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

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by  Georges Gras

English translation of the original article:
Les θ-r´egulateurs locaux d’un nombre alg´ebrique – Conjectures p-adiques
Canadian Journal of Mathematics, Vol. 68, 3 (2016), 571–624
LOCAL θ-REGULATORS OF AN ALGEBRAIC NUMBER
7
p-ADIC CONJECTURES
1
0
2
GEORGESGRAS

n
a Abstract. LetK/QbeGaloisandletη K×besuchthatthemultiplicative
J ∈
Z[G]-module generated by η is of Z-rank n. We define the local θ-regulators

0 ∆θp(η) ∈ Fp for the Qp-irreducible characters θ of G = Gal(K/Q). Let Vθ
T] 1 Eabseascothchie∆atθpeθ(d-ηir)wreyidtiheulcd∆isbθplRe(ηerg)eθppw(rηhe)soesmnetoandtuuilollonit.yp iiAns etlhqineueifvaaarclteronertpizrtaeotsieoδnnt≥ati1onR(TeLghθθpe(oη≃r)emδϕ(V13θ).9oi)sf.
Qθ (cid:0) (cid:1)
N RegG(η) := Regp(η) (normalized p-adic regulator of η), where ϕ θ is ab-
p p[K:Q] |
h. solutely irreducible. From the probability Prob ∆θp(η) = 0 & Lθ ≃ δVθ ≤
t p−fδ2 (f = residue degree of p in the field of(cid:0)values of ϕ) and the Bo(cid:1)rel–
a
Cantelliheuristic,weconjecture that,forplargeenough, RegG(η)isap-adic
m p
unitorthatpϕ(1) RegG(η) (existence ofasingleθ ofGwithf =δ=1and
[ noextra p-divisibilkity);tphisobstruction maybeliftedassumingthe existence

of a binomial probability law (Sec. 7) confirmed through numerical studies
1
v (with groups G = C3, C5, D6). This conjecture would imply that, for all p
large enough, Fermat quotients of rationals and normalized p-adic regulators
8
are p-adic units (Theorem. 1.1), whence the fact that number fields are p-
1
rational for p 0. We recall 8.7 some deep cohomological results, which
6 ≫ §
maystrengthen suchconjectures.
2
0
.
1 1. Introduction
0
7 Let K/Q be a Galois extension of degree n of Galois group G. Let η K . An
×
1 exponential notation is used for conjugation of η by σ G, which i∈mplies the
: ∈
v writing (ησ)τ =: ητσ for all σ,τ G (law of left G-module). We assume that the
∈
i multiplicative Z[G]-module generated by η is of Z-rank n (i.e., η Q Q[G]).
X Forplargeenough,weputRegG(η):=det 1log (ητσ−1) h(nioGrm⊗aliz≃edp-adic
r p −p p σ,τ G
a regulator of η). (cid:0) (cid:1) ∈
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 RegG(η) 0
p ≡
(mod p),isrelatedtoprimespsuchthatRegG(η)isexactelydivisiblebyaminimal
p
power of p; this is equivalent to RegG(η) pϕ(1) (equality up to a p-adic unit),
p ∼
wherethecharacterϕ(absolutelyirreducible)definesap-adiccharacterθsatisfying
certain conditions (Definition 4.1).
Date:January7,2017.
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–Jaulentconjecture;Abelianp-ramification;p-rationality;Fermatquotient;probabilistic
numbertheory.
1
2 GEORGESGRAS
O(1)
Such a situation is a priori of probability at most , only when η is consid-
p
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 and p :
→∞
Theorem 1.1. Let K/Q be a Galois extension of degree n and of Galois group G.
Let η K be fixed, η generating a multiplicative Z[G]-module of Z-rank n.
×
∈
(i) Under the Heuristic 7.4 (existence of a classical binomial law of probability),
C (η)
the probability to have RegGp(η) ≡ 0 (mod p) is at most plog2(p)/lo∞g(c0(η))−O(1) for
p , where c (η)=max (ησ ), e 1 C (η) 1, and log =log log.
