Local theta-regulators of an algebraic number -- p-adic Conjectures

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Local theta-regulators of an algebraic number – p-adic Conjectures

Georges Gras

To cite this version:

Georges Gras. Local theta-regulators of an algebraic number – p-adic Conjectures. Canadian Journal of Mathematics., 2016, Vol. 68 (3), pp.571-624. �hal-01429498�

English translation of the original article:

Lesθ-r´egulateurs locaux d’un nombre alg´ebrique – Conjecturesp-adiques Canadian Journal of Mathematics, Vol. 68, 3 (2016), 571–624 LOCAL θ-REGULATORS OF AN ALGEBRAIC NUMBER

p-ADIC CONJECTURES

GEORGES GRAS

Abstract. LetK/Qbe Galois and letη∈K^×be such that the multiplicative Z[G]-module generated byη is ofZ-rankn. We define the localθ-regulators

∆^θ_p(η) ∈ Fp for the Qp-irreducible characters θof G = Gal(K/Q). Let Vθ

be the θ-irreducible representation. A linear representation L^θ ≃ δ V_θ is associated with ∆^θ_p(η) whose nullity is equivalent toδ ≥ 1 (Theorem 3.9).

Each ∆^θ_p(η) yields Reg^θ_p(η) modulopin the factorizationQ

θ Reg^θ_p(η)ϕ(1)

of Reg^G_p(η) := ^Reg_p[K:Q]^p(η) (normalized p-adic regulator of η), where ϕ |θ is ab- solutely irreducible. From the probability Prob ∆^θ_p(η) = 0 &L^θ ≃δ V_θ

≤ p^{−f δ2} (f = residue degree of p in the field of values of ϕ) and the Borel–

Cantelli heuristic, we conjecture that, forplarge enough, Reg^G_p(η) is ap-adic unit or thatp^ϕ(1)kReg^G_p(η) (existence of a singleθofGwithf=δ= 1 and no extrap-divisibility); this obstruction may be lifted assuming the existence of a binomial probability law (Sec. 7) confirmed through numerical studies (with groupsG=C3,C5,D6). This conjecture would imply that, for allp large enough, Fermat quotients of rationals and normalizedp-adic regulators arep-adic units (Theorem. 1.1), whence the fact that number fields arep- rational forp ≫0. We recall§8.7 some deep cohomological results, which may strengthen such conjectures.

1. Introduction

Let K/Qbe a Galois extension of degree n of Galois groupG. Let η ∈ K^×. An exponential notation is used for conjugation of η by σ ∈ G, which implies the writing (η^σ)^τ =:η^{τ σ} for all σ, τ ∈G(law of leftG-module). We assume that the multiplicativeZ[G]-module generated by η is of Z-rankn (i.e.,hηiG⊗Q≃Q[G]).

Forplarge enough, we put Reg^G_p(η) := det ⁻_p¹log_p(η^{τ σ−1})

σ,τ∈G(normalizedp-adic regulator ofη).

We shall see that the unique obstruction, to apply the heuristic principle of Borel–

Cantelli leading (conjecturally) to a finite number of p such that Reg^G_p(η) ≡ 0 (modp), is related to primespsuch that Reg^G_p(η) is exactely divisible by a minimal power of p; this is equivalent to Reg^G_p(η) ∼ p^ϕ(1) (equality up to a p-adic unit), where the characterϕ(absolutely irreducible) defines ap-adic characterθsatisfying certain conditions (Definition 4.1).

Date: January 7, 2017.

1991 Mathematics Subject Classification. Primary 11F85; Secondary 11R04; 20C15; 11C20;

11R37; 11R27; 11Y40.

Key words and phrases. p-adic regulators; Frobenius group determinants; p-adic characters;

Leopoldt–Jaulent conjecture; Abelianp-ramification;p-rationality; Fermat quotient; probabilistic number theory.

1

Such a situation is a priori of probability at most ^O(1)

p , only when η is consid- ered as a random variable; it is the unique case where the Borel–Cantelli principle does not apply (see Section 9 for some enlightenment). We intend, from heuristics and numerical experiments, to remove this obstruction and to reach the following probabilistic result,when η is fixed andp→ ∞:

Theorem 1.1. LetK/Qbe a Galois extension of degree nand of Galois group G.

Let η∈K^× be fixed,η generating a multiplicativeZ[G]-module of Z-rankn.

(i) Under the Heuristic 7.4 (existence of a classical binomial law of probability), the probability to have Reg^G_p(η)≡0 (modp) is at most C_∞(η)

p^{log2(p)/log(c0(η))−O(1)} for p→ ∞, wherec0(η) = maxσ∈G(|η^σ|),e⁻¹≤C_∞(η)≤1, andlog₂= log◦log.

(ii) Under the previous heuristic 7.4 and the principle of Borel–Cantelli, the number of primes psuch that Reg^G_p(η)≡0 (modp)is finite.

We shall always suppose that the prime numberpthat we consider is large enough, in particular odd, not divisor of n, unramified in K, and prime to η, so that the normalizedp-adic regulator Reg^G_p(η) :=p^−[K:Q]·Reg_p(η) makes sense inZp, where Reg_p(η) is the usualp-adic regulator ofη (see§2.1.3).

Denote by ZK (resp. ZK,(p)) the ring of integers (resp. of p-integers) of K; for K = Q, one gets Z(resp. Z(p)). For all placev|p of K, we denote by pv|pthe prime ideal associated withv.

