering the isomorphism of ∆-modules Hom(G, Q/Z) = H1(G, Q/Z) =∼ H2(G, Z) given by the connecting homomorphism of the exact sequence 0 −→ Z −→ Q −→ Q/Z → 0 of G-modules with trivial action. Then, by [CE56, Chap. XII, §8, (14)],

cδ(x ∪ χ) = −(cδx) ∪ χ which establishes the result. Remark 2.2. The above proposition implies in particular that the Tate cohomology of a uniquely 2-divisible D-module is periodic of period 4. Indeed, if B is such a module and i ∈ Z, we have isomorphisms Hb i(D,B) =∼ Hb i(G, B)+ =∼ Hb i+2(G, B)− =∼ Hb i+4(G, B)+ =∼ Hb i+4(D,B). In fact, one can prove that the cohomology of every D-module is periodic of period 4 (cf. [CE56, Theorem XII.11.6]).

Remark 2.3. If B is a uniquely 2-divisible D-module such that Hb j(G, B) = 0 for some j ∈ Z, then j+2i Hb (D,B) = 0 for all i ∈ Z. Indeed, we have seen in the above proposition that Hb j+2i(D,B) ⊆ Hb j+2i(G, B). On the other hand, the periodicity of Tate cohomology for the cyclic group G implies that Hb j+2i(G, B) =∼ Hb j(G, B) = 0. We now give a description of some cohomology groups of D with values in a D-module B in terms of explicit subquotients of B. We start with the following lemma. CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 6

Lemma 2.4. Let B be a D-module. Then Σ Σ0 IGB ⊆ B + B . Proof. This is shown in [Lem05, Lemma 3.4] and we recall here the details of the proof. For every b ∈ B, we have (1 − ρ)b = −(1 + σ)ρb + (1 + σρ)b. It follows that Σ Σ0 IGB = (1 − ρ)B ⊆ (1 + σ)ρ B + (1 + σρ)B = 1 + σ B + (1 + σρ)B = NΣB + NΣ0 B ⊆ B + B . 0 Let B be a D-module. Observe that BΣ = ρt(BΣ) for every t ∈ Z such that 2t ≡ −1 (mod q). Indeed, to 0 see that ρt(BΣ) ⊆ BΣ , we have to check that ρt(BΣ) is ﬁxed by σρ. If b ∈ BΣ, σρρt(b) = ρ−t−1σ(b) = ρ−t−1(b) = ρt(b).

0 0 On the other hand ρ−t(BΣ ) ⊂ BΣ since, for every b ∈ BΣ , σρ−t(b) = ρt+1σρ(b) = ρt+1(b) = ρ−t(b). Σ Σ0 In particular, a generic element b1 + b2 ∈ B + B can be written as t 0 0 t 0 Σ b1 + b2 = b1 + ρ (b ) = b1 + b − (1 − ρ )b ∈ B + IGB for any t ∈ Z such that 2t ≡ −1 (mod q) and a suitable b0 ∈ BΣ. Using Lemma 2.4, we deduce that Σ Σ0 Σ Σ0 (2.4) B + B = B + IGB = B + IGB and, in particular, Σ Σ0 − − (2.5) B + B = (IGB) Σ since Σ acts trivially on B + IGB /IGB. Lemma 2.5. Let B be a uniquely 2-divisible D-module. There are isomorphisms of abelian groups Σ Σ0 . ∼ −1 B[NG] ∩ B + B IGB = Hb (D,B) and . Σ Σ0 ∼ 1 B[NG] B + B ∩ B[NG] = H (D,B).

Proof. Consider the short exact sequence deﬁning Hb −1(G, B): −1 0 −→ IGB −→ B[NG] −→ Hb (G, B) −→ 0. Since B is uniquely 2-divisible, taking Σ-invariants and using Proposition 2.1, we get an exact sequence Σ Σ −1 (2.6) 0 −→ IGB ∩ B −→ B[NG] ∩ B −→ Hb (D,B) −→ 0. Σ Σ0 Σ Σ0 Since IGB ⊆ B + B by Lemma 2.4, we can consider the quotient B[NG] ∩ (B + B ) IGB, as well as the map Σ Σ Σ Σ0 B[NG] ∩ B IGB ∩ B −→ B[NG] ∩ B + B IGB Σ Σ Σ0 induced by the inclusion B[NG] ∩ B ⊆ B[NG] ∩ B + B . The above map is clearly injective and is also surjective since Σ Σ Σ0 B + IGB = B + B by (2.4). This, together with the exact sequence in (2.6), proves the ﬁrst isomorphism of the lemma. For the second isomorphism, note that, by Proposition 2.1, we have an isomorphism of abelian groups 1 ∼ −1 −1 H (D,B) = Hb (G, B)/Hb (D,B). CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 7

Observe that the proof of Lemma 2.5 actually shows that −1 + Σ Σ0 . −1 − . Σ Σ0 (2.7) Hb (G, B) = B[NG] ∩ B + B IGB and Hb (G, B) = B[NG] B + B ∩ B[NG] . Another consequence of Lemma 2.4 is the following. Lemma 2.6. Let B be a uniquely q-divisible ﬁnite D-module. There is a direct-sum decomposition 0 B/BD =∼ BG/BD ⊕ BΣ + BΣ /BD which implies, moding out by BG/BD, 0 B/BG =∼ BΣ/BD ⊕ BΣ /BD.

Proof. Since B is q-divisible, the groups Hb 0(G, B) and Hb −1(G, B) are trivial. Therefore the tautological exact sequence 0 −→ B[NG] −→ B −→ NGB −→ 0 can be rewritten as NG G 0 −→ IGB −→ B −→ B −→ 0. Composing the inclusion BG ,→ B with multiplication by q−1 splits the above exact sequence and we find ∼ G (2.8) B = B ⊕ IGB. We now claim that D Σ Σ0 IGB + B = B + B : Σ Σ0 Σ indeed, equation (2.4) shows that IGB ⊆ B + B and one inclusion becomes clear. For the other, let b ∈ B : −1 we can write b = (1 − ρ)b1 + q NGb for some b1 ∈ B thanks to the splitting (2.8). Using that b is fixed by Σ −1 D D by assumption, we deduce that q NGb ∈ B , and thus b ∈ IGB + B . An analogous argument shows that Σ0 D B ⊆ IGB + B giving the other inclusion: in particular, D D Σ Σ0 D (2.9) IGB + B /B = B + B /B . D G Using once more the direct-sum decomposition (2.8), we see that IGB ∩ B ⊆ IGB ∩ B = 0. This implies G D G D G D D G D D D that (B ⊕ IGB)/B = B /B ⊕ IGB = B /B ⊕ IGB/(IGB ∩ B ) = B /B ⊕ (IGB + B )/B and we can combine (2.8) with (2.9) to deduce D ∼ G D G D D D G D Σ Σ0 D (2.10) B/B = B ⊕ IGB /B = B /B ⊕ IGB + B /B = B /B ⊕ B + B /B which is the first decomposition. To derive the second, moding out by BG/BD we obtain 0 0 B/BG =∼ BΣ + BΣ /BD = BΣ/BD + BΣ /BD and we are left with the proof that the above sum is direct. Now, an element in the intersection (BΣ/BD) ∩ 0 0 (BΣ /BD) is the class mod BD of an element x ∈ BΣ ∩ BΣ : since Σ and Σ0 generate D, this class is fixed by the whole D and it is therefore trivial. This finishes the proof. 1 To use the above results in the next sections, we systematically tensor with Z 2 the module of interest. When no ambiguity is possible, if f : B → B0 is a homomorphism of abelian groups we will still denote by f 1 1 0 1 the map f 2 : B 2 → B 2 . We constantly use that, for every D-module B and every subgroup H ⊆ D, there are isomorphisms i 1 ∼ i 1 Hb H,B 2 = Hb (H,B) 2 for all i ∈ Z. This is quite straightforward, but we sketch an argument, writing 0 1 Λ = Z[H] and Λ = Z[H] 2 . 1 0 1 1 1 1 The fact that Z 2 is Z-flat implies that H (H,B) 2 = HomΛ(Z,B) 2 coincides with HomΛ0 (Z 2 ,B 2 ). 0 1 1 On the other hand, a Λ -linear homomorphism from Z 2 to B 2 is uniquely determined by its value on 1, as is CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 8

1 1 1 1 0 1 every Λ-homomorphism from Z to B 2 , showing that HomΛ0 (Z 2 ,B 2 ) = HomΛ(Z,B 2 ) = H (H,B 2 ). 0 1 0 1 Therefore the right-derived functors of the functor F1 : B 7→ H (H,B 2 ) and of F2 : B 7→ H (H,B) 2 i i 1 i i 1 coincide; for the ﬁrst, they are R F1(B) = H (H,B 2 ) and for the second, they are R F2(B) = H (H,B) 2 , for i ≥ 0: this follows from a computation with the Grothendieck spectral sequence of a composition of functors, 1 using that −⊗Z 2 is acyclic by assumption. It follows that the two groups are isomorphic. A similar argument is valid for H-homology. The remaining cases of Tate cohomology in degrees −1, 0 can be treated directly by writing the Tate cohomology groups as a kernel and a cokernel, respectively (cf. [CE56, Chap. XII, § 2]). In the next lemma, for an abelian group B and a natural number n, we set B(n) = {b ∈ B : nb = 0}.

