On Frohlich's conjecture for tamely ramified quaternion extensions

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UNIVERSITÀ DEGLI STUDI DI PISA

Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Magistrale in Matematica

Tesi di Laurea

ON FRÖHLICH’S CONJECTURE FOR

TAMELY RAMIFIED QUATERNION

EXTENSIONS

Relatore: Candidato:

Prof. ILARIA DEL CORSO REYNOLD FREGOLI

Controrelatore:

Prof. ROBERTO DVORNICICH

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Introduction

Let N |K be a finite normal extension of number fields and let Γ be its Galois group. It is a well known result that N is a free rank 1 module over the ring K[Γ], i.e. there exists an element a ∈ N such that the set {σ(a) | σ ∈ Γ} is a K-basis of the vector space N . Let now ON (respectively OK) indicate the rings of integers of the field N (respectively

K). A natural question arises: what can be said about the structure of the ring ON

as an O_K[Γ]-module? It is easily seen that ON is rank 1 and torsion free over OK[Γ],

but in general it may not be free. If that is the case we shall say that the OK-basis

{σ(a) | σ ∈ Γ} with a ∈ O_K, is a normal integral basis for the extension N |K, and that N |K has the NIB property. The problem of establishing whether ON is a free module

over the ring OK[Γ] is known as the normal integral basis (or NIB) problem. Some of the

first remarkable results found in the attempt to face the NIB problem were the following: • if N |Q is a finite tame abelian extension, then N|Q has the NIB property. This is

known as the Hilbert-Speiser theorem;

• if N |Q is a finite and tame extension with Galois group the dihedral group Dp of

order 2p, where p is a prime number, then N |Q has the NIB property;

• there are tame extensions N |Q with Galois group the quaternion group H8, of order

8, which do not have the NIB property.

For the first result see [17], while the last two results are due to Martinet and can be found in [22] and [26]. In view of these facts, it seemed quite natural to deeper investigate the properties of finite tame extensions over the rational field with respect to the NIB problem, and especially the extensions having the quaternion group H8 as

Galois group. Recall also, that, by Noether’s Theorem, tameness is a necessary condition for an extension to the existence of a normal integral basis.

In this thesis I present some fundamental results concerning the study of the NIB problem, in the case of tamely ramified quaternion extensions over the rational field. These results, principally due to A. Fröhlich, are particularly relevant since they rep-resent the starting point of deep and substantial further discoveries on the subject. In 1972 Fröhlich introduced and subsequently settled a conjecture, establishing a surprising connection between the existence of a normal integral basis and the sign of the Artin root number of a certain character, in tame quaternion extensions over the field Q. Let us be more precise. Suppose that N |Q is a normal extension with Galois group the quaternion

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group of order 8, denoted by H8. Let then ON indicate the ring of integers of the field

N . It is easy to see that the group H8 may be described by the following relations, where

e is the identity element:

H8= hσ, τ | σ4= e, τ2 = σ2, στ σ−1 = τ−1i

Consider now T the representation of the group H8 defined by:

T (σ) = i 0 0 −i T (τ ) = 0 1 −1 0

The character ψ of T is such that ψ(e) = 2, ψ(σ2) = −2 and ψ = 0 elsewhere. It can be shown that ψ is the unique irreducible degree 2 character of the group H₈ and that it is symplectic, i.e. it preserves some skew-symmetric bilinear form. The group H8 possesses other four irreducible degree 1 characters, and, along with ψ, these are

the only irreducible characters of H₈: this gives the reason why ψ plays a key role in the conjecture. We shall indicate by w(ψ) the Artin root number associated with the L-function L(s, ψ, N |Q), as it is defined in 1.53. Recall that, being ψ a real valued character, we have w(ψ) = ±1. Moreover, it can be shown that inside the locally free class group Cl(Z[H8]), the class [ON] is of order at most 2 (actually the whole group

Cl(Z[H8]) is of order 2). We define now the constant UN by setting:

UN :=

(

1 if [ON] is the trivial class

−1 if [O_N] has order two (1) Fröhlich’s conjecture states that:

Theorem 0.1. Let N |Q be an at most tamely ramified quaternion extension. Then: w(ψ) = UN

A generalization to arbitrary base fields of this result was found by M.J. Taylor in 1985 (see [28]).

The thesis is based on a series of lectures held by Prof. Ph. Cassou-Noguès at the University of Pisa in the year 2014. The aim of those lectures was to give a rapid insight into the techniques developed so far to study Fröhlich’s conjecture and similar problems. I propose here two different proofs of Fröhlich’s conjecture. The first one runs through the results found by J.Martinet in [26] and by A.Fröhlich in [15]. Specifically, Martinet managed in giving a useful classification of the normal tame extensions of Q whose Galois group is the quaternion group of order 8, and found a purely arithmetic criterion to establish whether these extensions have a normal integral basis or not. On the other hand, Fröhlich developed an accurate study of the general quaternion extension (of order 8) over the rational field and gave an elementary proof of 0.1, taking as a starting point the work of Martinet. Details are dealt with in Chapters 2 and 4.

The second strategy of proof is far more instructive and linear, although it requires a huge theoretical background. It makes use of some innovative and brilliant theory developed

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CONTENTS v by Fröhlich and others, which allows an explicit description of the class group of any order via character theory, the so called Hom description. Before presenting the proof itself, I tried to summarize the prerequisites necessary to its understanding, but, as their amount is considerable, I decided to sacrifice the details in various proofs, aiming at an accurate comprehension of the context. Thus, the second part of this thesis must be intended as a sort of guide throughout the above mentioned topics, which the reader will be able to enrich with satisfactory proofs, where needed, by means of the given references. Details concerning the Hom description may be found in [2], while the proof of the conjecture may be found in [4].

Subsequently, a great deal of different results were attained even in the case of wild extensions. See for example [30] and [24], page 163.

I conclude by mentioning that this thesis owes much to Prof. I. Del Corso, my teacher, who took notes at Cassou-Noguès’ lectures, which I unfortunately could not attend.

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Part I

A handmade proof

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Chapter 1

Preliminary Notions

In this chapter we recall some general results, with the aim of sketching out the theoret-ical background necessary to the comprehension of Chapter 2. The reader feeling well acquainted with these concepts can skip directly to Chapter 2.

1.1 The locally free class group

Let K be a number field and let Γ be a finite group. We shall indicate by O_K the ring of integers of the field K.

Definition 1.1. A set Λ, contained in in the group ring K[Γ], is said an O_K-order in K[Γ], if the following two conditions hold:

• Λ is a subring of K[Γ]

• Λ is a finitely generated module over the ring O_K such that KΛ = K[Γ].

Let now P be a place of the field K. In the following discussion K_P will stay for the completion of K with respect to the absolute value associated with the prime P , and OKP for its ring of integers. Given Λ an OK order in K[Γ] and M a module over Λ, we

set:

MP := OKP ⊗OK M and ΛP := OKP ⊗OK Λ

It is plain that MP can be endowed with a ΛP-module structure. By means of this

notation, we shall now introduce the concept of locally free module over the order Λ. Definition 1.2. Let K be a number field and let Γ be a finite group. Let moreover Λ be an OK-order in K[Γ]. We shall say that a module M is locally free over Λ if:

1. M is a finitely generated module over Λ

2. For any finite prime P in K, the module MP is free over the ring ΛP.

In the particular case when Λ = O_K[Γ], the following result holds: 3

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Theorem 1.3. Let K be a number field and let Γ be a finite group. Then, a finitely generated module M over the order O_K[Γ] is locally free if and only if it is projective. Proof. From Theorem 8.1 of [32], we deduce that K ⊗OK M is K[Γ]-free. Let now P be

a finite prime of K. Tensoring by K_P, we get that K_P⊗OKM is free over KP[Γ]. Hence,

we have:

KP ⊗O_KP OKP ⊗OK M ∼= KP ⊗OK M ∼= KP[Γ]

m∼_{= K}

P ⊗O_KP OKP[Γ]

m

To conclude, just use Corollary 6.4 of [32]. Vice-versa, let R be a DVR, let P be its maximal ideal and let ˆR be the P -completion of R. For any R-algebra Λ, set ˆΛ = ˆR⊗RΛ.