If np is the common residue degree of the places v|pin K/Q, the multiplicative groups of the residue fields are of orderp^np−1 and for allv|pwe have the congruence η^pnp−1≡1 (modpv); hence finally, since Q

pv|ppv=p ZK, η^pnp−1= 1 +p αp(η), αp(η)∈ZK,(p), which leads, by Galois, to the relations

αp(η^σ) =αp(η)^σ for allσ∈G, and to the “logarithmic” properties

αp(η η^′)≡αp(η) +αp(η^′) (modp ZK,(p)) & αp(η^λ)≡λ αp(η) (modp ZK,(p)) (forη, η^′ ∈K^×,λ∈Z).

Thisgeneralized Fermat quotient αp(η) ofη is the key element of our study. More precisely, the properties of theG-module generated byαp(η) modulop ZK,(p)shall precise the properties of the normalized p-adic regulators of η, in particular for the search of the (rares) solutions p giving their divisibility by p. The numerical illustrations are obtained by means of PARI programs (from [P]).

2. Regulators and Representations – Local regulators

2.1. p-adic logarithm – p-adic regulators. Letpbe a fixed prime number sat- isfying the hypothesis given in the Introduction. We suppose that the number field Kis considered as a subfield ofCp. Thus, any “embedding” ofKintoCpis nothing else than aQ-automorphismσ∈G.

Letp0=pv0 be a prime ideal ofKabovepand letDp0 be its decomposition group.

The placesv|p, conjugates ofv0, correspond to the (G:Dp0) distinct prime ideals pv:=p^σ₀^v, where (σv)v|p is an exact system of representatives ofG/Dp0.

We consider theQp[G]-module Q

v|pKv whereKv=σv(K)Qp⊂Cp, is the comple- tion ofKatv; asK/Qis Galois,Kv/Qpis independent ofv|pbut the notationKv

recalls that this local extension isprovidedwith the embeddingσv : K→Kv⊂Cp, which allows the diagonal embedding with dense image

ip:= (σv)v|p:K−−−→ Q

v|pKv

whereip(x) := (σv(x))v|p, and which implies thatK⊗Qp≃ Q

v|pKv ≃Qp[G] (semi- local theory). By abuse, ifx∈K, we shall write x∈ Q

v|pKv,ip being understood.

2.1.1. p-adic logarithm onK^×. Thep-adic logarithm log_p:K^×→KQp is defined on the set

1 +p x, x∈Z_K,(p) , by means of the usual series (p >2) log_p(1 +p x) = P

i≥1(−1)^{i+1 (px)i}

i ≡p x (modp²), noting thatPN

i=1(−1)^i+1(px)i

i ∈Kfor anyN ≥1. In the case ofγ∈K_(p)^× , we use the functional relation

log_p(γ) = ¹

p^np−1log_p(γ^pnp−1) = ¹

p^np−1log_p(1 +p αp(γ))≡ −p αp(γ) (modp²).

More generally, this log_p function, seen modulo p^N+1, N ≥ 1, is represented by elements ofZK,(p)and is an homomorphism ofG-modules for the law defined, for all σ∈ G, by σ log_p(γ) (mod p^N+1)

:= log_p(γ^σ) (mod p^N+1), using the congruence (whereN^′ is an obvious function ofN)

σ log_p(γ) (modp^N+1)

≡ _pnp¹−1

P

1≤i≤N^′(−1)^{i+1(p αp(γ)σ)i}

i (modp^N+1), defining an element of Z_K,(p) which approximates log_p(γ^σ) modulo p^N+1. So σv(log_p(γ)) makes sense inKv for allv|p.

2.1.2. p-adic rank. Let Log_p := log_p◦ ip = (σv)v|p ◦ log_p be the homomorphism ofG-modules defined, on the subgroup of elements ofK^× prime top, by

Log_p(γ) = log_p(γ^σv)

v|p∈ Q

v|pKv.

Let η ∈ K^×, prime top, and letF be theZ[G]-module generated by η. We call p-adic rank ofF, the integer

rg_p(F) := dim_Qp(QpLog_p(F)).

The use of Log_p is a commodity since by conjugation by the elements of G, the knowledge of log_pimplies that of Log_pand conversely by projections Q

v^′|pKv^′ →Kv. To make a link with the concept ofp-adic regulator, we shall prove first the following two technical results:

Lemma 2.1. Let p be an odd prime, unramified in K, and let λ ∈ ZK,(p). If λ /∈p ZK,(p), there existsu∈K^×, prime top, such that TrK/Q(λu)6≡0 (modp).

Proof. For allu∈K^×, prime top, consider the diagonal embedding ofλuin Q

v|pKv, and let Trv be the local traces TrKv/Qp forv|p. Then

TrK/Q(λu) = P

v|pTrv(σv(λu)).

By assumption, there exists a non-empty set Σ of placesv|psuch thatσv(λ) (hence σv(λu) =σv(λ)σv(u) for alluprime top) is a unit ofKv.

Forv1∈ Σ, write TrK/Q(λu) = P

v|p, v6=v1

Trv(σv(λu)) + Trv1(σv1(λu)) =:a+ Trv1(σv1(λu)).