→∞ 0 σ∈G | | − ≤ ∞ ≤ 2 ◦
(ii)Underthepreviousheuristic7.4andtheprincipleofBorel–Cantelli, thenumber
of primes p such that RegG(η) 0 (mod p) is finite.
p ≡
Weshallalwayssupposethattheprimenumberpthatweconsiderislargeenough,
in particular odd, not divisor of n, unramified in K, and prime to η, so that the
normalizedp-adicregulatorRegG(η):=p [K:Q] Reg (η)makessenseinZ ,where
p − · p p
Reg (η) is the usual p-adic regulator of η (see 2.1.3).
p §
Denote by Z (resp. Z ) the ring of integers (resp. of p-integers) of K; for
K K,(p)
K = Q, one gets Z (resp. Z ). For all place v p of K, we denote by p p the
(p) v
| |
prime ideal associated with v.
If n is the common residue degree of the places v p in K/Q, the multiplicative
p
|
groupsoftheresiduefieldsareoforderpnp 1andforallv pwehavethecongruence
ηpnp 1 1(mod p ); hence finally, since − p =pZ , |
− v v K
≡ pQv|p
ηpnp 1 =1+pα (η), α (η) Z ,
− p p K,(p)
∈
which leads, by Galois, to the relations
α (ησ)=α (η)σ for all σ G,
p p
∈
and to the “logarithmic” properties
α (ηη ) α (η)+α (η ) (mod pZ ) & α (ηλ) λα (η) (mod pZ )
p ′ p p ′ K,(p) p p K,(p)
≡ ≡
(for η,η K , λ Z).
′ ×
∈ ∈
This generalized Fermat quotient α (η) of η is the key element of our study. More
p
precisely,the properties of the G-module generatedby α (η) modulo pZ shall
p K,(p)
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. Let p be a fixed prime number sat-
isfyingthe hypothesisgiveninthe Introduction. We suppose thatthe numberfield
K isconsideredasasubfieldofC . Thus,any“embedding”ofK intoC isnothing
p p
else than a Q-automorphism σ G.
∈
Letp =p beaprimeidealofK abovepandletD beitsdecompositiongroup.
0 v0 p0
The places v p, conjugates of v , correspondto the (G:D ) distinct prime ideals
p :=pσv, wh|ere (σ ) is an e0xact system of representatipv0es of G/D .
v 0 v v|p p0
LOCAL θ-REGULATORS – p-ADIC CONJECTURES 3
We consider the Q [G]-module K where K =σ (K)Q C , is the comple-
p v v v p p
vQp ⊂
|
tionofK atv;asK/QisGalois,K /Q isindependentofv pbutthenotationK
v p v
|
recallsthatthislocalextensionisprovidedwiththeembeddingσ : K K C ,
v v p
→ ⊂
which allows the diagonal embedding with dense image
i :=(σ ) :K K
p v v p v
| −−−→vQp
|
wherei (x):=(σ (x)) ,andwhichimpliesthatK Q K Q [G](semi-
p v v p p v p
| ⊗ ≃vQ|p ≃
local theory). By abuse, if x K, we shall write x K , i being understood.
v p
∈ ∈vQp
|
2.1.1. p-adic logarithm on K . The p-adic logarithm log :K KQ is defined
× p × → p
on the set 1+px, x Z , by means of the usual series (p>2)
K,(p)
∈
(cid:8) log (1+px)=(cid:9) ( 1)i+1 (px)i px (mod p2),
p iP≥1 − i ≡
notingthat Ni=1(−1)i+1 (pxi)i ∈K foranyN ≥1. In the caseofγ ∈K(×p), we use
the functionPal relation
log (γ)= 1 log (γpnp 1)= 1 log (1+pα (γ)) pα (γ) (mod p2).
p pnp 1 p − pnp 1 p p ≡− p
− −
More generally, this log function, seen modulo pN+1, N 1, is represented by
p ≥
elementsofZ andisanhomomorphismofG-modulesforthelawdefined,forall
K,(p)
σ G, by σ log (γ)(mod pN+1) := log (γσ)(mod pN+1), using the congruence
∈ p p
(where N′ is(cid:0)an obvious function(cid:1)of N)
σ log (γ) (mod pN+1) 1 ( 1)i+1 (pαp(γ)σ)i (mod pN+1),
(cid:0) p (cid:1)≡ pnp−11≤Pi≤N′ − i
defining an element of Z which approximates log (γσ) modulo pN+1. So
K,(p) p
σ (log (γ)) makes sense in K for all v p.
v p v |
2.1.2. p-adic rank. Let Log := log i = (σ ) log be the homomorphism
p p◦ p v v|p ◦ p
of G-modules defined, on the subgroup of elements of K prime to p, by
×
Log (γ)= log (γσv) K .
p (cid:0) p (cid:1)v|p∈vQ|p v
Let η K , prime to p, and let F be the Z[G]-module generated by η. We call
×
∈
p-adic rank of F, the integer
rg (F):=dim (Q Log (F)).
p Qp p p
The use of Log is a commodity since by conjugation by the elements of G, the
p
knowledgeoflogpimpliesthatofLogpandconverselybyprojectionsvQ′|pKv′ →Kv.
