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Pré-Publication, Document De Travail Année : 2014

Notion de F_p-régulateurs d'un nombre algébrique--Conjectures p-adiques

Georges Gras

Résumé

Let K/Q be a finite Galois extension of degree n, of Galois group G, and let η∈ K*. For all large enough prime p (in particular prime to n, Disc(K), and η), we define, by use of the Frobenius theorem on group determinants, the family Δ(p,θ)(η) ∈ Z/pZ of Z/pZ-regulators of η, indexed by the Q_p-irreducible characters θ of G. At each Δ(p,θ)(η) is associated a linear representation L(θ) ≃ δV(θ) (where V(θ) is the representation with irreducible caracter θ, 0≤ δ ≤ϕ(1) with ϕ I θ absolutely irreducible) characterising (see Thm. 3.13) the properties of Δ(p,θ)(η) including its nullity (equivalent to δ > 0). When η ∈ Q* and θ = 1, Δ(p,1)(η) is the p-Fermat quotient of η. When η is a ''Minkowski unit'' (for K real), each Δ(p,θ)(η), θ ≠ 1, gives the residue modulo p of the θ-regulator R(p,θ) in the decomposition Reg(K,p) =p^(n-1) ∏ R(p,θ)^ϕ(1) of the classical p-adic regulator Reg(K,p) of the field K. We suggest, starting from a general property of group determinants and the existence of the representation L(θ) that for θ of any degree, the ''probability'' of nullity of Δ(p,θ)(η) with L(θ) ≃ δ V(θ) is about O(1)/p^(δ^2 f), where f is the residue degree of p in the field of values of the characters ϕ I θ, and that the ''probability'' of nullity of several Δ(p,θ)(η) is equal to the product of the respective probabilities. We conjecture that p^(1-n) Reg(K,p), which measures the order of the p-torsion group in Abelian p-ramification over K, is a p-adic unit except perhaps for an infinite set of prime numbers of zero density. For this set of p said ''with minimal splitting'' (i.e. a single Δ(p,θ)(η) is zero with δ=f = 1, which expresses that L(θ) is of minimal Z/pZ-dimension ϕ(1)), it remains possible, η being then a ''partial local pth power'' at p, to justify and to propose a stronger conjecture leading to the same conclusion. Some other conjectural aspects related to the ABC conjecture are discussed (Section 7). We precise and verify these properties through numerical studies on particular cyclic cubic and quintic fields and a non-Abelian field (group D_6), and we publish the corresponding ''PARI'' programs.
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Dates et versions

hal-00936889 , version 1 (27-01-2014)
hal-00936889 , version 2 (01-03-2014)
hal-00936889 , version 3 (03-04-2014)

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Georges Gras. Notion de F_p-régulateurs d'un nombre algébrique--Conjectures p-adiques. 2014. ⟨hal-00936889v1⟩
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