If M is a Λ-module, then ˆM := ˆΛ ⊗ΛM is a ˆΛ-module. It can be shown that if M and

N are two finitely generated modules over Λ, then:

M ∼= N as Λ-modules ⇔ ˆM ∼= ˆN as ˆΛ-modules (1.1) Let now P be a prime of K and set R := O_K,P, i.e. localization of the ring O_K at the ideal P . If M is a locally free O_K[Γ]-module, then, by definition:

MP ∼= OrKP (1.2)

Since the completion of OK with respect to the ideal P coincides with the completion of

OK,P with respect to Pe:= P OK,P, by 1.2, we deduce that:

˜

MPe ∼= (O_K,P[Γ])r_Pe

where ˜M := M ⊗OKOK,P and the isomorphism is over OK,P[Γ]. From 1.1, it follows that

˜

M ∼= ˜Λr, i.e ˜M , the localization of M , is free over OK,P[Γ]. The proof is then complete

in view of [20], Theorems 7.12 and 2.5. In the general case, we have:

Proposition 1.4. Let Λ be an O_K-order in K[Γ], with K and Γ as above, and let M be a locally free Λ-module. Then, the rank of M is well defined as the rank of the free K[Γ]-module MK := K ⊗OK M . Moreover, this rank coincides with the rank of the modules

MP over the orders ΛP for all the places P of K.

Proof. Observe first that M is also finitely generated over O_K, since Λ is finitely generated over OK. Now, since M is locally free, by Theorem 1.3, it is also projective and hence,

by applying Theorem 8.1 of [32], we have that MK is free. The equality of the ranks

follows by tensoring by K_P (see the proof of Theorem 1.3).

Our aim now is to define the locally free class group over a fixed O_K-order Λ. To do so we define an equivalence relation over the set of locally free Λ-modules, as follows: Definition 1.5. We say that two locally free modules M and N over the OK-order Λ are

equivalent if and only if, by definition, there exist two free finitely generated Λ-modules L and L0 such that:

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1.1. THE LOCALLY FREE CLASS GROUP 5 Definition 1.6. The quotient set of all locally free Λ-modules with respect to the equiv-alence relation defined at 1.5, will be called the locally free class group over the O_K-order Λ, and will be denoted by Cl(Λ).

Remark 1.7. It is easily seen that this set becomes an abelian group when endowed with the operation +, defined by:

[M ] + [N ] = [M ⊕ N ]

for all pairs of locally free Λ-modules M and N . The identity of the class group is the element [0] and all the locally free Λ-modules belonging to the identity class will be called stably free.

We recall now a classical result, known as the Noether theorem:

Theorem 1.8. Let N |K be a finite normal extension of number fields and let Γ be its Galois group. Then, the following conditions are equivalent:

• N |K is tamely ramified • T r_{N |K}(ON) = OK

• O_N is a projective module over O_K[Γ]

where ON and OK indicate the rings of integers of N and K.

A proof may be found in [23], Theorem 4.1.

Remark 1.9. Suppose to be in the setting of Theorem 1.8, with K = Q and Λ = Z[Γ]. From Theorem 1.8, we may deduce that if N |Q is at most tamely ramified, then ON is a

projective Z[Γ]-order and thus, locally free by Theorem 1.3. We have then [ON] ∈ Cl(Λ).

This easy remark shows that becoming acquainted with the structure of the class group Cl(Z[Γ]) for a number field N , becomes useful when investigating the structure of the ring ON as a module over the order Z[Γ]. As a matter of fact, studying the class

group of this particular Z-order will become crucial in establishing the existence of a NIB in tame extensions of number fields and not only when the base field is Q. We conclude this section, by speaking of cancellation. The result stated below is strategic to our purpose and comes from the study of central simple algebras which we shall not deal with here. A definition first:

Definition 1.10. Let Λ be an OK-order in K[Γ], with K a number field and Γ a finite

group. We say that the order Λ has the cancellation property, if, for all pairs, M and N , of finitely generated modules over the ring Λ, it holds:

M ⊕ Λ ∼= N ⊕ Λ ⇒ M ∼= N

This means that every stably free module over the order Λ is free, whenever Λ pos-sesses the cancellation property. The following theorem gives us some examples of orders endowed with this property:

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Theorem 1.11. Let Γ be a finite group. The order Z[Γ] possesses cancellation property if:

i) Γ is abelian; ii) Γ is of odd order; iii) Γ is a dihedral group;

iv) Γ = H8, H16 (quaternion groups of order 8 and 16).

The reader requiring a deeper insight into these topics may consult [2] §3. More can be found also in [16] or [19]. It is worth to mention that the quaternion group of order 32 is the least quaternion group for which cancellation fails (note that H8 and H16 possess

the cancellation property without respecting Eichler’s condition). For this see [21] and [33].

1.2 Some class field theory

In this section we sketch briefly some results of class field theory, omitting the details which are unnecessary to the discussion carried out in Chapter 2. The reader interested in a more accurate account may consult [25], chapters X and XI.

Definition 1.12. Let I be a set and let {G_i}_i∈I and {H_i}_i∈I be families of groups, such that Hi is a subgroup of Gi, for each index i. Then, we define the restricted direct

product of the groups {G_i}_i∈I, with respect to the subgroups {H_i}_i∈I, to be the set: Y i∈I 0 Gi⊂ Y i∈I Gi

of all the I-tuples (g_i)i, such that gi ∈ Hi for all but a finite number of indices i. If

moreover {G_i}_i∈I is a family of topological groups, and H_i is open in G_i for all i ∈ I, then, the restricted product Q0

i∈IGi, is endowed with a topological structure. Namely,

the topology is generated by the sets: GS = Y i∈S Ai× Y i /∈S Hi

where S runs through the finite subsets of I and Ai is an open subset of Gi, for all i ∈ S.

Remark 1.13. With respect to this topology, it is easy to verify that: • the restricted productQ0

i∈IGiis a topological group with the usual component-wise

product.

• If the G_i’s are locally compact groups and the H_i’s are compact, then the restricted productQ0

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1.2. SOME CLASS FIELD THEORY 7 We recall now, the definition of adele ring of a number field k.

Definition 1.14. Let k be a number field and let M_k be the set of all the absolute values v of the field k, normalised to induce one of the standard absolute values on Q. We define the group Ad(k) as the restricted product of the topological groups {kv}v∈Mk

(completion of the field k with respect to the absolute value v), with respect to the family of subgroups {Okv}v∈Mk, of the rings of integers of the fields kv.

The adele ring is actually a topological ring with respect to the product operation, as usual defined component-wise. Moreover:

Definition 1.15. Let k be a number field. We call idele group of the field k, the group of units of the topological ring Ad(k).

Remark 1.16. Note that the idele group endowed with the induced topology, is not necessarily a topological group, since the inverse operation may fail to be continuous. Thus, we shall consider on it the topology induced by the injection J (k) → Ad(k) × Ad(k), where x 7→ (x, x−1). Clearly, as the map (x, y) 7→ (y, x) is continuous from Ad(k) × Ad(k) into itself and sends J (k) into J (k)−1= J (k), the inverse, x 7→ x−1, must be continuous on J (k) with this new topology. The product is obviously continuous, and thus, J (k) is now a topological group. It may be shown that J (k) endowed with this topology is isomorphic and omeomorphic to the restricted product of the topological groups {k_v×}_v∈M_k, with respect to the open subgroups {O×_k

v}.

Let now k be a number field. It is well known that the multiplicative group k× may be viewed as a discrete subgroup of the idele group J (k) via the diagonal embedding. We shall indicate by C_k the quotient group J (k)/k×.

Suppose now to be given a finite abelian extension K of the field k. We recall that an idele norm N_kK: J (K) → J (k) may be defined. This norm is given by:

N_kK((xw)w) = (

Y

w|v

NKw|kv(xw))v

where N_K_w|kv indicates the usual local norm. It may be shown that the subgroup N

K k (JK)

of J (k) is open (for a proof see [25], pag 208). Thus, by linearity of the norm, we may consider the subgroup N_kK(CK), which is open in Ck. With reference to this notation,

we state now the main result of global class field, whose importance and elegance are self-evident:

Theorem 1.17. Let k be a number field. Then, there exists a one to one correspondence between the open subgroups of the group Ck and the finite abelian extensions of the field

k, given by:

K|k 7→ N_kK(J (K))

Moreover, given K|k such an extension, there exists a homomorphism (called Artin map) from the group C_k onto the group Gal(K|k), whose kernel is precisely the subgroup asso-ciated with K|k by the above correspondence.