Tomakealinkwiththeconceptofp-adicregulator,weshallprovefirstthefollowing
two technical results:
Lemma 2.1. Let p be an odd prime, unramified in K, and let λ Z . If
K,(p)
∈
λ / pZ , there exists u K , prime to p, such that Tr (λu) 0 (mod p).
K,(p) × K/Q
∈ ∈ 6≡
Proof. Forallu K ,primetop,considerthediagonalembeddingofλuin K ,
× v
∈ vQp
|
and let Tr be the local traces Tr for v p. Then
v Kv/Qp |
Tr (λu)= Tr (σ (λu)).
K/Q v v
vPp
|
4 GEORGESGRAS
Byassumption,thereexistsanon-emptysetΣofplacesv psuchthatσ (λ)(hence
v
|
σ (λu)=σ (λ)σ (u) for all u prime to p) is a unit of K .
v v v v
For v Σ, write
1
∈
Tr (λu)= Tr (σ (λu))+Tr (σ (λu))=:a+Tr (σ (λu)).
K/Q v v v1 v1 v1 v1
v|pP,v6=v1
As p is unramified in K, the residue traces at p are surjective and since σ (λu)
v1
is a unit, it is sufficient to take a suitable u 1 (mod p ) (in which case
a∈Zp (mod p)doesnotdependonu)andu≡≡u1 (modQpvv,1v)6=svu1chvthatforinstance
Tr (σ (λu)) 1 a(mod p) if a 1(mod p)(resp. 1(mod p)if a 1 (mod p)).
Whv1encve1Tr ≡(λu−) 1 (resp. 2) (m6≡od p).1 ≡ (cid:3)
K/Q
≡
The following lemma, valid for any p>2 unramified, prime to η, will be especially
useful to us (from [Wa, 5.5, proof of Theorem5.31]):
§
Lemma2.2. Letη K , primetop, andlet λ(σ), σ G, bep-integercoefficients
×
∈ ∈
of KQ , not all divisible by p. Suppose that we have the relation of dependence
p
modulo pN+1, N 1, of the n vectors ℓ :=(...,log (ητσ−1),...) , σ G,
≥ σ p τ ∈
λ(σ)log (ητσ−1) 0(mod pN+1) for all τ G.
ThenthereexisσtP∈cGoefficientps λ(σ) ≡Z ,notalldivisible byp,∈fulfillingtherelation
′ (p)
∈
λ(σ)log (ητσ−1) 0 (mod pN+1) for all τ G.
σTP∈aGkin′g τ =1pyields th≡e relation λ′(σ)αp(η)σ∈−1 0 (mod p).
σP∈G ≡
Proof. Modulo pN+1, we can suppose that λ(σ) Z for all σ G. Here
K,(p)
∈ ∈
the log (ητσ−1) are also represented, modulo pN+1, by elements of Z and the
p K,(p)
corresponding linear algebra is a priori over the field K.
Weobtain(forinstance)Tr (λ(1)) 1 (mod p)bymultiplicationofthecongru-
K/Q
≡
enceby asuitable u K primeto p (Lemma 2.1). By conjugationwithν Gwe
×
obtain λ(σ)νlog∈(ηντσ−1) 0 (mod pN+1)forallτ G,whichisequiva∈lentto
λ(σσP)∈νGlog (ηsσ−1p) 0 (mo≡d pN+1) for all s G. T∈aking the trace in K/Q of
tσPh∈eGcoefficienpts (summ≡ation over ν), we obtain th∈e rational p-integers λ(σ) for all
′
σ G, with λ(1) 1 (mod p). (cid:3)
′
∈ ≡
We may suppose that such linear relations of dependence modulo pN+1Z Z , for
K p
N 1,arewithcoefficientsinZ becausethetwonotionsofrankcoincide. Taking
(p)
≥
the limit on N, one goes from the complete ring Z Z to the p-adic ring Z .