Lemma 2.7. Let B be D-module such that B(q) = B(q)G. Then

−1 −1 G G Ker Hb (G, B(q)) −→ Hb (G, B) = IGB ∩ B(q) = IGB ∩ B(q) = IGB ∩ B . If, moreover, B is uniquely 2-divisible then

−1 −1 D D Ker Hb (D,B(q)) −→ Hb (D,B) = IGB ∩ B(q) = IGB ∩ B . Proof. Consider the exact sequence

·q 0 −→ B(q) −→ B −→ qB −→ 0.

0 Since B(q) has trivial action of G by assumption, H (G, B(q)) = B(q) and IG(B(q)) = 0. Moreover, NG acts as multiplication by q on B(q), and so is the trivial map, hence H0(G, B(q)) = Hb 0(G, B(q)) = B(q). Similarly, we have Hb −1(G, B(q)) = B(q). Thus we obtain −1 −1 Ker Hb (G, B(q)) −→ Hb (G, B) = Ker B(q) −→ B[NG]/IGB = IGB ∩ B(q) which is the first equality of the statement. The second is clearly equivalent to the first since B(q)G = B(q) G G by hypothesis. As for the third, there is an obvious inclusion IGB ∩ B(q) ⊆ IGB ∩ B . To prove the reverse G G inclusion it is enough to observe that every x ∈ IGB∩B verifies qx = NGx (because x ∈ B ) but also NGx = 0 G G (since x ∈ IGB ⊆ B[NG]). Thus we have x ∈ IGB ∩ B ∩ B(q) = IGB ∩ B(q) . When B is uniquely 2-divisible, by taking plus parts in the above equation and applying Proposition 2.1 we obtain the second claim. We conclude this section with two lemmas which give some relations between the cohomology of D and that of its subgroups, without the assumption that the underlying module is uniquely 2-divisible.

Lemma 2.8. Let H ⊆ D be a subgroup, let B be an H-module and let j ∈ Z be an integer. Suppose that, j i for every prime p, Hb (Hp,B) = 0 where Hp is a p-Sylow subgroup of H. Then Hb (H,B) = 0 for every i ≡ j (mod 2).

i Proof. Under our assumptions, for every prime p, Hb (Hp,B) is trivial for every i ≡ j (mod 2) since p-Sylow subgroups of H are cyclic and the Tate cohomology of cyclic groups is periodic of period 2. Since the restriction map i i Hb (H,B) −→ Hb (Hp,B) i i is injective on the p-primary component of Hb (H,B) we deduce that Hb (H,B) = 0 for every i ≡ j (mod 2).

Lemma 2.9. Let B a D-module. Then, for every i ∈ Z, the restriction D i i resΣ : Hb (D,B) −→ Hb (Σ,B) maps the 2-primary component of Hb i(D,B) isomorphically onto Hb i(Σ,B). CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 9

Proof. As observed in the proof of Lemma 2.8, we only need to show surjectivity. The image of the 2-component is the subset of stable elements of Hb i(Σ,B) (see [CE56, Chapter XII, Theorem 10.1]). Recall that an element a ∈ Hb i(Σ,B) is called stable if, for every τ ∈ D, Σ τΣτ −1 resτΣτ −1∩Σ(a) = resτΣτ −1∩Σ cτ (a) i i −1 where cτ denotes the map Hb (Σ,B) → Hb (τΣτ ,B) induced by conjugation by τ. When τ∈ / Σ, then τΣτ −1 ∩ Σ is trivial and the above equality holds for each a ∈ Hb i(Σ,B). When τ ∈ Σ, then τΣτ −1 = Σ and the two restriction maps appearing in the above equality are in fact the identity map. The same holds for cτ i (see [CE56, Chapter XII, § 8, (5)]), proving that every element in Hb (Σ,B) is stable. 3. Class number formula We now come to the proof of the odd part of the class number formula. Still under the notation introduced in the previous section, we consider the following arithmetic setting. Let k be a number field and let L/k be a Galois extension whose Galois group is isomorphic to D. We fix such an isomorphism and we simply write D = Gal(L/k). Let F/k be the subextension of L/k fixed by G and let K/k (resp. K0/k) be the subextension of L/k fixed by Σ (resp. Σ0). In particular F/k is quadratic and we set ∆ := Gal(F/k). We begin by introducing a four-term exact sequence involving the class groups of L and K. By looking at the orders of groups in this sequence, we already get a formula involving the odd components of class numbers of L and K. The rest of the section is devoted to expressing the remaining factors in terms of the odd components of the class numbers of F and k and the orders of the cohomology group of units of L. As a corollary of the formula, using the value of the Herbrand quotient of units for the extension L/F , we easily obtain bounds on the ratio of class numbers of subfields of L.

Notation. For a number ﬁeld M, we write ClM for the class group of M. For an extension of number ﬁelds

M2/M1, ιM2/M1 : ClM1 → ClM2 and NM2/M1 : ClM2 → ClM1 denote the map induced by extending ideals from M1 to M2 and by taking the norm of ideals from M2 to M1, respectively. Consider the exact sequence η (3.1) 1 → Ker η −→ ClK ⊕ ClK0 −→ ClL −→ ClL/ Im η → 1 where the map η is deﬁned via the formula 0 0 0 (3.2) η (c, c ) = ιL/K (c)ιL/K0 (c ) for c ∈ ClK , c ∈ ClK0 . 1 Tensoring with Z 2 , we immediately obtain the formula 1 1 1 1 2 ClL 2 : Im η 2 (3.3) ClL 2 = ClK 2 1 . Ker η 2 1 1 1 Observe also that the map ιL/K 2 : ClK 2 → ClM 2 is injective, since NL/K ◦ ιL/K is multiplication by 2 = [L : K] on ClK . Remark 3.1. Let B be an abelian group and B0 ⊆ B a subgroup of B. Let f : B → C be a homomorphism of abelian groups. Then there is an exact sequence 0 −→ Ker f/Ker f ∩ B0 −→ B/B0 −→ f(B)/f(B0) −→ 0. It follows in particular that (3.4) (B : B0) = (f(B): f(B0)) · (Ker f : Ker f ∩ B0), meaning that if two of the above indices are ﬁnite, so is the third and the above equality holds.