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See [25], Chapter X, Theorem 5. We introduce now some more notation which will be used throughout.

Definition 1.18. Let k be a number field and let p be a finite place of k. We denote by Up the group of units of the local field kp. Moreover, for all the integers i > 0, we set:

U_pi := {x ∈ Up| x ≡ 1 (mod pi)}

where, by convention, U_p0 := Up. For the infinite places v of k, we set Uvi = kv for all

i ≥ 0.

Hereafter, if no further specification is needed, we assume these groups to be contained in the idele group J (k) via the usual embedding of k_p into J (k), i.e the map x 7→ (. . . 1, x, 1 . . . ), where x stands here in the position corresponding to kp.

Remark 1.19. The map quoted at 1.17 is the so called Artin map and is usually indicated by A_K|k. We recall here briefly how it acts on idele classes. The definition might appear a bit complicated, but it’s just a consequence of the lack of notation, which we are not going to introduce here (for a neat explanation see [25] chapter X).

1. Let a ∈ J (k). For all the finite places p of the field k, select an integer r_p ≥ 0, such that the open multiplicative group Uprp is contained in the norm group N_kK(J (K)).

It is known (see [25], pages 145-150) that there exists an element α ∈ k×, such that:

• for all the finite places p of k, we have:

(αa)p≡ 1 (mod prp) (1.3)

• for all the real infinite places v of k, we have:

(αa)v > 0 (1.4)

2. Let {p₁, . . . , pr} be the places p of the field k such that (αa)p is not a unit. We

define: I(αa) := r Y i=1 pei i

where ei := vpi((αa)pi) and vpi the valuation associated with the place pi.

3. Condition 1.3 implies that the ideal I(αa) is prime to the discriminant δ of the extension K|k, thus, I(αa)OK possesses a unique Frobenius automorphism

σ(I(αa)OK) in the group Gal(K|k). Recall that, if P is a prime ideal of K over p,

the automorphism σ(P ) is uniquely determined by the property:

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1.2. SOME CLASS FIELD THEORY 9 for all x ∈ OK. Moreover, given I = P1e1. . . Pses with Pi prime ideals of K, by

definition, we have: σ(I) = s Y i=1 σ(Pei₎ _(1.6)

With this notation, we set:

A_K|k(a) = σ(I(αa))

It may be shown that the map A_K|k is independent of the choices made in the definition and that it is a homomorphism which factorises through the class group Ck.

We also recall that the Artin map behaves well with respect to restriction. Namely, we have:

Lemma 1.20. Let K|k be an abelian extension of number fields and let k ⊂ L ⊂ K be an intermediate field such that the extension L|k is normal. Then, the following diagram is commutative:

Gal(K|k) Res //Gal(L|k)

J (k) AK|k OO J (k) AL|k OO

where A_L|k and A_K|k are the Artin maps and Res is the usual restriction.

Proof. This is clear from 1.19. In fact, since N_kK(J (K)) and N_kL(J (L)) are both open subgroups of J (k), for all the finite places p of k, we may choose an integer r_p ≥ 0, such that:

Urp

p ⊂ NkK(J (K)) ∩ NkL(J (L))

Hence, given a ∈ J (k), we may select α ∈ k×such that condition 1.3 holds for this choice of the rp’s, and it follows that:

AK|k(a) = σ(I(αa)OK)

AL|k(a) = σ(I(αa)OL)

We conclude by observing that:

σ(I(αa)OK)|L= σ(I(αa)OL)

which is a consequence of:

σ(I)|L= σ(I ∩ OL) (1.7)

for all ideals I of K. In particular, 1.7 is true for prime ideals, by 1.5, and may be extended to all the ideals, by 1.6.

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By means of Lemma 1.20, it is possible to extend the definition of Artin map to infinite extensions, since the Galois groups of these extensions are the direct limit of the Galois groups of their normal finite subextensions, with respect to restriction. However, we do not go into this here, as it is not necessary to our point.

The decomposition group has already been defined for finite places. We are going to extend this definition to infinite places.

Definition 1.21. Let K|k be a normal extension of number fields and let v be an infinite place of the field k. Let moreover w be a place of K over v. We shall say that the decomposition group of w over v is the subgroup of the group Gal(K|k) obtained by restricting the automorphisms in Gal(Kw|kv) to the field K. This group will be denoted

by D(w|v).

Remark 1.22. The subgroup D(w|v) is of course either trivial or of order 2 and that does not depend on the choice of the place w, being K|k a normal extension. Note that choosing a different place w over v gives a conjugate subgroup, thus, if the extension K|k is abelian, the decomposition group solely depends on the choice of the place v.

Now, as it is well known, the local field k_v may be embedded into the idele group J (k), via the map x 7→ (xv0)_v0, where:

xv0 :=

(

x if v0 = v 1 if v0 6= v

If v is finite, we will indicate by O_k_v the ring of integers of the field kv and by O_k×_v its

subgroup of units, both seen as multiplicative subgroups of J (k). If v is an infinite place, we set all this subgroups to be equal to kv. With reference to this notation, the following

important result holds:

Theorem 1.23. Let K|k be a finite abelian extension of number fields and let v be a place of k. Then, the Artin map A_K|k, defined at 1.19, is such that:

A_K|k(kv) = D(v)

and

AK|k(O×kv) = E (v)

where D(v) and E (v) are respectively the decomposition and inertia groups of the place v in the extension K|k (recall that K|k is abelian).

A proof may be found in [25], chapter XI, page 220.

1.3 On character theory

Let k be a number field. As above, we indicate by J (k) the idele group of k and by C(k) the group of idele classes, i.e. the quotient J (k)/k×.

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1.3. ON CHARACTER THEORY 11 Definition 1.24. We call abelian character of the field k any homomorphism χ from C(k) to C× whose kernel is of finite index.

Clearly, abelian characters may be viewed equivalently as homomorphisms from J (k) to C× which are trivial on k×.

As for primes in extensions of number fields, we may define a notion of ramification and tameness for characters. This is done as follows:

Definition 1.25. Let v be a place of the field k and let χ be an abelian character of k. We say that χ is unramified (respectively tamely ramified) at v if the group U_v (respectively U_v1) is contained in the kernel of χ.

The definition of ramified character is fairly connected with the definition of ramified place. Namely, we have:

Proposition 1.26. Let χ be an abelian character of the field k and let v be a place of k. Suppose moreover that K|k is the abelian extension associated with the open subgroup Ker(χ) of J (k), by means of the correspondence of Theorem 1.17. Then:

χ is ramified at v ⇔ v is ramified in K|k

Proof. By Theorem 1.23, the Artin map associated with the extension K|k sends the group of units U_v onto the inertia group of the prime v (this is well defined, the extension being abelian). Thus, χ is ramified at v if and only if, by definition, it is trivial on Uv,

if and only if Uv is contained in the group associated with the extension K|k in J (k), i.e

Ker(χ). Now, by Theorem 1.17, we know that the group associated with the extension K|k coincides with the kernel of the Artin map, and thus, we have:

Ker(χ) = Ker(AK|k)

Hence, the Artin map is trivial on U_v, i.e. the ramification group of v in K|k is trivial, and that is the required assertion. Clearly, in the infinite case, the ramification just reduces to see whether χ is trivial or not on the completion of k.

Observe that χ may be seen as an injective character defined on the group Gal(K|k) with values in C× via Artin map.

This simple observation shows that abelian characters are indeed part of a quite more general type of characters, the characters deriving from a representation, which we treat in the remaining part of this section. For this part, the reader may refer to [36]

Definition 1.27. Let Γ be a finite group. We call a complex representation of Γ any homomorphism T : Γ → GL(V ), for V a finite dimensional vector space over the field C. The dimension of the space V is said degree of the representation T .

It is well known that for any endomorphism φ ∈ GL(V ), the trace T r(φ) is well defined. This allows us to define the character associated with a representation T of a finite group, as follows:

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Definition 1.28. Let Γ be a finite group and let T be a complex representation of Γ. We define the character of T to be the function Γ → C, given by g 7→ T r(T (g)) for all g ∈ G.

Remark 1.29. Note that the character χ, associated with a representation T of Γ, need not necessarily be an homomorphism, but, from the properties of the trace function, it is easily deduced that χ is invariant under conjugation, i.e. for all g, h ∈ Γ we have:

χ(hgh−1) = χ(g)

Clearly, if T is a degree 1 representation, then T and its character χ are the same function and thus χ is itself an homomorphism.