K p p
2.1.3. Regulators. Let F be the Z[G]-module generated by η. Since Q Log (F) is
p p
theQ [G]-modulegeneratedbyLog (η)andsince K istherepresentationofG
p p v
vQ|p
inducedbytherepresentationK ofthedecompositiongroupD ,thep-adicrank
v0 p0
r (F) of F is equal to the Q -rank of the system of vectors (...,log (ητσ−1),...) ,
p p p τ
σ G, thentothe rank(inthe usualsensefromthelemmas)ofthe classicalp-adic
∈
regulator (η) (or Frobenius determinant) of η
p
R
(η):=FrobG(log (η)):=det log (ητσ−1) .
Rp p p σ,τ G
(cid:0) (cid:1) ∈
1Forp=2,K=Q(√17),λ=1+2√17,thereisnosolutionuprimeto2.
LOCAL θ-REGULATORS – p-ADIC CONJECTURES 5
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 (F) of F is equal to its Z-rank rg(F) := dim (F Q) (this is the
p Q ⊗
natural extension of the Leopoldt conjecture on the group of units of K).
We note that any minor of order r is divisible by pr since log (η) pα (η)
p ≡ − p
(mod p2) in Z . Hence the following definitions for η K prime to p:
p ×
∈
Definitions 2.3. (i) Consider (for p>2, unramified in K) the determinant
RegG(η):=FrobG −1log (η) :=det −1log (ητσ−1) ,
p (cid:16) p p (cid:17) (cid:16) p p (cid:17)σ,τ G
∈
with integer coefficients of KQ . This Frobenius determinant is called, in all the
p
paper, the normalized p-adic regulator of η. We have RegG(η) ∆G(η) (mod p),
p ≡ p
where
∆G(η):=FrobG(α (η))=det α (η)τσ−1
p p p σ,τ G
(cid:0) (cid:1) ∈
is called the local regulator of η (cf. 2.3).
§
(ii)ForarealGaloisfieldK,the usualp-adicregulator (K)ofthe units isgiven
p
by a minor of order n 1 of FrobG(log (ε))=det log (Rετσ−1) , where ε is a
− p p σ,τ G
suitable Minkowski unit, and the p-adic integer (cid:0) (cid:1) ∈
p−(n−1)·Rp(K)=det(cid:16)−p1logp(ετσ−1)(cid:17)σ=1,τ=1
6 6
is called the normalized p-adic regulator of K.
FromLemma2.2andafterdivisionbypofthelogarithms,wearereducedtolinear
algebra reasoning over Z/pNZ, N 1; in particular, rg (F) is the Z/pNZ-rank of
≥ p
the matrix 1log (ητσ−1)(mod pN) , for N large enough.
−p p σ,τ G
(cid:0) (cid:1) ∈
If a minor M of order rg(F) is nonzero modulo pN, then it gives rg (F), and it
p
is the chosen practical viewpoint that we shall limit to N = 1, hence to the α (η)
p
modulo p; in this case, rg (F) is a priori greater or equal to the Z/pZ-rank of
p
the matrix α (η)τσ−1(mod p) . If rg(F) = n, then the Leopoldt–Jaulent
p σ,τ G
conjecture g(cid:0)ives det 1log (ητσ(cid:1)−1)∈ pe, e 0.
−p p σ,τ G ∼ ≥
(cid:0) (cid:1) ∈
2.1.4. Strong form of the Leopoldt–Jaulent conjecture. The previous local point of
view(forallpexceptafinitenumber)canbeanalyzedinthefollowingtwomanners:
(a)Local analysis. Wemakenoassumptiononrg(F). IfthereexistsinF arelation
(ησ−1)λ(σ) =ζ (root of unity), λ(σ) Z not all zero (i.e., rg(F)<n), then for
σaQ∈llGp, prime to η, we have λ(σ)log (∈ησ−1)=0 (i.e., rg (F)<n). These global
p p
relations are transmitted σiPn∈tGo the weaker local relations λ(σ)α (η)σ−1 0
p
(mod p);theyaresaidtobetrivial(theydonotcomefromaσnP∈uGmericalcircumsta≡nce
withcoefficientsdependingontheconsideredprimep,buttotheexistenceofanon
trivial global relation in F given by some constants λ(σ) Z).
∈
Conversely, if we have for fixed integers λ(σ) Z, not all zero, the family of local
∈
conditions (for all p except a finite number)
λ(σ)α (η)σ−1 0 (mod p) , (*)
p
(cid:16)σP∈G ≡ (cid:17)p
6 GEORGESGRAS
the question is to know if this is globalisable under the form (ησ−1)λ(σ) =ζ.
