Local theta-regulators of an algebraic number -- p-adic Conjectures (1701.02618v1)

Abstract: Let K/Q be Galois and let eta in K* be such that the multiplicative Z[G]-module generated by eta is of Z-rank n.We define the local theta-regulators Delta_p^theta(eta) in F_p for the Q_p-irreducible characters theta of G=Gal(K/Q). Let V_theta be the theta-irreducible representation. A linear representation L^theta=delta.V_theta is associated withDelta_p^theta(eta) whose nullity is equivalent to delta$\ge$1 (Theorem 3.9). Each Delta_p^theta(eta) yields Reg_p^theta(eta) modulo p in the factorization $\prod$_theta (Reg_p^theta(eta))^phi(1) of Reg_p^G(eta) := Reg_p(eta)/p^[K : Q] (normalized p-adic regulator), where phi divides theta is absolutely irreducible.From the probability Prob(Delta_p^theta(eta) = 0 & L^theta=delta.V_theta)$\le$p^-f.delta^2 (f= residue degree of p in the field of values of phi) and the Borel--Cantelli heuristic, we conjecture that, for p large enough, Reg_p^G(eta) is a p-adic unit or that p^phi(1) divides exactly Reg_p^G(eta) (existence of a single theta with f=delta=1); this obstruction may be lifted assuming the existence of a binomial probability law (Sec. 7) confirmed through numerical studies (groups C_3, C_5, D_6). This conjecture would imply that, for all p large enough, Fermat quotients of rationals andnormalized p-adic regulators are p-adic units (Theorem 1.1), whence the fact that number fields are p-rational for p\textgreater{}\textgreater{}0. We recall \S8.7 some deep cohomological results, which may strengthen such conjectures.