Observe now that if χ and χ0 are characters of a finite group Γ, then χ + χ0 and χχ0 are as well characters of Γ. This may be seen by considering direct sums and tensor products of the vector spaces which appear in the definition of two representations of χ and χ0. In view of this fact, we may give the following:

Definition 1.30. We define the ring of virtual characters of the finite group Γ, to be the abelian group generated by all the characters of Γ. This ring will be denoted by RΓ.

Remark 1.31. By what we observed above, RΓis obviously a ring. Note that, although

the sum of two characters is a character, the difference may not, thus, the ring RΓ

will contain some functions which are characters of a representation and some virtual characters.

There exists a natural notion of isomorphism for representations:

Definition 1.32. Let Γ be a finite group, and let T : Γ → GL(V ) and T0 : Γ → GL(V0) be two representations of Γ, with V and V0 finite dimensional complex vector spaces. We say that T and T0 are isomorphic representations of Γ if V and V0 are isomorphic as modules over the ring C[Γ].

Clearly, two isomorphic representations have the same character. What can be said about the converse? The following important result holds:

Proposition 1.33. Suppose to be in the setting of Definition 1.32 and let χ be the character of T and χ0 the character of T0. Then:

T and T0are isomorphic ⇔ χ = χ0

For a proof see [36]. We define now irreducible characters, which are crucial in theory of representation:

Definition 1.34. Let T be a complex representation of a finite group Γ, with values in GL(V ). Then, the complex vector space V may be considered as a C[Γ]-module via the action of Γ on V , given by T . We say that T is an irreducible representation of the group Γ, if V is a simple C[Γ]-module, i.e. it doesn’t admit any proper submodule. A non virtual character χ of Γ is said to be irreducible if there is an irreducible representation T of Γ such that χ = T r(T ).

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1.3. ON CHARACTER THEORY 13 Remark 1.35. In Definition 1.34, by means of 1.33, we may equivalently require that every irreducible representation, having χ as a character, be irreducible. That allows us to define the degree of a character as the degree of any complex representation T of Γ whose character is χ.

A fundamental result in theory of representation, concerning irreducible characters, is the following:

Theorem 1.36. Let Γ be a finite group and let {χ_i}r

i=1 be the set of its irreducible

characters. Then, for all χ ∈ RΓ, there exist ni ∈ Z, uniquely determined, such that:

χ =

r

X

i=1

niχi

Moreover, χ 6= 0 is the character of a representation if and only if n_i ≥ 0 for all i = 1, . . . , r.

For a proof see [36]. We look now at subgroups and quotient groups of Γ. We shall see that characters defined on subgroups and quotient groups of Γ naturally induce a character defined on the whole group.

Definition 1.37. Let Γ be a finite group and let ∆ be any normal subgroup of Γ. Given a character χ of Γ/∆, the inflation Inf_Γ/∆Γ (χ) of χ to Γ is the function defined by:

x 7→ Inf_Γ/∆Γ (χ)(x) := χ(x∆) for all x ∈ Γ.

Remark 1.38. With notation as in Definition 1.37:

1. it is clear that Inf_Γ/∆Γ (χ) is a character. In fact, if T is the representation whose character is χ, then Inf_Γ/∆Γ (χ) is the character of the representation T ◦ π, where π : Γ → Γ/∆ is the usual projection.

2. Given χ and χ0 two characters of Γ/∆, we have:

Inf_Γ/∆Γ (χ) + Inf_Γ/∆Γ (χ0) = Inf_Γ/∆Γ (χ + χ0) thus, inflation induces an homomorphism R_Γ/∆ → RΓ.

Definition 1.39. Let Γ be a finite group and let ∆ be any subgroup of Γ. Let moreover χ be a character of ∆. If {c1, . . . cn} are any left coset representatives of the quotient

group Γ/∆ then, the character induced by χ on Γ is defined as the function: x 7→ IndΓ_∆(χ)(x) =

n

X

i=1

χ(cixc−1i )

for all x ∈ Γ, where at the right-hand side χ is extended to Γ by setting χ(y) = 0 for all y /∈ ∆.

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Remark 1.40. Observe that the definition is independent of the choice of the represen-tatives ci, as χ is invariant under conjugacy by elements of ∆.

We shall show now that:

Lemma 1.41. Let Γ, ∆ and χ be as in Definition 1.37. If χ is the character of a representation T of ∆, then IndΓ_∆(χ) is the character of a representation of Γ. Moreover, as in the case of inflation, IndΓ_∆ is a group homomorphism R_∆→ R_Γ.

Proof. Let T : ∆ → GL(W ), where W is a finite dimensional complex vector space, and consider the space:

V := M

σ∈G/H

Wσ

where Wσ are copies of W for all σ ∈ G/H. We shall define a representation Γ → GL(V ) of Γ, whose character is IndΓ_∆(χ). To do that, it is sufficient to define an action of the group Γ on V . Let {cσ}σ be a set of left coset representatives of the quotient G/H.

Observe first, that G acts on the set {c_σ}_σ. In fact, given g ∈ G and σ ∈ G/H, there exists a unique τ ∈ G/H such that:

gcσ = cτh

for some h ∈ H (unique as well). Then, we set:

g · cσ := cτ (1.8)

Formula 1.8 defines an action of the group G on {c_σ}_σ. In fact, given g, g0 ∈ G, we may write:

gcσ = cτh g0cτ = cρh0

where τ, ρ ∈ G/H and h, h0∈ H. Whence:

g0gcσ = g0cτh = cρh0h

It follows that:

g0· (g · cσ) = g0· cτ = cρ= g0g · cσ

Then clearly, the elements of the set {c_σ}_σ are permuted by G. In view of this fact, given (wσ)σ ∈ V , we may define:

g · (wσ)σ := (hwσ)τ (1.9)

provided that gcσ = cτh, with h ∈ H. The verification that 1.9 is an action is identical

to that of 1.8. Thus, we have defined a structure of C[Γ] module on V , that is a repre-sentation. Let now ˜T be the representation defined by 1.9. To see that the character of

˜

T is Ind∆_Γ(χ), we choose a basis {w1, . . . , wm} of the space W and observe that:

[

σ

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1.3. ON CHARACTER THEORY 15 is a basis of V , where w_iσ is the vector wi read in the space Wσ. Hence, given g ∈ G,

the representative matrix of T (g) in the basis 1.10 has only one m by m non null block each m consecutive columns. Moreover, as G permutes the set {c_σ}_σ, these blocks never occur in the same position, with respect to lines. Clearly, we have a diagonal block at all σ such that g · cσ = cσ, i.e. such that:

c−1_σ gcσ ∈ H (1.11)

and 1.11 is precisely the automorphism h of 1.9 acting on the σ-th component. This proves the required assertion. The fact that Ind∆_Γ is a homomorphism R∆→ RΓ, follows

immediately form definition 1.39.

The reader may see also [14]. In the sequel, we shall need the following powerful result of character theory, known as Brauer induction theorem:

Theorem 1.42. Let G be a finite group and let χ be a character of G. Then, there exists a collection {Hi}ni=1 of subgroups of the group G, and degree one characters χi of the

groups H_i, for i = 1 . . . n, such that it holds: χ =

n

X

i=1

ciIndH_Gi(χi)

where the ci’s are rational integers.

A proof may be found in [39], Chapter 2.

We conclude this section by stating a corollary of Brauer’s theorem:

Corollary 1.43. Let Γ be a finite group and let T be a degree n complex representation of the group Γ. Then, there exists a complex representation ˜T of Γ, taking values in the group GL_n_{(Q), isomorphic to T .}

Proof. If n = 1 this is clear, since the representation T will take values in the group of N -th roots of unity, where N = lcm(ord(γ) | γ ∈ Γ). Suppose now that n > 1. By Brauer’s theorem, for the character χ of T , it holds:

χ = n X i=1 IndHi G (χi) (1.12)

where H_i < G and χi are degree one characters of the groups Hi, for all i = 1, . . . , n.