For a group H and a H-module B, we denote by bhi(H,B) (resp. hi(H,B)) the order of Hb i(H,B) (resp. the order of Hi(H,B)), whenever this is deﬁned. CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 10

Proposition 3.2. We have an equality

1 G ClL 1 1 2 ClL : Im η = 2 2 −1 1 1 bh D, ClL 2 · NDClL 2 and an isomorphism 1 ∼ 0 1 Ker η 2 = H D, ClL 2 . In particular 1 G ClL 1 1 2 2 ClL = ClK . 2 2 0 1 −1 1 1 h D, ClL 2 bh D, ClL 2 · NDClL 2 0 Proof. Observe that, using equation (3.4) applied to f = NG, B = ClL, B = Im η, we get

(ClL : Im η) = (NGClL : NG(Im η)) · (ClL[NG] : Im η ∩ ClL[NG]). Now observe that

Im η = ιL/K (ClK )ιL/K0 (ClK0 ) and 0 1 Σ 1 1 Σ 1 (3.5) ClL 2 = ιL/K ClK 2 , ClL 2 = ιL/K0 ClK0 2 so that 0 1 1 Σ 1 Σ Im η 2 = ClL 2 ClL 2 . Therefore, using Lemma 2.5, we deduce that 1 1 1 1 1 1 ClL 2 : Im η 2 = NGClL 2 : NDClL 2 · h D, ClL 2 , because 1 1 1 Σ 1 1 NDClL 2 = NG NΣClL 2 = NG ClL 2 = NG ιL/K ClK 2 = NG ιL/K0 ClK0 2 , 0 1 since Hb (Σ, ClL 2 ) = 0. Now observe that

1 G ClL 1 2 NGClL = . 2 0 1 bh G, ClL 2 The first equality of the proposition follows, since h1D, Cl 1 1 L 2 = h1G, Cl 1 −1 1 L 2 bh D, ClL 2 0 1 1 1 1 by Proposition 2.1 and because bh (G, ClL 2 ) = h (G, ClL 2 ) since ClL 2 is finite. As for the second statement of the proposition, let θ be the map 1 0 1 θ : Ker η 2 −→ H D, ClL 2 0 0 −1 0 defined by θ(c, c ) = ιL/K (c). Since ιL/K (c) = ιL/K0 (c ) in ClL by definition of η, both Σ and Σ fix ιL/K (c), 0 so the map θ takes values in H (D, ClL). We claim that it is an isomorphism. First, θ is injective because 1 1 ιL/K : ClK 2 → ClL 2 is injective, as observed after equation (3.3). To check surjectivity, note that, since 0 1 0 0 every b ∈ H (D, ClL 2 ) is fixed by both Σ and Σ , we can write such b as b = ιL/K (c) = ιL/K0 (c ) for some 1 0 1 0 −1 1 0 −1 c ∈ ClK 2 , c ∈ ClK0 2 , using (3.5). In particular (c, (c ) ) ∈ Ker η 2 and b = θ(c, (c ) ), which shows the claimed isomorphism and hence the second statement of the proposition. The last formula of the proposition comes from (3.3) using the first two assertions. CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 11

1 × 1 We now want to translate cohomology groups of ClL 2 into cohomology groups of OL 2 . To achieve this we make intensive use of the following commutative diagram of Galois modules with exact rows and columns: 1 1 1

× × 1 / OM / M / PM / 1

(3.6) × 1 / UM / AM / IM / 1

1 / QM / CM / ClM / 1

1 1 1

Here we use the following notation for a number field M: OM is the ring of integers, IM is the group of fractional × ideals, PM is the group of principal ideals, AM is the idèlesgroup, CM is the idèlesclass group, UM is the group of idèlesof valuation 0 at every finite place and QM is defined by the exactness of the diagram. We refer the reader to [Tat67] for the class field theory results that we use. Proposition 3.3. Let H ⊆ D be any subgroup and j be an odd integer. Then j j j × j × Hb (H,IL) = Hb (H, CL) = Hb (H,L ) = Hb (H, AL ) = 0. 1 × × Proof. By Lemma 2.8, we only need to prove that H (H,B) = 0 where B is any of the modules IL, CL,L , AL × × and H is cyclic. This is Hilbert’s Theorem 90 when B = L , AL and follows from class field theory when B = CL. For B = IL, let M ⊆ L be the subfield of L fixed by H. There is an H-equivariant isomorphism ∼ M (3.7) IL = Z[H/H(L)] l⊂OM where l runs over the primes of M and H(L) denotes the decomposition group in L/M of a fixed prime L of L above l. Hence, by Shapiro’s lemma, 1 ∼ M 1 H (H,IL) = H (H(L), Z). l⊂OM 1 The statement follows, since H (H(L), Z) = Hom(H(L), Z) = 0. We can now move towards the proof of our main result, contained in Theorem 3.13. Proposition 3.4. We have isomorphisms 0 1 ∼ 1 1 −1 1 ∼ 0 1 Hb D, ClL 2 = H D,QL 2 and Hb D, ClL 2 = Hb D,QL 2 . Proof. From the long exact D-Tate cohomology sequence of the bottom row of Diagram 3.6, we get an exact sequence −1 0 0 0 1 (3.8) Hb (D, ClL) −→ Hb (D,QL) −→ Hb (D, CL) −→ Hb (D, ClL) −→ Hb (D,QL). The rightmost map is surjective and the leftmost map is injective thanks to Proposition 3.3. Tensoring with 1 Z 2 the middle term of (3.8) becomes trivial, since class field theory gives an isomorphism 0 ∼= ab ∼ (3.9) Hb (D, CL) −→ D = Z/2Z. CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 12

The following proposition is the dihedral analogue of Chevalley’s Ambiguous Class Number Formula (which holds for cyclic groups). Our proof is inspired by the cohomological proof of Chevalley’s formula (see [Lan90, Lemma 4.1]).

Proposition 3.5. Let H ⊆ D be a subgroup and let M be the subﬁeld of L ﬁxed by H. There is a long exact sequence

1 × 1 0 1 0 −→ Ker ιL/M −→ H (H, OL ) −→ H (H, UL) −→ H (H, ClL)/ιL/M (ClM ) −→ H (H,PL) −→ 0. In particular h1H, U 1 h1H,P 1 h1H, O× 1 L 2 L 2 = L 2 . 0 1 1 h H, ClL 2 ClM 2 Proof. From the rightmost column of Diagram 3.6 we get a commutative diagram with exact rows

0 / PM / IM / ClM / 0 _ _ ιL/M 0 0 0 1 0 / H (H,PL) / H (H,IL) / H (H, ClL) / H (H,PL) / 0.

The exactness of the bottom row follows from Proposition 3.3. Applying the snake lemma we get the exact sequence

0 0 0 −→ Ker ιL/M −→ H (H,PL)/PM −→ H (H,IL)/IM −→ 0 1 (3.10) −→ H (H, ClL)/ιL/M (ClM ) −→ H (H,PL) −→ 0. Now, from the upper row of Diagram 3.6, we get the exact sequence

× × 0 1 × 0 −→ M /OM = PM −→ H (H,PL) −→ H (H, OL ) −→ 0. We thus obtain an isomorphism 0 ∼ 1 × (3.11) H (H,PL)/PM = H (H, OL ).

1 × As for the middle term of (3.10), consider the central row of Diagram 3.6. Using that H (H, AL ) = 0 by Proposition 3.3, we get an exact sequence

× 0 1 0 −→ M /UM = IM −→ H (H,IL) −→ H (H, UL) −→ 0. This gives an isomorphism 0 ∼ 1 (3.12) H (H,IL)/IM = H (H, UL). Plugging (3.11) and (3.12) in the exact sequence (3.10), we obtain the exact sequence of the statement. The formula for the orders follows since

|Ker ιL/M | · |ιL/M (ClM )| = |ClM |.

1 1 Remark 3.6. Observe that the norm NL/k : CL 2 → Ck 2 is surjective (for this remark only, we extend the notation NL/K for the corresponding morphism on id`eleclass groups and similarly for ιL/K ). Indeed, Hilbert D 90 implies that CL = ιL/k(Ck) and in particular ιL/k : Ck → CL is injective . Since NDCL = ιL/k(NL/k(CL)), we deduce that 0 D ∼ Hb (D, CL) = CL /NDCL = ιL/k(Ck)/ιL/k(NL/k(CL)) = Ck/NL/k(CL). CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 13

0 1 1 1 In particular, since Hb (D, CL 2 ) = 0 by class ﬁeld theory (see (3.9)), we ﬁnd that Ck 2 = NL/k(CL 2 ). Looking at the commutative diagram with exact rows

CL / ClL / 0

NL/k NL/k Ck / Clk / 0

1 1 we deduce that NL/k : ClL 2 → Clk 2 must be surjective too. Moreover, since NDClL = ιL/k(NL/k(ClL)), we ﬁnd that

1 1 (3.13) NDClL 2 = ιL/k Clk 2 and in particular 0 1 0 1 1 Hb D, ClL 2 = H D, ClL 2 /ιL/k Clk 2 . Proposition 3.7. Let H ⊆ D be any subgroup and let M be the subﬁeld of L ﬁxed by H. There is an exact sequence