As p is unramified in K, the residue traces at pare surjective and since σv1(λu) is a unit, it is sufficient to take a suitableu≡1 (mod Q

v, v6=v1pv) (in which case a∈Zp (modp) does not depend onu) andu≡u1 (modpv1) such that for instance Trv1(σv1(λu))≡1−a(modp) ifa6≡1 (modp) (resp. 1 (modp) ifa≡1 (modp)).

Whence TrK/Q(λu)≡1 (resp. 2) (modp).¹

The following lemma, valid for anyp >2 unramified, prime to η, will be especially useful to us (from [Wa,§5.5, proof of Theorem 5.31]):

Lemma 2.2. Letη∈K^×, prime top, and letλ(σ),σ∈G, bep-integer coefficients of KQp, not all divisible by p. Suppose that we have the relation of dependence modulo p^N+1,N ≥1, of thenvectorsℓσ:= (. . . ,log_p(η^{τ σ−1}), . . .)τ,σ∈G,

P

σ∈Gλ(σ)log_p(η^{τ σ−1})≡0 (modp^N+1) for allτ∈G.

Then there exist coefficientsλ^′(σ)∈Z(p), not all divisible byp, fulfilling the relation P

σ∈Gλ^′(σ)log_p(η^{τ σ−1})≡0 (mod p^N+1)for allτ∈G.

Takingτ = 1yields the relation P

σ∈Gλ^′(σ)αp(η)^σ−1≡0 (modp).

Proof. Modulo p^N+1, we can suppose that λ(σ) ∈ ZK,(p) for all σ ∈ G. Here the log_p(η^{τ σ−1}) are also represented, modulop^N+1, by elements of ZK,(p) and the corresponding linear algebra is a priori over the fieldK.

We obtain (for instance) TrK/Q(λ(1))≡1 (modp) by multiplication of the congru- ence by a suitableu∈K^×prime top(Lemma 2.1). By conjugation withν ∈Gwe obtain P

σ∈Gλ(σ)^νlog_p(η^{ντ σ−1})≡0 (mod p^N+1) for allτ∈G, which is equivalent to P

σ∈Gλ(σ)^νlog_p(η^{s σ−1})≡0 (mod p^N+1) for alls∈G. Taking the trace inK/Qof the coefficients (summation over ν), we obtain the rationalp-integersλ^′(σ) for all

σ∈G, withλ^′(1)≡1 (modp).

We may suppose that such linear relations of dependence modulo p^N+1ZKZp, for N ≥1, are with coefficients inZ(p)because the two notions of rank coincide. Taking the limit onN, one goes from the complete ringZKZp to thep-adic ringZp. 2.1.3. Regulators. LetF be the Z[G]-module generated byη. SinceQpLog_p(F) is theQp[G]-module generated by Log_p(η) and since Q

v|pKvis the representation ofG induced by the representationKv0 of the decomposition groupDp0, thep-adic rank rp(F) ofF is equal to theQp-rank of the system of vectors (. . . ,log_p(η^{τ σ−1}), . . .)τ, σ∈G, then to the rank (in the usual sense from the lemmas) of the classicalp-adic regulatorR^p(η) (or Frobenius determinant) ofη

R^p(η) := Frob^G(log_p(η)) := det log_p(η^{τ σ−1})

σ,τ∈G.

1Forp= 2,K=Q(√

17),λ= 1 + 2√

17, there is no solutionuprime to 2.

The Z[G]-module F is monogenic in the framwork recalled in [J, §1], or [Gr1, III.3.1.2 (ii)], in which case the conjecture of Jaulent ([J, §2]), asserts that the p-adic rank rg_p(F) of F is equal to its Z-rank rg(F) := dim_Q(F⊗Q) (this is the natural extension of the Leopoldt conjecture on the group of units ofK).

We note that any minor of order r is divisible by p^r since log_p(η) ≡ −p αp(η) (mod p²) inZp. Hence the following definitions forη∈K^× prime top:

Definitions 2.3. (i) Consider (forp >2, unramified inK) the determinant Reg^G_p(η) := Frob^G₋₁

p log_p(η)

:= det₋₁

p log_p(η^{τ σ−1})

σ,τ∈G,

with integer coefficients of KQp. This Frobenius determinant is called, in all the paper, the normalized p-adic regulator ofη. We have Reg^G_p(η)≡∆^G_p(η) (modp), where

∆^G_p(η) := Frob^G(αp(η)) = det αp(η)^{τ σ−1}

σ,τ∈G

is called the local regulator ofη (cf. §2.3).

(ii) For a real Galois fieldK, the usualp-adic regulatorR^p(K) of the units is given by a minor of ordern−1 of Frob^G(log_p(ε)) = det log_p(ε^{τ σ−1})

σ,τ∈G, whereεis a suitable Minkowski unit, and the p-adic integer

p⁻⁽ⁿ⁻¹⁾· R^p(K) = det

−1

p log_p(ε^{τ σ−1})

σ6=1,τ6=1

is called the normalizedp-adic regulator ofK.

From Lemma 2.2 and after division bypof the logarithms, we are reduced to linear algebra reasoning overZ/p^NZ,N ≥1; in particular, rg_p(F) is theZ/p^NZ-rank of the matrix ⁻_p¹log_p(η^{τ σ−1}) (modp^N)

σ,τ∈G, forN large enough.

