Let, p 5 be primes such that p | (-1). Let Δ > 0 be the fundamental discriminant of a real quadratic field in which splits. We denote by h - (Δ) the order of the minus part (for the Galois action) of the ray class group of Q (Δ) of modulus. In this paper, we study the indivisibility of h - (Δ) by p, and prove that under the assumption that this set is non-empty. This lower bound is made unconditional if = 2p + 1, i. e. if p is a Sophie Germain prime. Our result can be viewed as being in the continuity of the results of Kohnen-Ono, Ono, Byeon, Beckwith etc. regarding the class numbers of quadratic fields, in the sense that we rely on techniques from the theory of half-integral weight modular forms. Significant difficulties however arise in our study, as we have to study Eisenstein congruences for cuspforms of weight 3 2, and use a generalized Shimura correspondence of Baruch-Mao. Combined with the results of Lecouturier-Wang, our result has implications eg. for the 5-part of BSD for even quadratic twists of X 0 (11).
Lecouturier et al. (Mon,) studied this question.