−1 0 × 0 0 0 −→ Hb (H,PL) −→ Hb (H, OL ) −→ Hb (H, UL) −→ Hb (H,QL) −→ Ker ιL/M −→ 0. In particular 0 1 −1 1 0 × 1 bh D, UL bh D,PL bh D, O 2 2 = L 2 . −1 1 1 |Cl 1 | bh D, ClL 2 · NDClL 2 k 2 Proof. From the long exact sequence of Tate cohomology of the left-hand column of Diagram 3.6, we get an exact sequence

0 × 0 0 1 × 1 Hb (H, OL ) −→ Hb (H, UL) −→ Hb (H,QL) −→ H (H, OL ) −→ H (H, UL). By Proposition 3.5 we know that 1 × 1 Ker H (H, OL ) → H (H, UL) = Ker ιL/M . Now observe that 0 × 0 0 × 0 × (3.14) Ker Hb (H, OL ) → Hb (H, UL) = Ker Hb (H, OL ) → Hb (H,L ) . This follows from the commutative diagram

Hb 0(H, O×) / Hb 0(H,L×) L _

0 0 × Hb (H, UL) / Hb (H, AL ) where the injectivity of bottom horizontal and right vertical arrows follow from Proposition 3.3. The exact sequence of the statement follows since

0 × 0 × −1 Ker Hb (H, OL ) → Hb (H,L ) = Hb (H,PL) by Proposition 3.3. The last assertion follows by Proposition 3.4 and (3.13).

Remark 3.8. Concerning the cokernel of ιL/M , similar techniques as in the above proof allow to give a cohomolog- 1 ical interpretation. Taking H-cohomology of the bottom row of diagram (3.6) and recalling that H (H, CL) = 0, CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 14 we get a commutative diagram

0 / QM / CM / ClM / 0

ιL/M H H H 1 0 / QL / CL / ClL / H (H,QL) / 0

Applying the snake lemma to the above diagram, we obtain an isomorphism

∼ 1 (3.15) Coker ιL/M = H (H,QL) which will be useful when considering the 2-part of our main result.

We now give an explicit description of the D-cohomology of UL, using local class ﬁeld theory. We need the following notation.

Notation. If M is a subfield of L, we denote by T (L/M) the set of primes in M which ramify in L/M. We also ` ` write T (L/k) = Ts(L/k) Tr(L/k) Ti(L/k) where the sets Ts(L/k),Tr(L/k) and Ti(L/k) contain the primes in T (L/k) which, respectively, split, ramify and remain inert in the subextension F/k. For each prime l of k we denote by el and fl its ramification index and inertia degree, respectively, in L/k. For a prime L of L, we × denote by D(L) the decomposition group of L in L/k. Given a prime r of M ⊆ L, we use the notation OM,r instead of O× to denote the units of the local field M , in order to lighten typography. Mr r We start with a lemma describing the cohomology of local units.

Lemma 3.9. Let l ∈ T (L/k) be a prime and let L ⊆ OL be a prime in L above l. 0 × a) If l ∈ Ti(L/k), then fl = 2 and Hb (D(L), OL,L) = 0; 0 × ∼ b) If l ∈ Tr(L/k), then fl = 1 and Hb (D(L), OL,L) = Z/2; 0 × ∼ c) If l ∈ Ts(L/k), then Hb (D(L), OL,L) = Z/el. 1 × ∼ Moreover, for any l ∈ T (L/k), we have Hb (D(L), OL,L) = Z/el. Proof. Consider the following diagram with exact rows

× × 0 / O / L / Z / 0 L,L L _

vL ·fl × × vl 0 / Ok,l / kl / Z / 0 where the left and central vertical maps are the norms and vl and vL are the l-adic and the L-adic valuation, respectively. The snake lemma, together with local class ﬁeld theory, gives an exact sequence

0 × ab 0 −→ Hb (D(L), OL,L) −→ D(L) −→ Z/fl −→ 0.

If l ∈ Ti(L/k) ∪ Tr(L/k), then D(L) has order divisible by 2 so it must be dihedral or cyclic of order 2. In both ab ∼ cases D(L) = Z/2 and therefore fl divides 2, by the above exact sequence. When l ∈ Ti(L/k), we must have 0 × fl = 2 and Hb (D(L), OL,L) = 0. When l ∈ Tr(L/k), we deduce fl = 1, because otherwise flel = |D(L)| would 0 × ∼ ab be divisible by 4, and so Hb (D(L), OL,L) = Z/2. Finally, if l ∈ Ts(L/k), then D(L) = D(L) is cyclic and, 0 × ∼ since flel = |D(L)|, we deduce that Hb (D(L), OL,L) = Z/el. CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 15

To prove the assertion on H1, apply the snake lemma to the commutative diagram with exact rows

× × vl 0 / Ok,l / kl / Z / 0 _

·el × D(L) ×D(L) vL 1 × 1 × 0 / OL,L / LL / Z / H (D(L), OL,L) / H (D(L),LL ) = 0 to deduce an isomorphism 1 × ∼ H (D(L), OL,L) = Z/el. Proposition 3.10. With the notation introduced before Lemma 3.9, there are isomorphisms 0 ∼ M M Hb (D, UL) = (Z/el) ⊕ (Z/2)

l∈Ts(L/k) l∈Tr (L/k) and 1 ∼ M M M Hb (D, UL) = (Z/el) ⊕ (Z/el) ⊕ (Z/el).

l∈Ts(L/k) l∈Ti(L/k) l∈Tr (L/k) Proof. For every prime l of k we fix a prime L | l in L. By combining [Tat67, Proposition 7.2] with the fact that local units are cohomologically trivial in unramified extension (see [Ser67, Chapter 1, Proposition 1]) we find i ∼ M i × ∼ M i × (3.16) Hb (D, UL) = Hb (D(L), OL,L) = Hb (D(L), OL,L) for all i ∈ Z. l⊆Ok l∈T (L/k) The proposition then follows from Lemma 3.9. Recall that, by Proposition 2.1, we have an isomorphism of ∆-modules 1 1 ∼ 1 1 −1 1 H G, UL 2 = H D, UL 2 ⊕ Hb D, UL 2 . 1 1 The following corollary gives another decomposition of H (G, UL 2 ), only as abelian groups, which turns out to be more useful than the above in the proof of Theorem 3.13. Corollary 3.11. There is an isomorphism of abelian groups 1 1 ∼ 1 1 0 1 H (G, UL 2 ) = H (D, UL 2 ) ⊕ Hb (D, UL 2 ). Proof. The same arguments leading to (3.16) give an isomorphism 1 ∼ M 1 × H (G, UL) = H (G(L), OL,L)

˜l⊂OF where for each prime ˜l in F we have fixed a prime L in L above ˜l. Therefore it is enough to show that, for each prime l of k, M 1 × 1 ∼ 1 × 1 0 × 1 (3.17) H (G(L), OL,L 2 ) = H (D(L), OL,L 2 ) ⊕ Hb (D(L), OL,L 2 ) . ˜l|l Arguing in a similar way as we did in the final part of the proof of Lemma 3.9 we get an isomorphism 1 × ∼ H (G(L), OL,L) = Z/e˜l ˜ where e˜l is the ramification index of l in L/F . Now if l ∈ Ts(L/k) ∪ Ti(L/k), then e˜l = el, while, if l ∈ Tr(L/k), then e˜l = el/2. Thus  /e ⊕ /e if l ∈ T (L/k) Z l Z l s M 1 × 1 ∼ (3.18) H (G(L), OL,L 2 ) = Z/el if l ∈ Ti(L/k) ˜l|l  el Z/ 2 if l ∈ Tr(L/k) CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 16

Therefore (3.17) follows by Lemma 3.9. We now come to the main result of this section. We begin with a result taken from [Wal77], which we reprove here for completeness; to state it, we introduce the notation B2 for the 2-primary component of an abelian group B: Proposition 3.12 (Walter, cf. [Wal77, Theorem 5.3]). There is an isomorphism ∼ 0 ClL/ClF 2 = ClK /Clk 2 ⊕ ClK /Clk 2. Proof. We ﬁrst claim that extending ideals to L induce isomorphisms Σ D Σ0 D ∼ ∼ 0 (3.19) ClL /ClL 2 = ClK /Clk 2 and ClL /ClL 2 = ClK /Clk 2.