If a minor M of order rg(F) is nonzero modulo p^N, then it gives rg_p(F), and it is the chosen practical viewpoint that we shall limit toN = 1, hence to theαp(η) modulo p; in this case, rg_p(F) is a priori greater or equal to the Z/pZ-rank of the matrix αp(η)^{τ σ−1}(mod p)

σ,τ∈G. If rg(F) = n, then the Leopoldt–Jaulent conjecture gives det ⁻_p¹log_p(η^{τ σ−1})

σ,τ∈G∼p^e,e≥0.

2.1.4. Strong form of the Leopoldt–Jaulent conjecture. The previous local point of view (for allpexcept a finite number) can be analyzed in the following two manners:

(a)QLocal analysis. We make no assumption on rg(F). If there exists inF a relation

σ∈G(η^σ−1)^λ(σ)=ζ (root of unity),λ(σ)∈Znot all zero (i.e., rg(F)< n), then for allp, prime toη, we have P

σ∈Gλ(σ) log_p(η^σ−1) = 0 (i.e., rg_p(F)< n). These global relations are transmitted into the weaker local relations P

σ∈Gλ(σ)αp(η)^σ−1 ≡ 0 (mod p); they are said to be trivial (they do not come from a numerical circumstance with coefficients depending on the considered primep, but to the existence of a non trivialglobal relationin F given by some constantsλ(σ)∈Z).

Conversely, if we have for fixed integers λ(σ)∈Z, not all zero, the family of local conditions (for allpexcept a finite number)

P

σ∈Gλ(σ)αp(η)^σ−1 ≡0 (modp)

p , (*)

the question is to know if this is globalisable under the form Q

σ∈G(η^σ−1)^λ(σ)=ζ.

We assume only the congruences P

σ∈Gλ(σ)αp(η)^σ−1 ≡0 (modp) for allpexcept a finite number, with someλ(σ)∈Z, not all zero and independent ofp.

Let η0 := Q

σ∈G(η^σ−1)^λ(σ) ∈ F; then log_p(η0) ≡ 0 (mod p²) (i.e., Log_p(η0) ≡ 0 (modp²)) andη0 is, in Q

v|pK_v^×, of the formξ(1 +β p)^p, β p-integer of Q

v|pKv and ξ of torsion (of prime toporder, for plarge enough); so η0 ∈ Q

v|pK_v^×p for almost allp. Conjecturaly,η0 is a root of unity ofK (from Conjecture 8.5).

(b)Global analysis. By comparison, suppose that, in a projective limit framework, we have coefficientsbλ(σ)∈Zb=Q

pZp, such that P

σ∈Gbλp(σ)Log_p(η^σ−1) = 0, for allpprime toη, where for allσ∈G,bλp(σ) is thep-component ofbλ(σ).

Leti:= (iv)v, v(η)=0 be the diagonal embeddingF⊗Zb→U, whereb Ub = Q

p,(p,η)=1

Q

v|pU_v¹ × Q

v∤p, v(η)=0µp(Kv) , µp(Kv) being the group of pth roots of unity in Kvand, for v|p,

U_v¹=µp(Kv)×U^′, whereU^′ isZp-free.

We putbη0:= Q

σ∈G η^σ−1bλ(σ)

∈F⊗Zb and we denote byηb0,p= Q

σ∈G η^σ−1bλp(σ)

the p-component of ηb0 (pprime to η). Since Log_p(bη0,p) = 0 for all pprime to η, we have for all placev prime toη,iv(ηb0) =ξv, where (generally)ξv is a root of unity of order a divisor of ℓ^nℓ −1, where ℓ is the residue characteristic ofv (the places v|pofK such thatξv is of order divisible bypare finite in number). We can write

i(ηb0)∈i(F⊗bZ)\ Q

p,(p,η)=1

Q

v, v(η)=0µp(Kv) .

By using the analogue forF of the local–global characterization of thep-adic con- jecture of Leopoldt–Jaulent ([J, §2]; see also [Gr1, III.3.6.6] in the case of units), we can state (under this conjecture, same reasoning) that we have

i(F⊗Z)b \ Q

p,(p,η)=1

Q

v, v(η)=0µp(Kv)

=i(µ(K)).

We deduce thatηb0,p is a root of unityζp∈K for allpprime toη.

If moreover we suppose that bλp(σ)≡λ(σ) (modp) for allσ∈G and allpprime toη, where theλ(σ) are given rational integers, thenη0∈F^×, defined by

η0:= Q

σ∈G η^σ−1λ(σ)

,

is equal toηb0,p up to a localpth power atp, thusη0ζ_p⁻¹∈ Q

v|pK_v^×p; we obtain the situation of§(a) since, for allplarge enough, ζp= 1 and there is coincidence.

We can see our approach as a very important weakening to this classical p-adic context concerning the Leopoldt–Jaulent conjecture for all primep; but as consid- eration, to have non empty information of a p-adic nature, we have been obliged

to suppose the existence of the family of rational integers (not all zero) (λ(σ))σ∈G

satisfying the relation (*).

2.1.5. General study project. Our purpose, in connection with the previousp-adic comments, is to see with what probability (a priori very small) the normalized regulators Reg^Gp(η) ofη (η fixed) are divisible byp(p→ ∞).