We assume only the congruences λ(σ)α (η)σ−1 0 (modσQ∈pG) for all p except a
p
finite number, with some λ(σ) ZσP∈,Gnot all zero and≡independent of p.
∈
Let η := (ησ−1)λ(σ) F; then log (η ) 0 (mod p2) (i.e., Log (η ) 0
0 σQ∈G ∈ p 0 ≡ p 0 ≡
(mod p2)) and η is, in K , of the form ξ(1+βp)p, β p-integer of K and
0 v× v
vQp vQp
| |
ξ of torsion (of prime to p order, for p large enough); so η0 ∈ vQ|pKv×p for almost
all p. Conjecturaly, η is a root of unity of K (from Conjecture 8.5).
0
(b) Global analysis. By comparison,suppose that, in a projective limit framework,
we have coefficients λ(σ) Z= Z , such that
∈ p p
bλ (σ)Lbog Q(ησ−1)=0, for all p prime to η,
p p
σP∈G
b
where for all σ G, λ (σ) is the p-component of λ(σ).
p
∈
Let i:=(i ) be the diagonal embedding F Z U, where
v v,v(η)=0 b b⊗ →
U = U1 µb (Kb) ,
p,(pQ,η)=1(cid:16)vQp v ×v∤p,vQ(η)=0 p v (cid:17)
|
µ (K ) being the grboup of pth roots of unity in K and, for v p,
p v v
|
U1 =µ (K ) U , where U is Z -free.
v p v × ′ ′ p
We put η := ησ−1 λb(σ) F Z and we denote by η = ησ−1 λbp(σ) the
0 0,p
p-compobnent oσQf∈Gη0(cid:0)(p p(cid:1)rime ∈to η)⊗. bSince Logp(η0,p) = 0b for aσlQl∈Gp(cid:0)prime(cid:1)to η, we
have for all place v prime to η, i (η )=ξ , where (generally) ξ is a root of unity
v 0 v v
of order a divisorbof ℓnℓ 1, where ℓ is the residbue characteristic of v (the places
v pofK suchthatξ is−oforderdibvisible by pare finite innumber). We canwrite
v
|
i(η ) i(F Z) µ (K ) .
0 p v
∈ ⊗ \ p,(pQ,η)=1(cid:16)v,vQ(η)=0 (cid:17)
By using the analogbue for F ofbthe local–globalcharacterizationof the p-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) µ (K ) =i(µ(K)).
p v
⊗ \ p,(pQ,η)=1(cid:16)v,vQ(η)=0 (cid:17)
b
We deduce that η is a root of unity ζ K for all p prime to η.
0,p p
∈
If moreover we suppose that λ (σ) λ(σ) (mod p) for all σ G and all p prime
b p ≡ ∈
to η, where the λ(σ) are given rational integers, then η F , defined by
b 0 ∈ ×
η := ησ−1 λ(σ),
0
σQ∈G(cid:0) (cid:1)
is equal to η up to a local pth power at p, thus η ζ 1 K p; we obtain the
0,p 0 p− ∈vQp v×
|
situation ofb (a) since, for all p large 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 prime p; but as consid-
eration, to have non empty information of a p-adic nature, we have been obliged
LOCAL θ-REGULATORS – p-ADIC CONJECTURES 7
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 previous p-adic
comments, is to see with what probability (a priori very small) the normalized
regulators RegG(η) of η (η fixed) are divisible by p (p ).
p →∞
A normalized regulator RegG(η) can be factorised by means of powers of χ-regu-
p
lators Regχ(η) (for the irreducible rational characters χ of G). This factorization
p
does not depend on p. On the other hand, one can factorize Regχ(η) by means of
p
θ-componentsRegθ(η)(fortheirreduciblep-adiccharactersθ χ);thisfactorization
p |
dependsontheresiduedegreeofpinthefieldofvaluesoftheabsolutelyirreducible
characters ϕ χ of G. Then we shall get the congruence
|
Regθ(η) ∆θ(η) (mod p)
p ≡ p
where the local θ-regulator ∆θ(η) is the θ-component of ∆G(η) = FrobG(α (η))
p p p
(cf. 2.3). We shall deduce a probabilistic study in order to apply the heuristic
§
principle of Borell–Cantelli.