We prove this for ιL/K , the proof for ιL/K0 being analogous. Consider the commutative diagram / / / / 0 (Clk)2 (ClK )2 ClK /Clk 2 0

(3.20) ιL/k ιL/K / D / Σ / Σ D / 0 ClL 2 ClL 2 ClL /ClL 2 0 Thanks to Proposition 3.5 we have isomorphisms ∼ 1 × 1 (Ker ιL/k)2 = Ker H (D, OL )2 −→ H (D, UL)2 and ∼ 1 × 1 (Ker ιL/K )2 = Ker H (Σ, OL )2 −→ H (Σ, UL)2 and Lemma 2.9 shows that these two kernels are isomorphic. Similarly, the isomorphisms ∼ 1 ∼ 1 (Coker ιL/k)2 = H (D,QL)2 and (Coker ιL/K )2 = H (Σ,QL)2 discussed in (3.15), again combined with Lemma 2.9, show that the cokernels are isomorphic. Applying the snake lemma to the commutative diagram (3.20), we establish (3.19). We now apply Lemma 2.6 to the module B = (ClL)2: we ﬁnd an isomorphism G ∼ Σ D Σ0 D ClL/ClL 2 = ClL /ClL 2 ⊕ ClL /ClL 2 which, in light of (3.19) we rewrite as G ∼ 0 ClL/ClL 2 = ClK /Clk 2 ⊕ ClK /Clk 2. G ∼ We conclude the proof by observing that the map ιL/F induces an isomorphism (ClL )2 = (ClF )2 since 2 is coprime to q = [L : F ].

For a number field M, we let hM = |ClM | denote the class number of M. Theorem 3.13. Let L/k be a Galois extension of number fields whose Galois group D = Gal(L/k) is dihedral of order 2q with q odd. Let Σ ⊂ D be a subgroup of order 2 and G ⊂ D be the subgroup of order q. Set K = LD and F = LG. Then 2 0 × 1 0 × −1 × hLh bh D, O bh (D, O ) bh (Σ, O ) k = L 2 = L L . 2 × 1 × × hF h −1 −1 0 K bh D, OL 2 bh (D, OL ) bh (Σ, OL ) Proof. We start by proving the first equality. Concerning the 2-part, this an immediate consequence of Propo- × 1 sition 3.12, observing that the cohomology groups with values in the uniquely 2-divisible module OL 2 have trivial 2-components. CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 17

We now focus on the odd part. From Proposition 3.2, we have

1 G ClL 1 1 2 2 (3.21) ClL = ClK . 2 2 0 1 −1 1 1 h D, ClL 2 bh D, ClL 2 NDClL 2 Note that Proposition 3.5 with H = G gives

1 1 1 1 G h G, U h G, P (3.22) Cl 1 = Cl 1 · L 2 L 2 , L 2 F 2 1 × 1 h G, OL 2

1 1 1 1 −1 1 and we can also replace the term h (G, PL 2 ) by h (D,PL 2 )bh (D,PL 2 ) in view of Proposition 2.1. By Corollary 3.11 we have

1 1 1 1 0 1 (3.23) h G, UL 2 = h D, UL 2 bh D, UL 2 . Thus (3.21) becomes

G 1 Cl 1 ClL L 2 2 = 1 2 h0D, Cl 1 h−1D, Cl 1 · N Cl 1 ClK 2 L 2 b L 2 D L 2 1 1 1 1 −1 1 1 h G, UL 2 h D,PL 2 bh D,PL 2 (by (3.22)) = ClF · 2 1 × 0 1 −1 1 1 h G, OL h D, ClL 2 bh D, ClL 2 · NDClL 2 1 1 1 1 0 1 −1 1 1 h D, UL 2 h D,PL 2 bh D, UL 2 bh D,PL 2 (by (3.23)) = ClF · · 2 h1G, O× 1 h0D, Cl 1 −1 1 1 L 2 L 2 bh D, ClL 2 · NDClL 2 1 1 1 1 0 × 1 h D, UL h D,PL bh D, O (by Proposition 3.7) = Cl 1 · 2 2 · L 2 F 2 1 × 1 0 1 1 h G, OL 2 h D, ClL 2 Clk 2 1 1 × 1 0 × 1 ClF h D, O bh D, O (by Proposition 3.5 with H = D) = 2 · L 2 · L 2 1 × 1 1 1 h G, OL 2 Clk 2 Clk 2 1 0 × 1 ClF bh D, O (by Proposition 2.1) = 2 · L 2 . 1 2 h−1D, O× 1 Clk 2 b L 2 This completes the proof of the ﬁrst equality of the theorem. To obtain the second equality in the statement, observe that i × 1 ∼ i × i × Hb D, OL 2 = Hb (D, OL )/Hb (D, OL )2

i × i × where Hb (D, OL )2 is the 2-primary component of Hb (D, OL ). We conclude by Lemma 2.9.

Now we recall the deﬁnition of the Herbrand quotient of a G-module B as

bh0(G, B) h(G, B) = h1(G, B) whenever the quotient is well-deﬁned. The following result will be crucial for our computations. Herbrand quotients of units (see [Lan94, Corollary 2 of Theorem 1, Chapter IX]). The Herbrand quotient × of the G-module OL is 1 h(G, O×) = . L q CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 18

Using the value of the Herbrand quotient of units, we can easily obtain bounds on the value of the ratio of class numbers of subﬁelds of L. For a number ﬁeld M we set ( 0 if µM (q) is trivial, βM (q) = 1 otherwise where µM denotes the group of roots of unity of Mand µM (n) is the subgroup of roots of unity of M killed by n ≥ 1. In what follows, for a prime number `, v` denotes the `-adic valuation. Corollary 3.14. Let notation be as in Theorem 3.13. For every prime `, the following bounds hold 2 hLhk −av`(q) ≤ v` 2 ≤ bv`(q) hF hK × × where a = rkZ(OF )+βF (q)+1 and b = rkZ(Ok )+βk(q). In particular, if F is totally real of degree [F : Q] = 2d, then 2 hLhk −2dv`(q) ≤ v` 2 ≤ (d − 1)v`(q). hF hK Proof. Thanks to Theorem 3.13, we only have to prove that the prescribed bounds hold for the ratio

0 × 1 −1 × 1 bh D, OL 2 /bh D, OL 2 . 0 × 1 × 1 × 1 × 1 Observe that Hb (G, OL 2 ) = OF 2 /NGOL 2 is a quotient of OF 2 annihilated by q. In particular its rk (O×)+β (q) order divides q Z F F . Using the value of the Herbrand quotient of units, we therefore have

−1 × 1 −1 × 1 0 × 1 v` bh D, OL 2 ≤ v` bh G, OL 2 = v`(q) + v` bh (G, OL 2 ) × ≤ v`(q) + βF (q) + rkZ(OF ) v`(q) = av`(q). We deduce that 0 × 1 ! bh D, OL 2 0 × 1 −1 × 1 −1 × 1 v` = v` bh D, O − v` bh D, O ≥ −v` bh D, O ≥ −av`(q). −1 × 1 L 2 L 2 L 2 bh D, OL 2 0 × 1 × 1 × 1 × 1 Similarly, Hb (D, OL 2 ) = Ok 2 /NDOL 2 is a quotient of Ok 2 annihilated by q. In particular its order rk (O×)+β (q) divides q Z k k and therefore

0 × rk (O×)+β (q) 1 Z k k v` bh D, OL 2 ≤ v`(q ) = bv`(q). We deduce that 0 × 1 ! bh D, OL 2 0 × 1 −1 × 1 0 × 1 v` = v` bh D, O − v` bh D, O ≤ v` bh D, O ≤ bv`(q). −1 × 1 L 2 L 2 L 2 bh D, OL 2 When F is totally real, it contains no roots of unity diﬀerent from ±1 so that a = 2d and b = (d − 1). Remark 3.15. It would be interesting to understand to what extent the bounds of Corollary 3.14 are optimal. To be more precise, for given a, b ∈ Z, consider the set

S(a, b) = {c ∈ Q such that −av`(q) ≤ v`(c) ≤ bv`(q), for every prime `}.

