We give a criterion of the vanishing of $\lambda_p$-invariants for relative cyclic extensions of totally real number fields with degree $p$ using Iwasawa's result of Riemann-Hurwitz type which is an analogue to Kida's formula.
"On Iwasawa $\lambda_p$-Invariants of Relative Real Cyclic Extensions of Degree $p$." Tokyo J. Math. 20 (2) 475 - 480, December 1997. https://doi.org/10.3836/tjm/1270042120