A normalized regulator Reg^G_p(η) can be factorised by means of powers ofχ-regu- lators Reg^χ_p(η) (for the irreducible rational characters χ ofG). This factorization does not depend onp. On the other hand, one can factorize Reg^χ_p(η) by means of θ-components Reg^θ_p(η) (for the irreduciblep-adic charactersθ|χ); this factorization depends on the residue degree ofpin the field of values of the absolutely irreducible charactersϕ|χofG. Then we shall get the congruence

Reg^θ_p(η)≡∆^θ_p(η) (modp)

where the local θ-regulator ∆^θ_p(η) is the θ-component of ∆^G_p(η) = Frob^G(αp(η)) (cf. §2.3). We shall deduce a probabilistic study in order to apply the heuristic principle of Borell–Cantelli.

Remark 2.4. Lemma 3.8 shall allow us to reduce (moduloQ^×) to anη∈ZK, what we suppose in numerical and Diophantine studies. When the integer η is fixed or varies in a small numerical neighborhood (in an Archimedean meaning and notp- adic)and whenp→ ∞, we shall speak ofprobability, for instance Prob Reg^G_p(η)≡0 (modp)

; on the other hand, whenpis fixed andηis the variable (defined modulop² in our study), the probability coincides with the density of the numbers η ∈K^× (prime top) satisfying the property.

It is clear that densities are canonical and are computed by means of algebraic calculations. As probabilities are linked to densities, one can confuse the two no- tions as soon as they are at most ^O(1)_p2 , and then “excluded” from probabilistic considerations of the Borell–Cantelli principle.

On the other hand, in the case of densities ^O(1)_p , the distinction is necessary. The idea (developed in [Gr2] for ordinary Fermat quotients) is that, conjecturally, when η is given, these probabilities are less thandensities whenp→ ∞and that, under the existence of a binomial law for Prob ∆^θ_p(z)≡0 (modp)

(zrunning through a suitable set of residues modulop), this probability is ^O(1)

p^{log2(p)/log(c0(η))−O(1)} instead of ^O(1)

p , wherec0(η) = maxν∈G(|η^ν|), which suggests the finiteness of the number of cases (Theorem 1.1).

2.2. Representations and group determinants (Frobenius determinants).

We make no assumption on the Galois group G; for this, we begin by a general recall in terms of representations (for a comprehensive course on representations and characters, see [Se1]; for the Abelian case, see [Wa] or [C]).

2.2.1. General notation. As C[G] is the regular representation, we have the iso- morphismC[G]≃L

ρdeg(ρ). Vρ, where (ρ, Vρ) runs through the set of absolutely irreducible representations ofGand where deg(ρ) is the degree (C-dimension ofVρ).

We denote by ϕ the character of ρ; consequently, deg(ρ) = ϕ(1). We choose to index objects depending on ρ by the letter ϕ (e.g. Vϕ) and to keep ρ =ρϕ as a homomorphism ofGinto End(Vϕ).

For the algebraC[G] of endomorphismsE ∈C[G], acting on the basis{ν, ν∈G} by multiplication ν 7→E . ν, we have the isomorphismC[G] ≃L

ϕEnd(Vϕ), with End(Vϕ)≃ eϕC[G], where the eϕ = ^ϕ(1)

|G|

P

ν∈Gϕ(ν⁻¹)ν are the central orthogonal idempotents ofC[G].

For the decomposition of eϕC[G] into a direct sum ofϕ(1) irreducible representa- tions, isomorphic toVϕ, we use the projectors comming from a matrix representa- tionM(ρϕ(ν)) = a^ϕ_ij(ν)

i,j ([Se1, §I.2.7]) π_i^ϕ= ^ϕ(1)

n

P

ν∈Ga^ϕ_ii(ν⁻¹)ν, i= 1, . . . , ϕ(1),

giving a system of (non central) orthogonal idempotents such thateϕ=P

i π_i^ϕ. 2.2.2. Recalls on group determinants (from [C]). Let G be a finite group and let Frob^G(X) = det Xτ σ⁻¹

σ,τ∈G be the determinant of the group G, or Frobenius determinant, with indeterminatesX := (Xν)ν∈G. We then have the formula

Frob^G(X) =Q

ϕdet P

ν∈GXνρϕ(ν⁻¹)ϕ(1)

.

Hence the existence of homogeneous polynomialsP^ϕ(X), of degreesϕ(1), such that Frob^G(X) =Q

ϕP^ϕ(X)^ϕ(1). The specializationXν7→ ⁻_p¹log_p(η^ν) leads to (Definitions 2.3)

Reg^G_p(η) := Frob^G ₋₁

p log_p(η)

=Q

ϕ det P

ν∈G

−1

p log_p(η^ν)ρϕ(ν⁻¹)ϕ(1)

,

and from Reg^ϕ_p(η) :=P^ϕ . . . ,⁻¹

p log_p(η^ν), . . .

= det P

ν∈G

−1

p log_p(η^ν)ρϕ(ν⁻¹) , we group into partial products associated with the characters χand θ irreducible overQandQp, respectively

Reg^χ_p(η) = Q

ϕ|χReg^ϕ_p(η) & Reg^θ_p(η) = Q

ϕ|θReg^ϕ_p(η).

2.2.3. Practical calculation of theP^ϕ(X). The polynomialsP^ϕ(X) are obtained in the following way: from the vectorial spaceV =C[G] (provided with the basisG), we consider the endomorphism ofV[X],L(X) = P

ν∈GXνν⁻¹, which is such that P

ν∈GXνν⁻¹

. τ = P

ν∈GXνν⁻¹τ = P

σ∈GX_{τ σ}⁻¹σ, ∀τ∈G.