One could ask whether, for every a ≥ 1 and b ≥ 0 and every c ∈ S(a, b), there exists a number field kc and a quadratic extension Fc/kc satisfying a = rk (O× ) + β (q) + 1 and b = rk (O× ) + β (q) Z Fc Fc Z kc kc CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 19 and a dihedral extensions Lc/kc of degree 2q such that Fc ⊆ Lc and 2 hL h c kc = c, h h2 Fc Kc where Kc/kc is a subextension of Lc/kc of degree q. The bounds of Corollary 3.14 hold in full generality, i. e. for an arbitrary base field k. When k is totally real and F is a CM field, these bounds can be improved, as we show in Corollary 3.17, which generalizes [Bar12, Example 6.3]. We start with the following lemma, whose first statement is well-known. Lemma 3.16. We have

µF = µL and µk = µK . × × Moreover, the intersection IGOL ∩ OF is a ﬁnite cyclic group and there is a ∆-antiequivariant isomorphism × q × × q ∼ × × (OL ) ∩ OF /(OF ) = IGOL ∩ OF . In particular we have isomorphisms of abelian groups

∓ ± × q × × q ∼ × × (OL ) ∩ OF /(OF ) = IGOL ∩ OF .

Proof. Let ζ be a root of unity of L, and set M = k(ζ). Then M/k is an abelian extension of k contained in L and therefore M ⊆ F . In particular µF = µL and, taking Σ-invariants, µk = µK , so that the ﬁrst statement is proved. To prove the rest of the lemma, consider the Kummer sequence

q × (·) × q (3.24) 1 −→ µL(q) −→ OL −→ (OL ) −→ 1. Taking G-cohomology, we obtain a ∆-equivariant isomorphism

× q × × q 1 1 × (3.25) (OL ) ∩ OF /(OF ) = Ker H (G, µL(q)) → H (G, OL ) . Using Proposition 2.1 we deduce a ∆-antiequivariant isomorphism × q × × q ∼ −1 −1 × (OL ) ∩ OF /(OF ) = Ker Hb (G, µL(q)) → Hb (G, OL ) .

× 1 G By the first statement of the lemma, the D-module B = OL 2 verifies B(q) = B(q) and we can apply Lemma 2.7. In particular we obtain a ∆-antiequivariant isomorphism × q × × q ∼ × × (OL ) ∩ OF /(OF ) = IGOL ∩ OF . × × × Observe that, again by Lemma 2.7, IGOL ∩ OF is equal to IGOL ∩ µF (q) ⊆ µF , whence the claim that it is cyclic. We can now state the following result which sharpens Corollary 3.14 under the assumption that F is a CM + ∞ qm field and F = k. We write µF (q ) to denote the roots of unity ζ ∈ µF such that ζ = 1 for some m ≥ 0: it is clearly a finite group.

Corollary 3.17. With notation as in Theorem 3.13, assume moreover that F is a CM ﬁeld of degree [F : Q] = 2d × q × × q and that k is its totally real subﬁeld. Let s be the order of the quotient (OK ) ∩ Ok /(Ok ) and let t be the order ∞ of µF (q )/µF (q). Then, for every prime `, 2 hLhk −v`(q) − v`(s) − v`(t) ≤ v` 2 ≤ (d − 1)v`(q). hF hK CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 20

0 × 1 −1 × 1 Proof. Again, we need to study the ratio bh (D, OL 2 )/bh (D, OL 2 ), thanks to Theorem 3.13. For i = −1, 0 i × 1 i × i × − i × − we write bh (D, OL 2 ) = bh (G, OL )/bh (G, OL ) , where we denote by bh (G, OL ) the order of the minus-part i × i × i × 1 of Hb (G, OL ) (observe that Hb (G, OL ) = Hb (G, OL 2 ) since G is of odd order). We find 0 × 1 × × × bh D, O bh0(G, O ) bh−1(G, O )− 1 bh−1(G, O )− (3.26) L 2 = L · L = · L . −1 × 1 −1 × 0 × − q 0 × − bh D, OL 2 bh (G, OL ) bh (G, OL ) bh (G, OL ) We now use that F is a CM field and F = k+. Starting from the tautological exact sequence defining 0 × Hb (G, OL ), we can consider its minus-part to obtain × 1 − × 1 − 0 × − 0 −→ NL/F OL 2 −→ OF 2 −→ Hb (G, OL ) −→ 0. × × × 1 − The quotient OF /Ok µF is of order 1 or 2 by [Lan90, Chapter XIII, §2, Lemma 1] and, in particular, OF 2 = 1 µF 2 . Since the rightmost term of the above sequence is killed by q, the sequence can be rewritten as × 1 − ∞ 0 × − (3.27) 0 −→ NL/F OL 2 −→ µF (q ) −→ Hb (G, OL ) −→ 0.

On the other hand, thanks to Lemma 3.16, the G-action on µL is trivial and (as in the proof of Lemma 2.7) −1 0 we ﬁnd µF (q) = Hb (G, µL(q)) = Hb (G, µL(q)). By considering the Kummer sequence (3.24) and taking the minus part of its G-cohomology we obtain

× −1 − −1 × − 0 / IGOL ∩ µF (q) / Hb (G, µL(q)) = µF (q) / Hb (G, OL )

(3.28)

−1 × q− 0 − / Hb G, (OL ) / µF (q) = Hb (G, µL(q)) / Y / 0

0 × − where Y ⊆ Hb (G, OL ) is deﬁned by the exactness of the diagram. Taking orders in the above long exact sequence we deduce −1 × q − −1 × − bh (G, (OL ) ) · |Y | bh (G, OL ) = × |IGOL ∩ µF (q)| 0 × − ∞ while (3.27) implies that bh (G, OL ) ≤ |µF (q )|. Therefore (3.26) becomes 0 × 1 × bh D, O 1 bh−1(G, (O )q)− · |Y | 1 L 2 ≥ · L · . × 1 × ∞ −1 q |I O ∩ µ (q)| |µF (q )| bh D, OL 2 G L F −1 × q − The second line of the long exact sequence (3.28) shows that bh (G, (OL ) ) · |Y | ≥ |µF (q)| so that we obtain 0 × 1 bh D, O 1 1 1 (3.29) L 2 ≥ · · . −1 × 1 q |I O× ∩ µ (q)| t bh D, OL 2 G L F 1 × 1 − × × − × ×− Since µF 2 = OF 2 , we have IGOL ∩ µF (q) = IGOL ∩ µF = IGOL ∩ OF , by Lemma 2.7, and × applying Lemma 3.16 we deduce that IGOL ∩ µF (q) is of order s because + ×q × ×q × q × ×q OL ∩ OF / OF = OK ∩ Ok / Ok . Taking `-adic valuations in (3.29), we obtain the bound 2 hLhk −v`(q) − v`(s) − v`(t) ≤ v` 2 . hF hK CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 21

