Indivisibility of relative class numbers of totally imaginary quadratic extensions and vanishing of these relative Iwasawa invariants

We study an indivisibility problem of the relative class numbers of CM fields. For prime p>3, Kohnen-Ono gave a lower bound of the number of the imaginary quadratic fields whose class numbers are prime to p by using modular forms of half-integral weight. We generalize their method to Hilbert modular forms and give a lower bound of the number of CM quadratic extensions K/F whose relative class numbers prime to p for totally real number field F which is Galois over Q and sufficiently large prime p. Combining the indivisibility result with the decomposition condition of p, we show a result on vanishing of relative Iwasawa invariants.