So, the determinant of this endomorphism in the basis{τ, τ ∈G}is the Frobenius determinant (defined up to the sign).

Let (ρϕ, Vϕ) be the family of non isomorphic absolutely irreductible representations.

We shall take for End(Vϕ) the componenteϕC[G] associated with the characterϕ.

We use the algebra isomorphismρe:V →Q

ϕEnd(Vϕ) defined by P

ν∈Ga(ν)ν⁻¹7−→ P

ν∈Ga(ν)ρϕ(ν⁻¹)

ϕ.

whereρϕ(ν⁻¹) =eϕν⁻¹in the previous identification. From the Maschke theorem, we get for the endomorphismL(X)

detV(L(X)) =Q

ϕ detVϕ(L^ϕ(X))ϕ(1)

, whereL^ϕ(X) = P

ν∈GXνρϕ(ν⁻¹)∈End(Vϕ[X]). We put P^ϕ(X) := detVϕ(L^ϕ(X)).

With a matrix realizationM(ρϕ(ν)) = a^ϕ_ij(ν)

i,j of theρϕ(ν), the matrix associ- ated withL^ϕ(X) isM^ϕ(X) = P

ν∈Ga^ϕ_ij(ν⁻¹)Xν

i,j, of determinantP^ϕ(X).

Letgbe the least common multiple of the orders of the elements ofG; it is known that representations are realizable over the fieldCg =Q(µg) of gth roots of unity ([Se1,§12.3]). So, we may suppose that thea^ϕ_ij(ν) arep-integer algebraic numbers, for allplarge enough (i.e.,P^ϕ(X)∈ZCg,(p)[X] for allϕ).

Let Γ := Gal(Cg/Q) (commutative). Given an absolutely irreducible representation ρϕ:G7→EndCg(Vϕ), we define its conjugates in the following Galois manner so that for alls∈Γ,ρ^s_ϕis the representationG7→EndCg(Vϕ^s)≃eϕ^sCg[G] of characterϕ^s defined byϕ^s(ν) = (ϕ(ν))^s, for alls∈Γ. We have, for alls∈Γ, ϕ^s(ν) =ϕ(ν^ω(s)), where ω is the character Γ → (Z/gZ)^× of the action of Γ on µg. We also put ϕ^t(ν) :=ϕ(ν^t) for all integert prime tog (Γ-conjugation).

2.2.4. Rational and p-adic characters – Idempotents. We recall their practical de- termination.

(i)Rational characters. We put, forϕfixed

χ= P

s∈Gal(C/Q)ϕ^s=: P

ϕ|χϕ and P^χ(X) := Q

s∈Gal(C/Q)P^ϕs(X) =: Q

ϕ|χP^ϕs(X), whereC⊆Cgis the field of values of anyQ-conjugate ofϕ.

(ii) p-adic characters. If p∤ g, denote, for χ fixed, by L and D the field and the decomposition group ofpinC/Q. Let f =|D|be the residue degree ofpin C/Q andh= [L:Q] the number of prime idealspabovepinC(orL); thus [C:Q] =h f. Letϕ|χ. We put

θ(ν) := P

s∈Dϕ^s(ν)∈L, for allν∈G & P^θ(X) := Q

s∈DP^ϕs(X) =: Q

ϕ|θP^ϕ(X).

We fix one of thehprime idealsp|pofL(we shall say thatθandpare associated).

AsLp^t =Qpfor allt∈Gal(C/Q)/D, we have congruences of the formθ(ν)≡rp^t(ν) (modp^t) in L, rp^t(ν)∈Z; the rationals rp^t(ν) depend numerically of the residue images atp^tof the trace inC/Lof theϕ(ν).

Ifθ= P

s∈Dϕ^sandpare associated, thehconjugates ofθare theθ^t= P

s∈D(ϕ^t)^sand we have θ^t(ν) ≡ r_pt−1(ν) (modp) (or θ^t−1(ν) ≡ rp^t(ν) (modp)). As the θ^t are seen inZp⊂Lp, we shall write by abuseθ^t(ν)≡r_pt−1(ν) (modp).

For p fixed, the integer f depends only on χ and is called the residue degree of the charactersϕ, θ and χ. We have, by Γ-conjugation,ϕ^pi(ν) =ϕ(ν^pi) =ϕ(ν)^sip, wheresp is the Frobenius automorphism (of orderf) inC/Q.

(iii) Idempotents. We put eχ = P

ϕ|χeϕ and eθ = P

ϕ|θeϕ; thuseχ = P

θ|χeθ. Theeθ

(resp. eχ) give a fundamental system of orthogonal idempotents of Qp[G] (resp.

Q[G]). We can replaceQp (resp. Q) byZp (resp. Z(p)) becausep∤g.

From P^ϕ(X) = detVϕ(L^ϕ(X)) we deduce that P^ϕs(X) = detV_ϕs(L^ϕs(X)) where L^ϕs(X) = P

ν∈GXνρ^s_ϕ(ν⁻¹) is given via the (a^ϕ_ij(ν⁻¹))^s, which defines the conjugate bysof the polynomialP^ϕ(X) (i.e., of its coefficients).

