In Z_p-extensions of totally p-adic imaginary quadratic fields, the p-valuation of a Fermat quotient of the fundamental p-unit governs the orders of logarithmic class groups and the quotients of the first two layers of p-class group filtrations for large n.
Gras, Les -r\'egulateurs locaux d'un nombre alg\'ebrique : Conjectures p -adiques, Canad
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On the $\mathbb{Z}_p$-extensions of a totally $p$-adic imaginary quadratic field -- With an appendix by Jean-Fran\c{c}ois Jaulent
In Z_p-extensions of totally p-adic imaginary quadratic fields, the p-valuation of a Fermat quotient of the fundamental p-unit governs the orders of logarithmic class groups and the quotients of the first two layers of p-class group filtrations for large n.