Clearly, the χi’s are characters of a representation and thus, by Lemma 1.41, IndH_Gi(χi)

are as well characters of a representation. For all i = 1, . . . , n, let Ti : Hi → Wi be a

representation whose character is χ. Then, in view of 1.12 and of Proposition 1.33, the representation: n M i=1 ni M j=1 Ti: G → n M i=1 ni M j=1 Wi

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is isomorphic to T . Hence, it suffices to show that Tiadmits an isomorphic representation

with values in GLmi(Q), for all i = 1, . . . , n, where mi := degTi. Now, while proving

1.41, we observed that, if degχ_i = m0_i, the matrix representing IndHi

G (χi)(g), for any

g ∈ G, has at each cluster of m0_i columns starting from the first, one and only one non null m0_i-dimensional square block. Since here m0_i = 1, all the entries of the matrix are either null or in Im(χ_i). This concludes the proof.

Remark 1.44. Note that the proof of Corollary 1.43, yields actually that, for all the representations T of Γ, there exists a representation ˜T , isomorphic to T , with values in GL(CN), where CN is the group of N -th roots of unity in C× and N := lcm(ord(γ) |

γ ∈ Γ).

1.4 Real-valued characters

We recall now some properties of real-valued characters and in particular of symplectic characters. These notions will be useful in Chapter 8.

Definition 1.45. Let K be a subfield of the field C and let Γ be a finite group. We shall say that a complex representation T of Γ is K-realizable, if there exist a K vector space V0

and a homomorphism T0: Γ → GL(V ), such that the Γ-representation T0⊗1 : Γ → V0⊗C

is isomorphic to T .

Remark 1.46. Note that Definition 1.45 depends only on the isomorphism class of the representation T and hence, only on its character. We shall adopt the same terminol-ogy for characters and the subgroup of R_Γ generated by K-realizable characters will be indicated by RK

Γ

In the sequel, we shall be concerned with the case K = R. In the R-realizable case, there are two important types of characters:

Definition 1.47. Let Γ be a finite group and let T be a representation of Γ with values in the group GL(V ), where V is a finite dimensional vector space on C. We say that the character χ of T is orthogonal (respectively symplectic), if χ leaves invariant a non-degenerate symmetric (respectively alternating) bilinear form on the space V , i.e. there exists a non degenerate symmetric (respectively alternating) bilinear form B on the space V , such that:

B(T (γ)v, w) = B(v, T (γ)w) for all γ ∈ Γ and for all v, w ∈ V .

The subgroup of R_Γ generated by all the orthogonal (respectively symplectic) char-acters of Γ is denoted by Ro_Γ ( respectively Rs_Γ) and is called the subgroup of virtual orthogonal (respectively symplectic) characters. We shall write RΓ(R) for the subgroup

of the real-valued characters in R_Γ. With this notation, the following result holds: Proposition 1.48. Let Γ be a finite group, then:

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1.5. ABELIAN AND NON ABELIAN L-FUNCTIONS 17 i) RR Γ = RoΓ ii) RΓ(R) = Ro_Γ+ Rs_Γ iii) R_Γo ∩ Rs Γ= {χ + χ | χ ∈ RΓ}

For a proof see [36], page 121.

1.5 Abelian and non Abelian L-functions

Abelian L-functions were the first kind of L-functions introduced in algebraic number theory. As in the analytic case, they are a weighted version of the Dedekind zeta function ζk, a generalization of the classical Riemann ζ to the case of a number field k larger than

Q. In [13] Artin further enlarged the concept of L-function to a more general non abelian case. Both these cases case be discussed in this section. The reader requiring a deeper insight into these topics may see [25], chapter XII.

Let k be a number field and let χ be an abelian character of k. We shall now define the conductor of χ. Note first that for every finite place p of k, the powers of the ideal pOkp

are a fundamental system of neighbourhoods at the point 0 in kp, and thus, the sets Upi

with i running through the positive integers are a fundamental system of neighbourhoods at the point 1. It follows that, given any open subgroup H of J (k), there exists an index i such that the set H ∩ kp contains the subgroup Upi. Moreover, being H a finite index

subgroup of J (k), it is plain that for almost all the finite places p of k, the set H ∩ k_p contains the ring Okp, and thus, the group Up. We are now in the position to define the

conductor of χ.

Definition 1.49. Let k be a number field and let χ be an abelian character of k. Let moreover n(χ, p) be the least integer (≥ 0) such that the set Upn(χ,p) is contained in the

kernel of χ, for all finite places p of k. Then, we define the conductor of the character χ as the ideal:

f (χ) =Y

p

pn(χ,p) (1.13)

where p runs through the finite places of k. Remark 1.50. Some remarks:

1. note that almost all the factors in 1.13 are trivial and thus, f (χ) is an ideal of the field k. Moreover, in view of Theorem 1.23, the primes dividing f (χ) are exactly those ramified in the abelian extension induced by the character χ via class field. 2. It is important to mention that some authors define the conductor of a character χ

as a cycle, i.e an ideal having some infinite factors, and take into consideration the ramification of χ at the infinite places of k (see for example [25], pag. 229). This is unnecessary to our purposes and thus, we restrict our definition of conductor to its finite components.

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Let now p be fixed prime ideal of k and let χ be an abelian character. Suppose that χ is unramified at p. We set by definition χ(p) := χ(π), where π is a uniformizing element for the ideal p in k_p. As usual π is seen as an element of J (k). This definition is independent of the choice of π, as the character χ is trivial on the group of units Up by

hypothesis. On the other hand, if χ is ramified at p, we define χ(p) = 0. With reference to this convention, we give the following definition:

Definition 1.51. Let k be a number field and let χ be an abelian character of k. We call abelian L-function associated with the character χ, the following complex function, defined for Re(s) > 1:

L(s, χ) =Y

p

1 1 −_{N (p)}χ(p)s

(1.14) where p runs through the finite primes of k and N stands for the norm in the extension k|Q.

Remark 1.52. Observe that:

1. when computing product 1.14, only the primes not dividing the conductor need to be taken into consideration, i.e. the primes at which χ is unramified (see 1.50, point 1).

2. Product 1.14 is convergent in the half-plane Re(s) > 1. For a proof of this fact, see [25] pag.162.

3. Similarly to the L-functions over Q, these new L-functions admit a meromorphic continuation defined on the whole complex plane, which satisfies a functional equa-tion. We do not go further into this here, since these topics will be properly discussed for the non abelian case, which we now proceed to introduce.

Let K|k be a finite normal extension of number fields, and let Γ be its Galois group. Let moreover p be a finite place of k. It is well known that for each place P of K over p, there exists an automorphism σ of K|k, such that for each x ∈ O_K it holds:

σ(x) ≡ xN (p) (mod P )

where N stands for the norm in the extension k|Q. Such a map is called a Frobenius automorphism and is uniquely determined modulo the inertia group of the prime P over p. Now, given χ a character of Γ, for each integer m ≥ 0, we set:

χ(pm) = 1 e(P |p)· χ X τ ∈T (P |p) σmτ ! (1.15) where e(P |p) is the ramification index of P over p and T (P |p) is the inertia group. Clearly, being χ invariant under conjugation, the right-hand side is independent of the choice of P and thus, the value of χ on the powers of p is well defined. Note that 1.15 agrees with the abelian definition (see 1.55). Using the notation just introduced we are able to define the non abelian L-series of a character χ of the group Γ. This is done via complex logarithm.

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1.5. ABELIAN AND NON ABELIAN L-FUNCTIONS 19 Definition 1.53. Let K|k be a normal extension of number fields and let χ be a character of the Galois group of the extension K|k. Then, with notation as in 1.15, on the half-plane Re(s) > 1 we define: logL(s, χ, K|k) = −X p X m≥0 χ(pm) mN (p)ms (1.16)

where log is any branch of the complex logarithm on the half-plane Re(s) > 1 and p runs through the finite places of the field k.

Remark 1.54.

1.16 gives a well defined function L(s, χ, K|k), obtained by exponentiating the logarithm, called the non abelian L-function associated with the character χ.

Once again, it is easily seen that the series at 1.16 converges uniformly on the half-plane Re(s) > 1 + δ for all δ > 0, by comparison with the usual zeta function.