Remark2.4. Lemma3.8shallallowustoreduce(moduloQ )toanη Z ,what
× K
∈
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 not p-
adic)andwhenp ,weshallspeakofprobability,forinstanceProb RegG(η) 0
→∞ p ≡
(mod p) ;ontheotherhand,whenpisfixedandηisthevariable(defin(cid:0)edmodulop2
in our st(cid:1)udy), the probability coincides with the density of the numbers η K×
∈
(prime to p) 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), and then “excluded” from probabilistic
p2
considerations of the Borell–Cantelli principle.
On the other hand, in the case of densities O(1), the distinction is necessary. The
p
idea(developedin[Gr2]forordinaryFermatquotients)isthat,conjecturally,when
η is given, these probabilities are less than densities when p and that, under
→ ∞
the existenceofabinomiallawforProb ∆θ(z) 0 (mod p) (z runningthrougha
p ≡
(cid:0) (cid:1) O(1)
suitable set of residues modulo p), this probability is instead
plog2(p)/log(c0(η))−O(1)
of O(1), where c (η) = max (ην ), which suggests the finiteness of the number
0 ν G
p ∈ | |
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-
morphism C[G] deg(ρ).V , where (ρ,V ) runs through the set of absolutely
≃ ρ ρ ρ
irreduciblerepresenLtationsofGandwheredeg(ρ)isthedegree(C-dimensionofVρ).
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. V ) and to keep ρ = ρ as a
ϕ ϕ
homomorphism of G into End(V ).
ϕ
8 GEORGESGRAS
For the algebra C[G] of endomorphisms E C[G], acting on the basis ν, ν G
∈ { ∈ }
by multiplication ν E.ν, we have the isomorphism C[G] End(V ), with
7→ ≃ ϕ ϕ
End(V ) e C[G], where the e = ϕ(1) ϕ(ν 1)ν are theLcentral orthogonal
ϕ ϕ ϕ −
idempoten≃ts of C[G]. |G| νP∈G
For the decomposition of e C[G] into a direct sum of ϕ(1) irreducible representa-
ϕ
tions, isomorphic to V , we use the projectors comming from a matrix representa-
ϕ
tion M(ρ (ν))= aϕ(ν) ([Se1, I.2.7])
ϕ ij i,j §
(cid:0) (cid:1)
πϕ = ϕ(1) aϕ(ν 1)ν, i=1,...,ϕ(1),
giving a system of (noin cenntraνlP)∈Gortihiog−onalidempotents such that e = πϕ.
ϕ i
Pi
2.2.2. Recalls on group determinants (from [C]). Let G be a finite group and let
FrobG(X) = det Xτσ−1 σ,τ G be the determinant of the group G, or Frobenius
determinant, with(cid:0) indete(cid:1)rmi∈nates X :=(X ) . We then have the formula
ν ν G
∈
ϕ(1)
FrobG(X)= det Xνρϕ(ν−1) .
Qϕ (cid:16)νP∈G (cid:17)
HencetheexistenceofhomogeneouspolynomialsPϕ(X),ofdegreesϕ(1),suchthat
FrobG(X)= Pϕ(X)ϕ(1).
Qϕ
The specialization Xν 7→ −p1logp(ην) leads to (Definitions 2.3)
RegGp(η):=FrobG(cid:16) −p1 logp(η)(cid:17)=Qϕ det(cid:16)νP∈G −p1 logp(ην)ρϕ(ν−1)(cid:17)ϕ(1),
and from Regϕp(η):=Pϕ(cid:16)...,−p1logp(ην),...(cid:17)=det(cid:16)νP∈G −p1logp(ην)ρϕ(ν−1)(cid:17),
we group into partial products associated with the characters χ and θ irreducible
over Q and Q , respectively
p
Regχ(η)= Regϕ(η) & Regθ(η)= Regϕ(η).
p p p p
ϕQ|χ ϕQ|θ
2.2.3. Practical calculation of the Pϕ(X). The polynomialsPϕ(X)areobtainedin
the following way: from the vectorialspace V =C[G] (provided with the basis G),
we consider the endomorphism of V[X], L(X)= Xνν−1, which is such that
νP∈G
(cid:16)νP∈GXνν−1(cid:17).τ =νP∈GXνν−1τ =σP∈GXτσ−1σ, ∀τ ∈G.
So,the determinantofthis endomorphismin the basis τ, τ G is the Frobenius
{ ∈ }
determinant (defined up to the sign).
Let(ρ ,V )bethefamilyofnonisomorphicabsolutelyirreductiblerepresentations.