As for the other inequality of the corollary, we argue as in Corollary 3.14 observing that since k is totally real, × βk(q) = 1 and rkZ(Ok ) = d − 1. Remark 3.18. The quotient (O× )q ∩ O×/(O×)q has exponent dividing q and is cyclic (by Lemma 3.16), hence K k k √ q × its order s divides q. We claim that s = q if and only if K = k( u) for some unit u ∈ Ok . This is equivalent × q × × q × × q to the fact that, given u ∈ (OK ) ∩ Ok , the class [u] ∈ (OK ) ∩ Ok /(Ok ) has order strictly less than q if × q q and only if k(x) ( K, where x ∈ OK is such that x = u. Note that k(x) ( K precisely when X − u is × p reducible, a condition equivalent to the existence of a prime p | q such that u ∈ (Ok ) (see [Lan02, Chapter VI, × q × × q Theorem 9.1]). We are thus reduced to show that the class [u] ∈ (OK ) ∩ Ok /(Ok ) has order strictly less than × p q if and only if there is p | q such that u ∈ (Ok ) . × p q/p × q If u ∈ (Ok ) , then clearly u ∈ (Ok ) and the order of [u] strictly divides q. Suppose, conversely, that q/p × q × q/p q [u] has order q/p for some prime p, so u ∈ (Ok ) , and let v ∈ Ok be such that u = v . We deduce that q q/p q q/p q/p × v = (x ) or, equivalently, x = vζ for some ζ ∈ µK . Hence x ∈ Ok , by Lemma 3.16, and therefore q q/p p × p u = x = (x ) ∈ (Ok ) . This shows our claim. ∞ When q = p is a prime and µF (p ) = µF (p), we can restate Corollary 3.17 by saying that the√ ratio of class numbers equals pm for some −2 ≤ m ≤ (d − 1) or −1 ≤ m ≤ (d − 1) according as whether K = k( p u) for some × u ∈ Ok , or not. The next corollary (which generalizes [HK77, §4] and [Mos79, ThéorèmeIV.1]) shows that if k = Q even sharper bounds hold. Corollary 3.19. Let L/Q be a dihedral extension of degree 2q where q is odd, and let F/Q be the quadratic extension contained in L. Denote by K the field fixed by a subgroup of order 2 in Gal(L/Q). Then, for every prime `, ( hL −2v`(q) if F is real quadratic 0 ≥ v` 2 ≥ hF hK −v`(q) if F is imaginary quadratic. Proof. The case when F is real quadratic is already contained in Corollary 3.14. If F is totally imaginary, it is a CM field so that we can apply Corollary 3.17: we need to prove that s = t = 1. Recall that s was defined as the order of the quotient (O× )q ∩ {±1}/{±1}, K √ ∞ which is trivial. As for t, the only case in which µF (q ) 6= 1 occurs when F = Q( −3) and 3 | q, but in this ∞ case µF (q ) = µF (q), so t = 1. 4. Class number formula in terms of a unit index In this section we rewrite the formula of Theorem 3.13 in a form more similar to the one often found in the literature, namely replacing the orders of cohomology groups by the index of a subgroup of the group of units of L. Notation. Recall that for an abelian group B and a natural number n, we write B(n) = {b ∈ B : nb = 0}. We set [ torZ(B) = B(n) n∈N for the torsion subgroup of B and B = B/ torZ(B) for the maximal torsion-free quotient of B. Proposition 4.1. When B is a D-module which is Z-finitely generated the following formula holds: 0 1 D bh D,B 2 Σ Σ0 G IGB ∩ B rk (BD )−rk (BG) = (B : B + B + B )h(G, B) q Z Z . 1 G −1 |IGB ∩ B | bh D,B 2 Proposition 4.1, as well as its proof, is purely algebraic and requires no arithmetic. We postpone its proof after Theorem 4.3 and focus here on its arithmetic application. CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 22

Corollary 4.2. With notation as in Theorem 3.13, it holds

2 (rk (O×)−rk (O×)−1) hLhk × × × × q Z k Z F = O : O O 0 O . 2 L K K F × q × × q hF hK (OK ) ∩ Ok :(Ok ) × q × × q and the term ((OK ) ∩ Ok :(Ok ) ) divides q. Proof. The formula follows by combining Theorem 3.13, Proposition 4.1, the value of the Herbrand quotient of units and Lemma 3.16. The formula of Corollary 4.2 recovers and generalizes those previously appeared in the literature (for precise references, see the Introduction). For instance, we prove below that Bartel’s formula coincides with ours when q is a prime: note that Bartel shows (see [Bar12, Corollaries 6.1 and 6.2]) that his formula implies the ones of [HK77] and [Lem05]. Theorem 4.3 (Bartel, see [Bar12, Theorem 1.1]). Let L/k be a Galois extensions of number fields with Galois 0 group D2p for p an odd prime, let F be the intermediate quadratic extension and K,K be distinct intermediate extensions of degree p. Set δ to be 3 if K/k is obtained by adjoining the pth root of a non-torsion unit (thus so is L/F ) and 1 otherwise. Then we have 2 hLhk α/2 × × × × 2 = p · OL : OK OK0 OF hF hK rk (O×)−rk (O×) × × Z L Z F where α = 2 rkZ(Ok ) − rkZ(OF ) − p−1 − δ. Remark 4.4. Observe that Bartel’s theorem assumes that q = p is an odd prime. However Bartel’s formulation is more general than the one given above, since he obtains a formula for general S-class numbers where S is any finite set of primes in k containing the archimedean ones. Since the proof of Theorem 3.13 ultimately relies on the study of G-cohomology of Diagram 3.6 and a S-version of the diagram exists, our main result should easily generalize to S-class numbers and S-units. × × Proof. Observe that since L/F is Galois of odd order, we have rkZ(OL ) = p rkZ(OF ) + (p − 1) and so rk (O×) − rk (O×) (p − 1)(rk (O×) + 1) Z L Z F = Z F = rk (O×) + 1. p − 1 p − 1 Z F Therefore × × α = 2 rkZ(Ok ) − 2 rkZ(OF ) − 1 − δ. Inserting this in the definition of α and comparing Corollary 4.2 with the statement of the theorem, we simply need to show that δ−1 2 × p × × p p = (OK ) ∩ Ok :(Ok ) . × p × × p By definition of δ, the above reduces, using that (OK ) ∩ Ok /(Ok ) is cyclic by Lemma 3.16, to prove that √ × p × × p p × (OK ) ∩ Ok :(Ok ) 6= 0 ⇐⇒ K = k( u) for some u ∈ Ok and this is clear (see also Remark 3.18). We now prove Proposition 4.1. The strategy follows very closely the one of [Cap13, Section 2], which, in turn, is partly inspired by [Lem05, Section 5].

0 Lemma 4.5. Let B be a D-module which is Z-ﬁnitely generated. Then the index of BΣ + BΣ + BG in B is ﬁnite and

Σ Σ0 G Σ Σ0 G Σ Σ0 G (B : B + B + B ) = NGB : NG(B + B + B ) · B[NG]: B[NG] ∩ (B + B + B ) . Moreover, the three terms appearing in the above equality are coprime to any prime not diving q. CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 23

0 Σ Σ0 G Proof. By equation (3.4) with f = NG and B = B + B + B , we only need to verify that the above indices are finite. Since B is Z-finitely generated, all G-Tate cohomology groups of B are finite of exponent dividing q. Σ Σ0 G In particular, B[NG] ∩ B + B + B is of finite index in B[NG] since it contains IGB by Lemma 2.4. This Σ Σ0 G argument also shows that (B[NG]: B[NG] ∩ (B + B + B )) is coprime to any prime not dividing q. On the Σ Σ0 G G G other hand, NG B + B + B contains q(B ), so its index in B is finite and coprime to any prime not Σ Σ0 G G dividing q. The same holds therefore for the index of NG B + B + B in NGB, since NGB ⊆ B . This finishes the proof. We now analyze separately the two factors of the right-hand side of the equality of Lemma 4.5.

Lemma 4.6. Let B be a D-module which is Z-finitely generated. Then the following equality holds: 0 1 G G Σ Σ0 G bh D,B 2 B : q(B ) NGB : NG(B + B + B ) = . 0 1 D D bh G, B 2 B : q(B ) Proof. Every factor appearing in the formula of the statement is coprime to any prime not dividing q by 1 Lemma 4.5. This means that, replacing B by B 2 , we can assume that B is uniquely 2-divisible (note that Tate cohomology groups remain finite) so that Σ-Tate cohomology of B is trivial, and similarly for Σ0. In particular Σ NG(B ) = NG(NΣB) = NDB Σ0 and similarly NG(B ) = NG(NΣ0 B) = NDB. Therefore Σ Σ0 G G NGB : NG(B + B + B ) = (NGB : q(B ) + NDB) BG : q(BG) + N B = D bh0(G, B) BG : q(BG) = G G 0 q(B ) + NDB : q(B ) · bh (G, B) BG : qBG = G 0 (NDB : qB ∩ NDB) · bh (G, B) BG : q(BG) (see (4.1) below) = D 0 NDB : q(B ) · bh (G, B) G G 0 B : q(B ) · bh (D,B) = . BD : q(BD) · bh0(G, B) We have used that G D (4.1) q(B ) ∩ NDB = q(B ). To see this, first observe that (4.2) q(BG) ∩ BD = q(BD). q because the surjective map BG −→ q(BG) stays surjective after taking Σ-invariants, since BG is uniquely G D 2-divisible (see (2.2)). In particular, we have q(B ) ∩ NDB ⊆ q(B ) which implies G D q(B ) ∩ NDB = q(B ) ∩ NDB. Now (4.1) follows since D D q(B ) ∩ NDB = q(B ) because D D G G q(B ) = NG(B ) = NG(NΣ(B )) = ND(B ) ⊆ NDB. CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 24

