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

Let $k$ be an imaginary quadratic field and let $p \ne 2$ be a prime number, split in $k$ into ${\mathfrak p}{\overline {\mathfrak p}}$. We assume that the $p$-class group of $k$ is trivial. Let $δ\geq 0$ be the ${\mathfrak p}$-valuation of the ${\overline {\mathfrak p}}$-Fermat quotient of the fund