Chevalley-Herbrand formulas and Z p -extensions of a p-principal imaginary quadratic field

Let k be an imaginary quadratic field and let p>2 be a prime number, split in k into PP’. We assume that the p-class group of k is trivial. Let $\delta$ $\ge$ 0 be the P’-valuation of the P-Fermat quotient of the fundamental P-unit x of k. Let K/k be any Zp-extension and let p^e be the degree of the inertia field of P, P’ being totally ramified. We prove that if e $\ge$ $\delta$, then $\lambda$(K/k) = 1, $\mu$(K/k) = 0; if e < $\delta$ a characterization is obtained from the S^P-Iwasawa invariants (Theorems 6.1, 7.6). These results only use generalizations of Chevalley-Herbrand formulas and the non-nullity of a p-adic regulator R^p_$\delta$ in incomplete p-ramification. They provide effective and computable results that complement some aspects of Iwasawa’s theory. A pari/gp program computes $\delta$ and R^p_$\delta$ , for p = 3, e = 1.