The non abelian L-functions coincide with the abelian L-functions on abelian exten-sions, taking into account the fact that abelian characters are actually degree one non abelian characters, via composition by the Artin map. In fact, we have:

Lemma 1.55. Let K|k be an abelian extension of number fields and let χ be a character of the group Gal(K|k), whose kernel correspond to the extension K|k via Theorem 1.17. Then, χ ◦ A_K|k, where A_K|k is the Artin map defined at 1.19, is an abelian character of the field k. Let now L(s, χ) be the abelian L-function attached to the character χ ◦ A_K|k and let L(s, χ, K|k) be the corresponding non abelian L-function. Then, for Re(s) > 1, it holds:

L(s, χ) = L(s, χ, K|k) Proof. Clearly, for all s ∈ {Re(s) > 1}, we have:

logL(s, χ) =X p log 1 1 −_{N (p)}χ(p)s = −X p X m≥0 χ(p)m mN (p)ms (1.17)

where in the first sum p runs through the finite unramified places of the field k and the value χ(p) is calculated according to Definition 1.51. Fix now p an unramified prime of k and π a uniformizing parameter in the completion of k with respect to p. Recalling the definition of A_K|k given at 1.19, we have:

χ(π) = χ ◦ A−1_K|k(A_K|k(π)) = χ ◦ A−1_K|k(σ) (1.18) where σ is the Frobenius automorphism of the prime p (note that this automorphism is uniquely determined, as p is unramified in K|k and the decomposition groups of the primes lying over p are coincident). Here the element α mentioned in Remark 1.19 may be taken to be 1. Thus, 1.18 shows that the abelian and non abelian definitions of χ(p) (see 1.51 and 1.53) coincide in the unramified case, since the inertia group of the prime

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p is trivial. Suppose now that p is ramified in K|k. Then, recalling that χ ◦ A−1_K|k is a homomorphism, for all the primes P of K over p, we have:

χ X

τ ∈T (P |p)

σmτ !

= 0 (1.19)

by the well-known relation of orthogonality (note that it is essential that the inertia group is non trivial). Hence, again the two definitions coincide. Now, in view of 1.18 and of 1.19, for all the places p of k, it holds χ(pm) = χ(p)m, according to the non abelian definition. Thus, 1.17 yields the required equality.

Let’s go back now to the general case. We shall show here an alternative way to define non abelian L-series. Let T be a representation of Γ = Gal(K|k). For each subset S of Γ we set:

T (S) :=X

τ ∈S

T (τ )

Note that the right-hand side might not be an invertible endomorphism. Chosen now p a finite place of the field k and P a place of the field K over p, we define:

T (pm) := 1 e(P |p)T (σ

m_{)T (E (P |p))}

where σ is any Frobenius automorphism of P over p and E (P |p) stands for the inertia group of the place P . It is a straightforward verification to see that this definition is independent of the choice of the prime P . Now, the series:

X p X m≥0 T (pm) mN (p)ms

is convergent in norm, and if χ is the character of T , we find that for all the finite places p of k, it holds: X m≥0 χ(pm) mN (p)ms = T r X m≥0 T (pm) mN (p)ms ! = −log det I − T (p) N (p)s

where the last equality is an easy generalization of the logarithm expansion for matrices. Let now VE(P |p) indicate the subspace of V fixed by the inertia group E (P |p) of P over p, via the action of T . The reader may easily verify that the endomorphism:

1 e(P |p)

X

τ ∈T (P |p)

T (τ )

is the orthogonal projection of the space V onto the subspace VE(P |p) with respect to any non degenerate scalar product induced by a basis adapted to VE(P |p). This implies that: det I − T (p) N (p)s = det I − T (σ) N (p)s |VE(P |p)

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1.5. ABELIAN AND NON ABELIAN L-FUNCTIONS 21 for any Frobenius automorphism σ of P over p. Note that the restriction of σ to the sub-space VT (P |p) is independent of the choice of σ. Thus, we have the following alternative way to write the L-series associated with the extension K|k and the character χ:

L(s, χ, K|k) =Y p 1 det I −_{N (p)}T (σ)s |VE(P |p)

where p runs through the finite places of the field k and P is any place of K over p, for all p. Moreover, σ indicates here any Frobenius automorphism of P over p.

We are now interested in understanding how L-functions change when they are calculated on inflated and induced characters (see 1.37 and 1.39), as such modifications turn out to be natural when considering characters of fields extensions. Namely, we have:

Proposition 1.56. Let K|k be a normal extension of number fields and let Γ be its Galois group. Then, for Re(s) > 1:

1. For all characters χ and χ0 of the group Γ, we have:

L(s, χ + χ0, K|k) = L(s, χ, K|k) · L(s, χ0, K|k) (1.20) 2. Let K0|k be a finite normal extension such that K0 ⊃ K and let Γ0 be its Galois

group. For all characters χ of Γ, we have:

L(s, Inf_ΓΓ0(χ), K0|k) = L(s, χ, K|k) (1.21) 3. Let K0|k be a finite normal extension such that K0 _{⊂ K and let ∆ be the Galois}

group of the extension K|K0. For all characters χ of ∆, we have:

L(s, IndΓ_∆(χ), K|k) = L(s, χ, K|K0) (1.22) Proof. 1.20 and 1.21 are obvious, while 1.22 requires a non trivial proof (the reader may find it in [25], pagg. 236-239). Note that 1.20 allows us to define L-functions on virtual characters.

We conclude this section by introducing Artin root numbers. We shall show how to enlarge the functions we defined at 1.53, to the whole complex plane. Thus, property of symmetry will arise, showing that the behaviour of these new L-functions is not so different from that of the classical L-functions seen in analytic theory. The Artin root numbers will be the coefficients of the functional equation expressing this symmetry. To do so, we first need to define a non abelian version of the conductor, the so called Artin conductor (more can be found in [7] and in [27] pages 13,14). Consider K|k a Galois extension of number fields and let χ be a character of the Galois group Gal(K|k). Suppose that χ derives from a representation T with values in GL(V ), where V is a finite dimensional vector space on the field C. Fix then p a prime in k and P a prime in K

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over p and let Ti be the ramification groups of P over p for i ≥ 0, where it is understood

that T0 is the inertia group. We set:

n(χ, p) =X

i≥0

|T_i|

|T₀|· codimV

Ti _(1.23)

here, VTi _{is the vector subspace of V fixed by the group T}

i via the action of T . Clearly

this is a finite sum, as the Ti’s are definitely trivial. Note that n(χ, p) is independent

of the choice of the prime P over p. Moreover, it can be shown that n(χ, p) is always an integer (this is done by using Brauer induction theorem, see [35], chap. VI, § 1-3). In view of the preceding considerations, we are now in the position to give the following definition:

Definition 1.57. Let K|k be a normal extension of number fields and let χ be a character of Gal(K|k). We define the Artin conductor of the character χ to be the ideal:

f (χ) =Y

p

pn(χ,p)

where p runs through the finite primes of the field k and n(χ, p) is the integer defined at 1.23.

By means of Theorem 1.23 it can be shown that the Artin conductor is a generalization of the abelian conductor defined at 1.49. For a proof of this fact, see [35], page 228, Corollary 2. We set now:

A(χ) = |dk|χ(1)Nk|Q(f (χ)) (1.24)

where d_k indicates the discriminant of the field k and N_k|Q the norm in the extension k|Q. Moreover, given v an infinite place of the field k, if v is complex, we define:

γ_χv(s) = (γ(s)γ(s + 1))χ(1) (1.25) where γ(s) = π−s/2Γ(s/2) and Γ is the Euler gamma function. If v is real, we select w a place of K over v and consider the decomposition group of w over v, defined at 1.21. The generator of this subgroup σ_w, depends up to conjugation solely on v and hence, its image under the representation T does. Now, T (σw) is at most an order 2 automorphism

of the vector space V and thus, its eigenvalues are ±1. We can then decompose the vector space V into the direct sum:

V = V_v+⊕ V_v−

where V_v+is the eigenspace associated with the eigenvalue 1 and V_v−that associated with the eigenvalue −1. We define then:

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1.5. ABELIAN AND NON ABELIAN L-FUNCTIONS 23 Finally, we set: γχ(s) = Y v γ_χv(s)

where v ranges over the infinite places of k. With this notation, we are able to define the enlarged L-function:

Definition 1.58. Let K|k be a normal extension of number fields and let χ be a character of the group Gal(K|k), then with notation as above, we define the enlarged L-function on the half-plane Re(s) > 1 as:

Λ(s, χ, K|k) = A(χ)s/2· γ_χ(s) · L(s, χ, K|k) (1.27) where L(s, χ, K|k) is the non abelian L-function defined at 1.53.

We invite the reader to compare 1.27 with the function: ξ(s) = 1

2s(s − 1)π

−s/2_Γ(s/2)ζ(s) _(1.28)

used in analytic theory to define a meromorphic continuation of the Riemann zeta func-tion. The constant A(χ) defined at 1.24 plays the role of π in 1.28, while the so called "gamma factors", defined at 1.25 and 1.26, substitute the classical gamma function.

