Les θ-régulateurs locaux d'un nombre algébrique : Conjectures p-adiques

Let K/ℚ be Galois and let η K ×be such that Reg∞(η)=0 .We define the local θ–regulator for the ℚp–irreducible characters θ of G = Gal(Kℚ). Let Vθ be the θ-irreducible representation. A linear representation is associated with whose nullity is equivalent to δ≥1. Each yields Regθp modulo p in the factorization of (normalized p–adic regulator). From Prob f ≥ 1 is a residue degree) and the Borel–Cantelli heuristic, we conjecture that for p large enough, RegGp(η) is a p–adic unit (a single with f = δ=1); this obstruction may be led assuming the existence of a binomial probability law confirmed through numerical studies (groups C3, C5, D6) is conjecture would imply that for all p large enough, Fermat quotients, normalized p–adic regulators are p–adic units and that number fields are p-rational.We recall some deep cohomological results that may strengthen such conjectures.

Some congruences related to the q-Fermat quotients

We give q-analogues of the following congruences by Z.-W. Sun: [Formula: see text] where p is an odd prime, [Formula: see text] are the Delannoy numbers, and [Formula: see text] are the harmonic numbers. We also prove that, for any positive integer m and prime p > m + 1, [Formula: see text] which is a multiple generalization of Kohnen's congruence. Furthermore, a q-analogue of this congruence is established.