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

Abstract

Let k be an imaginary quadratic field and let p>2 be a prime number, split in k into PP&#39;. We assume that the p-class group of k is trivial. Let $δ$ $\ge$ 0 be the P&#39;-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&#39; being totally ramified. We prove that if e $\ge$ $δ$, then $λ$(K/k) = 1, $μ$(K/k) = 0; if e < $δ$ 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\_$δ$ in incomplete p-ramification. They provide effective and computable results that complement some aspects of Iwasawa&#39;s theory. A pari/gp program computes $δ$ and R^p\_$δ$ , for p = 3, e = 1.