Theorem 2.5. (i) For allp large enough, the polynomials P^χ(X)(resp. P^θ(X)) have rational p-integer coefficients (resp. p-adic integer coefficients).

(ii) For all irreducible characterϕ, we haveP^ϕ(. . . , Xπν, . . .) =ζπP^ϕ(. . . , Xν, . . .) for allπ∈G, where ζ_π^g= 1.

Proof. (i) AsP^ϕ(X)∈ZC,(p)[X] for allϕ|χ,P^χ(X) = Q

s∈Gal(C/Q)P^ϕs(X) is invari- ant by Galois. LikewiseP^θ(X) = Q

s∈DP^ϕs(X)∈L[X]⊂Lp[X] =Qp[X].

(ii) Forπ∈Gcall [π] the operator defined by [π]Xν =X(πν)for allν∈G. Then [π]

andρe:V[X]→Q

ϕEnd(Vϕ[X]) commute; moreover, sinceρϕis a homomorphism, we have the following formula

[π] P

ν∈GXνρϕ(ν⁻¹)

= P

ν∈GXπνρϕ(ν⁻¹) = P

ν∈GXνρϕ(ν⁻¹) ρϕ(π).

Then, since the determinant ofρϕ(π)∈End(Vϕ) is that of a diagonal matrix whose diagonal is formed of roots of unity, we get

det [π] P

ν∈GXνρϕ(ν⁻¹)

=ζπdet P

ν∈GXνρϕ(ν⁻¹) ,

whereζπ is of order a divisor of the order ofρϕ(π) which is a divisor ofg.

Corollary 2.6. For all π∈G and all absolutely irreducible characterϕ, we have P^ϕ(. . . , α^πν, . . .) =ζπP^ϕ(. . . , α^ν, . . .)by the specialization Xν7→α^ν,α∈ZK. Consequentely,P^χ(. . . , α^πν, . . .) =±P^χ(. . . , α^ν, . . .) for allπ∈G.²

In the same way, P^θ(. . . , α^πν, . . .) =ζ_π^′P^θ(. . . , α^ν, . . .)for all π∈G, where ζ_π^′ is of order a divisor of g.c.d.(g, p−1).

2.2.5. Numerical determinants. In this section, there is no reference to a prime numberpand the characters that we consider are absolutely irreducible or rational.

The above leads to define the numericalχ-determinants of Frobenius of anyα∈ZK

(i.e., independent of the givenη ∈K^×).

Definition 2.7. LetGbe a finite group and let Frob^G(X) be the associated group determinant. Theχ-determinants (with indeterminates and numerical) are by def- inition the expressions

Frob^χ(X) = Q

ϕ|χP^ϕ(X) and Frob^χ(α) = Q

ϕ|χP^ϕ(. . . , α^ν, . . .), so that Frob^G(α) =Q

χ (Frob^χ(α))^ϕ(1) (whereϕ|χfor eachχ).

Example 2.8. In the case of the group D6 = {1, σ, σ², τ, τ σ, τ σ²}, we have the following numericalχ-determinants

Frob¹(α) =α+α^σ+α^σ2+α^τ+α^{τ σ}+α^{τ σ2}, Frob^χ1(α) =α+α^σ+α^σ2−α^τ−α^{τ σ}−α^{τ σ2},

Frob^χ2(α) =α²+α^2σ+α^2σ2−α^2τ−α^{2τ σ}−α^{2τ σ2}−αα^σ−α^σα^σ2−α^σ2α

+α^τα^{τ σ}+α^{τ σ}α^{τ σ2}+α^{τ σ2}α^τ. 2Sign + except ifχ=ϕis quadratic andϕ(π) =−1.

The two last one are of the form Frob^′.√m, Frob^′∈Q, wherek=Q(√m) is the quadratic subfield of K and we neglect the factor √m; but Frob^χ2(α) appears to the square in the determinant Frob^G(α) and the result is rational, which is not the case of Frob^χ1(α). This is specific of quadratic characters.

For computations, we can return to the matrix realizations (C=Q,ϕ=χ2)

ρϕ(1) = 1 0

0 1

, ρϕ(σ) =

−1 −1

1 0

, ρϕ(σ²) =

0 1

−1 −1

,

ρϕ(τ) =

1 0

−1 −1

, ρϕ(τ σ) =

−1 −1

0 1

, ρϕ(τ σ²) = 0 1

1 0

,

which leads (by specialization and by taking the determinant) to

P

ν∈GXνρϕ(ν⁻¹) =

X1−X_σ2+Xτ−Xτ σ Xσ−X_σ2−Xτ σ+X_{τ σ}2

−Xσ+X_σ2−Xτ+X_{τ σ}2 X1−Xσ−Xτ+Xτ σ

,

Frob^χ2(α) =

α−α^σ2+α^τ−α^{τ σ} α^σ−α^σ2−α^{τ σ}+α^{τ σ2}

−α^σ+α^σ2−α^τ+α^{τ σ2} α−α^σ−α^τ+α^{τ σ} .

Still for χ2 (of degree 2) and the representation eχ2Q[G] ≃2Vϕ, there exist two orthogonal projectorsπ1, π2, of sumeχ2 = ¹₃(2−σ−σ²) (§2.2.1), which yields here

π1= ¹

3(1−σ²+τ−τ σ) & π2= ¹

3(1−σ−τ+τ σ).