ϕ ϕ
We shall take for End(V ) the component e C[G] associated with the characterϕ.
ϕ ϕ
We use the algebra isomorphism ρ:V End(V ) defined by
ϕ
→Qϕ
a(ν)ν−1e a(ν)ρϕ(ν−1) .
νP∈G 7−→(cid:16)νP∈G (cid:17)ϕ
whereρ (ν 1)=e ν 1 inthepreviousidentification. FromtheMaschketheorem,
ϕ − ϕ −
we get for the endomorphism L(X)
LOCAL θ-REGULATORS – p-ADIC CONJECTURES 9
det (L(X))= det (Lϕ(X)) ϕ(1),
V Qϕ (cid:0) Vϕ (cid:1)
where Lϕ(X)= X ρ (ν 1) End(V [X]). We put
ν ϕ − ϕ
νP∈G ∈
Pϕ(X):=det (Lϕ(X)).
Vϕ
With a matrix realization M(ρ (ν)) = aϕ(ν) of the ρ (ν), the matrix associ-
ϕ ij i,j ϕ
ated with Lϕ(X) is Mϕ(X)= aϕ((cid:0)ν 1)X(cid:1) , of determinant Pϕ(X).
ij − ν
Let g be the least common mul(cid:16)tiνpP∈leGof the orders(cid:17)io,jf the elements of G; it is known
that representations are realizable over the field C = Q(µ ) of gth roots of unity
g g
([Se1, 12.3]). So,wemaysupposethatthe aϕ(ν)arep-integeralgebraicnumbers,
§ ij
for all p large enough (i.e., Pϕ(X) Z [X] for all ϕ).
∈ Cg,(p)
LetΓ:=Gal(C /Q)(commutative). Givenanabsolutelyirreduciblerepresentation
g
ρ :G End (V ),wedefineitsconjugatesinthefollowingGaloismannersothat
ϕ 7→ Cg ϕ
for all s Γ, ρs is the representationG End (V ) e C [G] ofcharacterϕs
∈ ϕ 7→ Cg ϕs ≃ ϕs g
defined by ϕs(ν)=(ϕ(ν))s, for all s Γ. We have,for all s Γ, ϕs(ν)=ϕ(νω(s)),
∈ ∈
where ω is the character Γ (Z/gZ) of the action of Γ on µ . We also put
× g
→
ϕt(ν):=ϕ(νt) for all integer t prime to g (Γ-conjugation).
2.2.4. Rational and p-adic characters – Idempotents. We recall their practical de-
termination.
(i) Rational characters. We put, for ϕ fixed
χ= ϕs =: ϕ and Pχ(X):= Pϕs(X)=: Pϕs(X),
s∈GaPl(C/Q) ϕP|χ s∈GaQl(C/Q) ϕQ|χ
where C C is the field of values of any Q-conjugate of ϕ.
g
⊆
(ii) p-adic characters. If p ∤ g, denote, for χ fixed, by L and D the field and the
decomposition group of p in C/Q. Let f = D be the residue degree of p in C/Q
| |
andh=[L:Q]thenumberofprimeidealspabovepinC (orL);thus[C :Q]=hf.
Let ϕ χ. We put
|
θ(ν):= ϕs(ν) L, for all ν G & Pθ(X):= Pϕs(X)=: Pϕ(X).
sP∈D ∈ ∈ sQ∈D ϕQ|θ
Wefixoneofthehprimeidealsp pofL(weshallsaythatθ andpareassociated).
|
AsL =Q forallt Gal(C/Q)/D,wehavecongruencesoftheformθ(ν) r (ν)
pt p pt
∈ ≡
(mod pt) in L, r (ν) Z; the rationals r (ν) depend numerically of the residue
pt pt
∈
images at pt of the trace in C/L of the ϕ(ν).
If θ = ϕs andp areassociated,the h conjugates of θ are the θt = (ϕt)s and
we havsPe∈Dθt(ν) ≡ rpt−1(ν) (mod p) (or θt−1(ν) ≡ rpt(ν) (mod p)). AsP∈sDthe θt are
seen in Zp ⊂Lp, we shall write by abuse θt(ν)≡rpt−1(ν) (mod p).