Now, observe that if χ is the character of a representation of some finite group Γ, then the same holds for χ, i.e. the complex conjugate of χ. In fact, if T : Γ → GL(V ) is a representation such that χ is its character,then χ may be seen as the character of the representationT : Γ → GL(V∗) defined by:

T (g)(f )(v) = f (T (g)−1(v))

for all f ∈ V∗, v ∈ V and g ∈ Γ. In the definition above V∗ stands for the dual space of the vector space V . The fundamental result concerning enlarged L-functions is the following:

Theorem 1.59. Let K|k be a normal extension of number fields and let χ be a character of the group Gal(K|k). Then, the enlarged L-function Λ(s, χ, K|k), defined at 1.58, possesses a meromorphic continuation defined on the whole complex plane, which satisfies the functional equation:

Λ(1 − s, χ, K|k) = w(χ) · Λ(s, χ, K|k)

where w(χ) is a constant of absolute value 1, called Artin root number of the character χ.

For a proof see [27], pages 14-18.

An immediate corollary to Theorem 1.59 is the following: Corollary 1.60. In the hypotheses of Theorem 1.59, we have:

w(χ)w(χ) = 1 (1.29) Moreover, if χ is real-valued, then necessarily w(χ) = ±1.

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Now, enlarged L-functions satisfy equalities similar to 1.20, 1.21 and 1.22, and from these equalities we deduce analogous properties for the Artin root numbers. Namely, we have:

Proposition 1.61. Let K|k be a normal extension of number fields and let Γ be its Galois group. Then:

1. For all characters χ and χ0 of the group Γ, we have:

w(χ + χ0) = w(χ) · w(χ0) (1.30) 2. Let K0|k be a finite Galois extension such that K0 _{⊃ K and let Γ}0 _{be its Galois}

group. For all characters χ of Γ, we have:

w(Inf_ΓΓ0(χ)) = w(χ) (1.31) 3. Let K0|k be a finite normal extension such that K0 _{⊂ K and let ∆ be the Galois}

group of the extension K|K0. For all characters χ of ∆, we have:

w(IndΓ_∆(χ)) = w(χ) (1.32)

1.6 Gauss sums and local Artin root numbers

In this section we introduce a local version of the Artin root number for abelian charac-ters, defined by means of the so called local Gauss sums, yet to be defined. The reader wishing a deeper insight into these topics, may refer to [27], chapter II, pages 26-33. We restrict ourselves to the abelian case, as the notion of local abelian root number will be necessary to prove Fröhlich’s conjecture. A non abelian generalization of these local constants was provided by Deligne and Langland (see Chapter 7 and also [37]), but we do not go deeper into this now.

Let k be a finite extension of the field Qp, for a fixed rational prime p. We shall build

up a non trivial additive homomorphism ψ from k to C×, which will be used to define local Gauss sums for abelian characters. This homomorphism is the composition of the following four maps:

k (1) //Qp (2)_// Qp/Zp (3) _// Q/Z (4) _// C× (1.33) where: • (1) is the trace T r_k|Q_p _{: k → Q}p;

• (2) is the canonical projection;

• (3) is the canonical injection, i.e. the map cutting out positive powers in the p-adic series expansion;

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1.6. GAUSS SUMS AND LOCAL ARTIN ROOT NUMBERS 25 • (4) is the exponential map x 7→ e2πix_.

Let now k be a number field and let χ be an abelian character of k. We shall indicate by f (χ) the conductor of the character χ and by n(χ, p) the exponent of the prime p in the factorisation of f (χ), as to 1.49. We shall moreover denote by D_k, the different of the extension k|Q. We now give the following definition:

Definition 1.62. Let k be a number field and let χ be an abelian character of k. Given p a finite place of k, we shall call local Gauss sum of the character χ with respect to the place p, the number:

τp(χ) := X x∈Up/Upn(χ,p) χ x c ψ x c (1.34)

where c is a generator of the ideal D_kf (χ) · Okp. The map ψ is defined at 1.33 and U

i p is

the group at 1.18.

Local Gauss sums satisfy the two following properties:

Proposition 1.63. Let k be a number field and let χ be an abelian character of k. For any prime p of k, we have:

• |τ_p(χ)| =pN (fp(χ))

• τp(χ)τp(χ) = χ(−1) · N (fp(χ))

where fp(χ) indicates the p-component of the conductor f (χ) and N stands for the norm

in the extension kp|Q.

For a proof see [27], pages 30-32.

Remark 1.64. Note that τp(χ) = 1 for almost all the primes p of k. In particular, if χ

is unramified at p, then, by definition, χ is trivial on the group U_p and the sum at 1.34 reduces to 1 since in this case the p-component of the conductor is trivial. By Theorem 1.23, this happens for all the unramified primes in the extension associated with the kernel of χ, which are all but a finite number.

In order to introduce local components for the Artin root numbers, we need to extend the index n(χ, p), defined at 1.49, to the infinite places of k. This is done as follows: Definition 1.65. Let k be a number field and let v an infinite place of k. Let moreover χ be an abelian character of the field k. Suppose that K|k is the abelian extension associated with the group Ker(χ) by means of the class field correspondence, and let w be a place of K over v. If v is complex, we set n(χ, v) := 0. If v is real, we set n(χ, v) := 0 when w is real and n(χ, v) := 1 when w is complex.

It is clear that this definition is independent of the choice of w, being K|k a Galois extension. This is a way to define the infinite components of the conductor, as it was recalled at 1.50, point 2.

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Remark 1.66. Let K|k be the abelian extension associated with the character χ and let v be a place of the field k. If w is any place of K over v and σv is the generator of

the inertia group of w|v (independent of w, see 1.21), we have: n(χ, v) = 1

2(χ(1) − χ(σv)) The proof is straightforward.

We are now able to define local root numbers:

Definition 1.67. Let k be a number field and let χ be an abelian character of k. Let moreover v be a place of k. If v is finite, we define:

wv(χ) :=

τv(χ)

pN (fv(χ))

while, if v is infinite, we set:

wv(χ) := i−n(χ,v)

where fv(χ) is the v-component of the conductor f (χ) and N is the norm in the extension

kv|Qv. Here, Qv stands for the completion of the field Q with respect to the restriction

of the place v to Q.

Remark 1.68. Note that w_v(χ) is trivial for almost all the places v of k, since by Remarks 1.64 and 1.50, point 2, wv(χ) is trivial for all the finite unramified places v.

Global and local root numbers are linked by the following fundamental relation: Theorem 1.69. Let k be a number field and let χ be an abelian character of k. Then it holds:

w(χ) =Y

v

wv(χ) (1.35)

where v runs through all the places of k. For a proof see [37].

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Chapter 2

Modules over the quaternion ring

In this chapter we present a first proof of Fröhlich’s conjecture, to be completed in Chapter 4. As it was recalled in the Introduction, this proof is essentially based on [26] and [15]. We start with a classification of the projective rank 1 modules over the ring Z[H8], carried out by Martinet in [26]. This classification will allow us to determine

the number [O_N] (recall that [ON] is of order 2, see Proposition 4.4) in terms of some

purely arithmetic invariants of the extension N |Q. The proof of 0.1 will be completed by evaluating the Artin root number at the right-hand side, as to Fröhlich’s calculations in [15].

2.1 Classification of projective rank one modules

We know from 1.8 that if N |Q is a finite, normal and at most tamely ramified extension, the ring of integers O_N of the field N , is a rank 1 projective module over the quaternion ring Z[H8] and thus, [ON] ∈ Cl(Z[H8]). This explains why we are led to investigate the

nature of projective rank 1 modules over the ring Z[H8].

Let now M be a projective rank 1 module over the ring Z[H8]. We consider the two

following submodules of M :

M0= {x ∈ M | σ2x = x} M00= {x ∈ M | σ2x = −x} (2.1) Clearly M0 _{is also a module over the quotient ring Z}0 _{= Z[H}₈]/(σ2_{− e) and M}00 _{is a}

module over the ring Z00= Z[H8]/(σ2+ e). In his paper Martinet shows that M0 and M00

are actually free modules respectively over Z0 and over Z00. For a proof see [26], page 400. Here the fact that M is a projective module turns out to be crucial. From this result it follows that:

M0 = (e + σ2)M (2.2) M00= (e − σ2)M (2.3) To prove this, just observe that M0 _{is isomorphic to Z}0, which is actually isomorphic to Z[H8]0, it is then a straightforward calculation to see that Z[H8]0 = (e + σ2)Z[H8]. An

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analogous proof works for M00.