2.3. The local θ-regulators. Letη∈K^× be given and letpbe large enough so thatpis unramified inK, prime ton= [K:Q] andη.

2.3.1. Generalities. We fix an algebraic integer α ∈ ZK defined by α ≡ αp(η) (modp). We obtain the determinant, with coefficients inZK, defined modulop

∆^G_p(η) := Frob^G(α) = det α^{τ σ−1}

σ,τ∈G=Q

χ

Q

θ|χ

Q

ϕ|θP^ϕ(. . . , α^ν, . . .)^ϕ(1). If ∆^G_p(η)∈/ Q, we find again the existence of a factor√mwhich comes from the resolvant of a quadratic character of G and that we neglect in the definitions of regulators.

Definition 2.9. For allplarge enough and for eachQp-irreducible characterθ of G, we call localθ-regulator ofη, thep-adic integer defined by

∆^θ_p(η) := Q

ϕ|θP^ϕ(. . . , α^ν, . . .), forα≡αp(η) :=¹_p(η^pnp−1−1) (mod p).

Forθ|χ (χ fixed), the corresponding localθ-regulators depend on the splitting of pin C/Qand there areh= ^[C:Q]_f such regulators, wheref is their residue degree (§2.2.4 (ii)). These regulators are only defined modulo p.

Remark 2.10. In the same manner, we may write (for plarge enough) that the normalized regulator Reg^G_p(η) is equal to

Q

χ Reg^χ_p(η)^ϕ(1) =Q

θ Reg^θ_p(η)^ϕ(1), where

Reg^θ_p(η) = Q

ϕ|θP^ϕ . . . ,⁻_p¹log_p(η^ν), . . . . We then have the congruences

Reg^θ_p(η)≡∆^θ_p(η) (mod p);

so pdivides Reg^θ_p(η) if and only if ∆^θ_p(η) ≡ 0 (modp); in this case, there exists e≥1 such thatp^eϕ(1) divides Reg^θ_p(η) where at each timeϕ|θ(§2.2.2).

We shall speak of an extrap-divisibility if e≥2.

2.3.2. Particular remarks. (i) We have

∆^χ_p(η) := NC/Q P^ϕ(. . . , α^ν, . . .)

∈Z (ϕ|χ fixed),

with the convention on the notation NC/Q, especially whenK andC are not lin- eairely disjoint. We recall the quadratic exception forχ.

In the same way

∆^θ_p(η) := Np P^ϕ(. . . , α^ν, . . .) ,

where forp|pinC,passociated withθ, Np denotes the absolute local norm (issued from NC/L) in the completion ofC at p; we find again ∆^χ_p(η) as a product of the correspondent local norms atp.

Same normic relations by replacing ∆pby Reg_pand αby ⁻_p¹log_p(η).

(ii) IfH={ν∈G, ϕ(ν) =ϕ(1)}is the kernel ofϕ|θ|χ(which only depends onχ) and ifK^′ is the subfield ofK fixed byH, we have

∆^θ_p(η) = ∆^θ_p^′(NK/K^′(η))

whereθ^′is the faithful character resulting fromθ. By replacingηbyη^′:= N_K/K^′(η) one always can suppose thatθ is a faithful character.

2.3.3. Characters χ of degree 1, of order 1 or 2. Let η ∈ K^× and letα≡ αp(η) (mod p),α∈ZK.

(i) If χ=θ= 1, theθ-regulator corresponds to NK/Q(η) =a∈Q^×and is given by TrK/Q(α), in other words

∆¹_p(η)≡⁻_p¹log_p(a)≡ ¹_p(a^p−1−1)≡qp(a) (modp)

(Fermat quotient of a); for classical properties and use of Fermat quotients, see, e.g., [EM], [GM], [Gr2], [Hat], [H-B], [KR], [OS], [Si].

Fora= 659 andp≤10⁹, we only find the solutionsp= 23, 131, 2221, 9161, 65983.

See [Gr4, Pr. A-1]. Fora= 47 anda= 72, we find no solution forp≤10¹¹. (ii) Ifχ=θis quadratic and ifk=Q(√m) is the quadratic subfield ofKfixed by the kernel ofχ, we obtain aθ-regulator corresponding to the case NK/k(η)∈k^×\Q^×; if TrK/k(α) =:u+v√m∈k, it is given by

∆^θ_p(η)≡(1−τ)(u+v√

m)≡2v√

m (mod p).

If K is a real quadratic field with the fundamental unit ε, because of the multi- plicative relation of dependenceε^1+σ=±1, the 1-regulators ∆¹_p(ε) are trivialy zero modulop. The θ-regulator of the quadratic character is ∆^θp(ε)≡2v√

m (mod p) (computed viaε^pnp−1≡1 +p v√m (mod p²)).

We compute the θ-regulator ∆^θ_p(ε) of the fondamental unit ε= 5 + 2√

6, for all p ≤ 10⁹ (p 6= 2,3) (see [Gr4, Pr. A-2] valuable for any quadratic integer). We find aθ-regulator equal to zero moduloponly forp= 7,523, which gives a second observation on the rarity of the phenomenon.

Letη = 1+√

6 of norm−5. We have rg(F) = 2 (no trivial nullities). We verifiy that Fermat quotients ∆¹_p(η) of−5 are all nonzero modulopin the tested interval. The