For p fixed, the integer f depends only on χ and is called the residue degree of
the characters ϕ,θ and χ. We have, by Γ-conjugation, ϕpi(ν) = ϕ(νpi) = ϕ(ν)sip,
where s is the Frobenius automorphism (of order f) in C/Q.
p
(iii) Idempotents. We put e = e and e = e ; thus e = e . The e
χ ϕ θ ϕ χ θ θ
ϕPχ ϕPθ θPχ
| | |
(resp. e ) give a fundamental system of orthogonal idempotents of Q [G] (resp.
χ p
Q[G]). We can replace Q (resp. Q) by Z (resp. Z ) because p∤g.
p p (p)
10 GEORGESGRAS
From Pϕ(X) = det (Lϕ(X)) we deduce that Pϕs(X) = det (Lϕs(X)) where
Vϕ Vϕs
Lϕs(X)= X ρs(ν 1) is givenvia the (aϕ(ν 1))s,which defines the conjugate
ν ϕ − ij −
by s of theνPp∈oGlynomial Pϕ(X) (i.e., of its coefficients).
Theorem 2.5. (i) For all p large enough, the polynomials Pχ(X) (resp. Pθ(X))
have rational p-integer coefficients (resp. p-adic integer coefficients).
(ii) For all irreducible character ϕ, we have Pϕ(...,X ,...)=ζ Pϕ(...,X ,...)
πν π ν
for all π G, where ζg =1.
∈ π
Proof. (i) As Pϕ(X) Z [X] for all ϕ χ, Pχ(X) = Pϕs(X) is invari-
C,(p)
∈ | s GaQl(C/Q)
ant by Galois. Likewise Pθ(X)= Pϕs(X) L[X] ∈L [X]=Q [X].
p p
sQ∈D ∈ ⊂
(ii)Forπ Gcall[π]theoperatordefinedby[π]X =X forallν G. Then[π]
ν (πν)
∈ ∈
andρ:V[X] End(V [X])commute; moreover,since ρ is a homomorphism,
→ ϕ ϕ ϕ
we have the folloQwing formula
e
[π] X ρ (ν 1) = X ρ (ν 1)= X ρ (ν 1) ρ (π).
ν ϕ − πν ϕ − ν ϕ − ϕ
(cid:16)νP∈G (cid:17) νP∈G (cid:16)νP∈G (cid:17)
Then,sincethedeterminantofρ (π) End(V )isthatofadiagonalmatrixwhose
ϕ ϕ
∈
diagonal is formed of roots of unity, we get
det [π] X ρ (ν 1) =ζ det X ρ (ν 1) ,
ν ϕ − π ν ϕ −
(cid:16) (cid:16)νP∈G (cid:17)(cid:17) (cid:16)νP∈G (cid:17)
where ζ is of order a divisor of the order of ρ (π) which is a divisor of g. (cid:3)
π ϕ
Corollary 2.6. For all π G and all absolutely irreducible character ϕ, we have
∈
Pϕ(...,απν,...)=ζ Pϕ(...,αν,...) by the specialization X αν, α Z .
π ν K
7→ ∈
Consequentely, Pχ(...,απν,...)= Pχ(...,αν,...) for all π G.2
± ∈
In the same way, Pθ(...,απν,...) = ζ Pθ(...,αν,...) for all π G, where ζ is
π′ ∈ π′
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
numberpandthecharactersthatweconsiderareabsolutelyirreducibleorrational.
Theaboveleadstodefinethenumericalχ-determinantsofFrobeniusofanyα Z
K
∈
(i.e., independent of the given η K ).
×
∈
Definition 2.7. LetGbeafinite groupandletFrobG(X)bethe associatedgroup
determinant. The χ-determinants (with indeterminates and numerical)are by def-
inition the expressions
Frobχ(X)= Pϕ(X) and Frobχ(α)= Pϕ(...,αν,...),
ϕQχ ϕQχ
| |
so that FrobG(α)= (Frobχ(α))ϕ(1) (where ϕ χ for each χ).
Qχ |
Example 2.8. In the case of the group D = 1,σ,σ2,τ,τσ,τσ2 , we have the
6
{ }
following numerical χ-determinants
Frob1(α)=α+ασ+ασ2+ατ +ατσ+ατσ2,
Frobχ1(α)=α+ασ+ασ2 ατ ατσ ατσ2,
− − −
Frobχ2(α)=α2+α2σ+α2σ2 α2τ α2τσ α2τσ2 αασ ασασ2 ασ2α
− − − − − +ατα−τσ+ατσατσ2+ατσ2ατ.
2Sign+except ifχ=ϕisquadraticandϕ(π)= 1.
−