Then, we have the following important result:

Proposition 2.1. Let M be a projective rank 1 module over the ring Z[H8] and let M0

and M00 be as in 2.1. Then, one and only one of the following conditions hold:

a) there exist φ ∈ M0, basis of M0 _{over Z}0, and ψ ∈ M00, basis of M00 _{over Z}00, such that:

ψ ≡ φ (mod 2M )

b) there exist φ ∈ M0, basis of M0 _{over Z}0, and ψ ∈ M00, basis of M00 _{over Z}00, such that:

ψ ≡ (σ + τ + στ )φ (mod 2M ) Proof. Let m0 ∈ M0, then, by 2.2, for some m ∈ M we have:

m0 = (e + σ2)m = (e − σ2)m + 2σ2m Hence, if π : M → M/2M is the canonical projection, it holds:

π(M0) = π(M00) (2.4) Moreover, we have that:

M0∩ 2M = 2M0 (2.5) To see that, write (e + σ2)m = 2 ˜m, for some m, ˜m ∈ M , and then multiply both sides by e − σ2. It follows that:

2(e − σ2) ˜m = 0

Looking at this relation in M00, we have that (e − σ2) ˜m must be null (recall that M00 is a free module over the ring Z00), and thus, ˜m ∈ M0. The same holds for M00. From 2.4 and 2.5 and from the analogous equations for M00, we deduce that:

M0 2M0 ∼= M 2M and M00 2M00 ∼= M 2M (2.6) via the injection m0+ 2M0 7→ m0_{+ 2M and the analogous injection for M}00 _{(recall that}

M = M0+ M00)

It is an easy exercise to see that Z0 = Z[C2× C2], where C2 is the cyclic group of order

2. An obvious consequence of this fact is that:

Z0/2Z0= F2[C2× C2]

Moreover, as (2, e − σ2) = (2, e + σ2) in Z[H8], we also have that:

Z00/2Z00= F2[C2× C2]

It is then clear that M0/2M0 and M00/2M00 are free modules over F2[C2× C2]. Equality

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2.1. CLASSIFICATION OF PROJECTIVE RANK ONE MODULES 29 the respective modules over Z0 and Z00. Then, clearly their projections φ and ψ are bases of M0/2M0 and M00/2M00 as modules over F2[C2× C2]. Hence, there exists an invertible

element λ ∈ F2[C2× C2] such that ψ = λφ. Set now:

λ = a0e + a1s + a2t + a3st (2.7)

where a0, . . . , a3 ∈ F2 and C2× C2= hs, ti. Then:

λ2 = a0+ a1+ a2+ a3∈ F2

but as λ is invertible, it must hold λ2 = 1, whence, there are two possible cases: a) in 2.7 three of the ai’s are null;

b) in 2.7 only one of the a_i’s is null.

In case (a), up to substituting φ by σφ, τ φ or στ φ, we have φ ≡ ψ (mod 2M ). In case (b), up to the same substitutions, we have ψ ≡ (σ +τ +στ )φ (mod 2M ). It remains to be proved that cases (a) and (b) are incompatible. Suppose by contradiction that there are two couples of bases φ, ψ and φ0, ψ0, such that ψ ≡ φ (mod 2M ) and ψ0 ≡ (s + t + st)φ0 (mod 2M ). By hypothesis, φ and φ0 differ by an invertible element in F2[C2× C2] and

the same holds for ψ and ψ0. Hence, in M0/2M0, we have: φ = ψ

(s + t + st)µφ = µ0ψ

where µ, µ0 _{∈ F}2[C2× C2]× (we omit to indicate the projection, by abuse of notation).

Whence:

(−µ0µ−1+ s + t + st)µφ = 0 (2.8) Observe now, that Z0∼= Z[C2× C2] via the homomorphism induced by σ 7→ s and τ 7→ t.

Moreover, if Q(H8) is the quaternion division ring over the field Q, generated by the

elements i, j, k, such that i2 = j2 = k2 = −1 and ij = −ji = k, then Z00 ∼= Z(H8), the

subring of Q(H8) formed by the integral combinations, via the homomorphism induced

by σ 7→ i and τ 7→ j. The unique invertible elements in Z[C2 × C2] are ±e, ±s, ±t and

±st (see [26]), while in Z(H8) the unique invertible elements are ±1, ±i, ±j, ±k. Hence,

µ, µ0 = ±e, ±s, ±t, ±st. This shows that 2.8 is a contradiction, since inside M0 (recall that M0/2M0 ,→ M/2M ) the left-hand side has at least one odd coefficient, while the right-hand side does not.

We proceed now with the classification of the projective rank 1 modules over the quaternion ring Z[H8], by creating a set of representatives up to isomorphism. For each

odd integer a, consider the free abelian group Pa generated by the elements {e0, eγ| γ ∈

H8, γ 6= e}, endowed with the Z[H8]-module structure defined by:

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αeγ=

(

eαγ if α 6= γ−1

e1:= ae0−P_γ6=eeγ if α = γ−1

for all α, γ ∈ H₈. An easy verification shows that αe₁ = eα for all α ∈ H8\ {e}. Now,

the following result holds:

Proposition 2.2. For all odd integers a, let P_a _{be the Z[H}₈]-modules defined by the relations above, then:

1. Pa∼= Pa0 if and only if a ≡ ±a0 (mod 8);

2. P_a _{is a projective rank 1 module over the ring Z[H}₈];

3. For all projective rank 1 modules M over the ring Z[H8], there exists an odd integer

a such that M ∼= Pa.

Proof. Let’s start with point (1). Suppose that a0 _{= ±a + 8k for some k ∈ Z, then, a} straightforward verification yields that the application induced by:

(

e0 7→ e0

eγ 7→ ±eγ+ ke0 γ 6= e

is an isomorphism P_a → P_a0. We may then reduce to consider only P₁ and P₃ as models.

To conclude the proof of (1) we need to show that P1 P3. This is done as follows. It is

clear that P1 and P3 are rank 1 (the rank was defined in Proposition 1.4), as αe1 = eα

for all α 6= e and a becomes invertible when making tensor product by Q. Note that this argument shows that P1 is actually free over Z[H8] (it is immediate to see that e1 is

torsion free). It follows then, that P₁0 is generated by (e + σ2)e1 over the ring Z0 and that

P₁00 is generated by (e − σ2)e1 over the ring Z00. These two elements respect condition

(a) of Proposition 2.1 and thus, to prove the claim, it is sufficient to find two elements in P3 respecting condition (b) of the same proposition. Now, P3 is generated by e0 and

e1, hence we have:

P₃0 = (e + σ2)P3 = h(e + σ2)e0, (e + σ2)e1i

and an easy check shows that the element φ = e0− e1− eσ2 generates both these elements

over Z0. Hence, φ is a basis of P₃0. On the other hand:

P₃00 = (e − σ2)P3= h(e − σ2)e0, (e − σ2)e1i

and it is easy to see that ψ = e₁− e₀ is a basis of P₃00 _{over Z}00. Clearly (s + t + st)φ ≡ ψ (mod 2M ) and the proof of (1) is concluded.

We now prove (2). It remains to be shown that P3 is projective. By Swan’s Theorem

(1.3), this is equivalent to prove that P₃ is locally free. Fix then p a prime integer. We have to look at P3,p = Zp⊗ZP3 as a module over the ring Zp⊗ZZ[H8], where Zp denotes

the ring of p-adic integers. If p 6= 3 then, as for P1, the element e1 is a generator of the

On Frohlich's conjecture for tamely ramified quaternion extensions

UNIVERSITÀ DEGLI STUDI DI PISA

Tesi di Laurea

ON FRÖHLICH’S CONJECTURE FOR

TAMELY RAMIFIED QUATERNION

EXTENSIONS

Contents

Introduction

Part I

A handmade proof

Chapter 1

Preliminary Notions

1.1

The locally free class group

1.2

Some class field theory

1.3

On character theory

1.4

Real-valued characters

1.5

Abelian and non Abelian L-functions

1.6

Gauss sums and local Artin root numbers

Chapter 2

Modules over the quaternion ring

2.1

Classification of projective rank one modules