Lemma 4.7. Let B be a D-module which is Z-ﬁnitely generated. Then

1 1 G D Σ Σ0 G bh G, B 2 · IGB ∩ B (q): IGB ∩ B (q) B[NG]: B[NG] ∩ B + B + B = . −1 1 G D bh D,B 2 · (B (q): B (q))

Σ Σ0 G Proof. Observe that B[NG]: B[NG] ∩ B + B + B is coprime to any prime not dividing q (by Lemma 4.5). The same holds for all the factors of the right-hand side of the equality of the lemma. This means that, 1 replacing B by B 2 , we can assume that B is uniquely 2-divisible. Observe ﬁrst that

0 B[N ]: B[N ] ∩ BΣ + BΣ 0 G G B[N ]: B[N ] ∩ BΣ + BΣ + BG = G G Σ Σ0 G Σ Σ0 (B[NG] ∩ B + B + B : B[NG] ∩ B + B ) bh1(D,B) (by Lemma 2.5) = Σ Σ0 G Σ Σ0 B[NG] ∩ B + B + B : B[NG] ∩ B + B

Since bh1(D,B) = bh−1(G, B)/bh−1(D,B) by Proposition 2.1, we only need to prove that

G D Σ Σ0 G Σ Σ0 B (q): B (q) B[NG] ∩ B + B + B : B[NG] ∩ B + B = G D . (IGB ∩ B (q): IGB ∩ B (q)) To prove the above equality, observe that Σ acts as −1 on the quotient

Σ Σ0 G B[NG] ∩ B + B + B Σ Σ0 B[NG] ∩ B + B by (2.7). In particular we have

0 0 !− B[N ] ∩ BΣ + BΣ + BG B[N ] ∩ BΣ + BΣ + BG G = G Σ Σ0 Σ Σ0 B[NG] ∩ B + B B[NG] ∩ B + B − Σ Σ0 G B[NG] ∩ B + B + B = Σ Σ0 − B[NG] ∩ B + B − Σ Σ0 G B[NG] ∩ B + B + B (by (2.5)) = − . (IGB)

Σ Σ0 Σ Now note that B + B = IGB + B by (2.4), so that

Σ Σ0 G Σ G B[NG] ∩ B + B + B = B[NG] ∩ IGB + B + B Σ G = IGB + B[NG] ∩ B + B and taking minus parts we obtain

Σ G − − Σ G − IGB + B[NG] ∩ (B + B ) = (IGB) + B[NG] ∩ ((B + B ) − G− = (IGB) + B[NG] ∩ B − G − = (IGB) + B (q) G − = IGB + B (q) . CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 25

We have therefore that

0 − B[N ] ∩ BΣ + BΣ + BG I B + BG(q) G = G Σ Σ0 − B[NG] ∩ B + B (IGB) − BG(q) =∼ G − (IGB ∩ B (q)) BG(q)/BD(q) = G D (IGB ∩ B (q))/(IGB ∩ B (q)) and this concludes the proof of the lemma.

Let B be a ﬁnitely generated Z-module and let m be a positive natural number. Then we have

rk (B) |B(m)| · m Z = (B : mB).

Indeed, let π : B → B = B/ torZ(B) be the projection map. It induces a commutative diagram with exact rows

0 / B(m) / B m / B / B/m / 0

π π πm 0 / B m / B / B/m / 0 where πm is the map induced by π on B/m. The snake lemma gives an exact sequence of ﬁnite abelian groups

m 0 −→ B(m) −→ torZ(B) −→ torZ(B) −→ Ker πm −→ 0. On the other hand we have the tautological exact sequence

0 −→ Ker πm −→ B/m −→ B/m −→ 0. Taking cardinalities we obtain

(B : mB) −rk (B) (4.3) |B(m)| = |Ker πm| = = (B : mB)m Z , (B : mB) as claimed.

Proof of Proposition 4.1. Combining Lemmas 4.5, 4.6 and 4.7 we get

Σ Σ0 G Σ Σ0 G Σ Σ0 G B : B + B + B = NGB : NG(B + B + B ) · B[NG]: B[NG] ∩ (B + B + B )

0 1 G G 1 1 G D bh D,B · B : q(B ) bh G, B · (IGB ∩ B (q): IGB ∩ B (q)) = 2 2 0 1 D D −1 1 G D bh G, B 2 · B : q(B ) bh D,B 2 · (B (q): B (q)) G G 0 1 G D B : q(B ) bh D,B IGB ∩ B (q): IGB ∩ B (q) = 2 . BD : q(BD) · (BG(q): BD(q)) −1 1 h(G, B) bh D,B 2 Now observe that, in light of (4.3), we have

G G B : q(B ) rk (BG)−rk (BD ) G D = q Z Z B (q): B (q) . BD : q(BD) CLASS NUMBER FORMULA FOR DIHEDRAL EXTENSIONS 26

References [Bar] Alex Bartel, Class groups in dihedral extensions – some sort of Spiegelungssatz?, https://mathoverflow.net/q/73033. [Bar12] , On Brauer-Kuroda type relations of S-class numbers in dihedral extensions, J. Reine Angew. Math. 668 (2012), 211–244. [Bra51] Richard Brauer, Beziehungen zwischen Klassenzahlen von Teilkörpern eines galoisschen Körpers, Math. Nachr. 4 (1951), 158–174. [Cap13] Luca Caputo, The Brauer–Kuroda formula for higher S-class numbers in dihedral extensions of number fields, Acta Arith. 151 (2013), no. 3, 217–239. [CE56] Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. [CN] Luca Caputo and Filippo A. E. Nuccio Mortarino Majno di Capriglio, On fake Zp-extensions of number fields, in prepa- ration. [CN09] Luca Caputo and Filippo A. E. Nuccio Mortarino Majno di Capriglio, On fake Zp-extensions of number fields, 2009, pp. 1–40. [Dir42] G. Lejeune Dirichlet, Recherches sur les formes quadratiques àcoëfficientset àindéterminées complexes. Première partie, J. Reine Angew. Math. 24 (1842), 291–371. [Hil94] David Hilbert, Ueber den Dirichlet’schen biquadratischen Zahlkörper, Math. Ann. 45 (1894), no. 3, 309–340. [HK77] Franz Halter-Koch, Einheiten und Divisorenklassen in Galois’schen algebraischen Zahlkörpern mit Diedergruppe der Ord- nung 2l füreine ungerade Primzahl l, Acta Arith. 33 (1977), no. 4, 355–364. [HR63] A. Heller and I. Reiner, Representations of Cyclic Groups in Ring of Integers, II, Ann. of Math. 77 (1963), no. 2, 318–328. [Jau81] Jean-Fran¸cois Jaulent, Unités et classes dans les extensions métabéliennes de degré nls sur un corps de nombres algébriques, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 1, ix–x, 39–62. [Lan90] Serge Lang, Cyclotomic Fields I and II, combined second ed., Graduate Texts in Mathematics, vol. 121, Springer-Verlag, New York, 1990, With an Appendix by Karl Rubin. [Lan94] , Algebraic number theory, second ed., Graduate Texts in Mathematics, vol. 110, Springer-Verlag, New York, 1994. [Lan02] , Algebra, third ed., Graduate Texts in Mathematics, vol. 211, Springer-Verlag, New York, 2002. [Lem94] Franz Lemmermeyer, Kuroda’s class number formula, Acta Arith. 66 (1994), no. 3, 245–260. [Lem05] , Class groups of dihedral extensions, Math. Nachr. 278 (2005), no. 6, 679–691. [Mos79] Nicole Moser, Unitéset nombre de classes d’une extension galoisienne diédrale de Q, Abh. Math. Sem. Univ. Hamburg 48 (1979), 54–75. [Nuc09] Filippo A. E. Nuccio Mortarino Majno di Capriglio, On Zp-extensions of real abelian number fields, Ph.D. thesis, University of Rome La Sapienza, 2009. [Ser67] Jean-Pierre Serre, Local class field theory, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thomp- son, Washington, D.C., 1967, pp. 128–161. [Tat67] John T. Tate, Global class field theory, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 162–203. [Wal77] Colin D. Walter, A class number relation in Frobenius extensions of number fields, Mathematika 24 (1977), no. 2, 216–225. MR 0472768 Email address: [email protected]

Email address: [email protected]

Univ Lyon, Universite´ Jean Monnet Saint-Etienne,´ CNRS UMR 5208, Institut Camille Jordan, F-42023 Saint- Etienne,´ France