Indivisibility of ray class groups of real quadratic fields

Abstract

Let ${\ell}$, p $\ge$ 5 be primes such that p | (${\ell}$ -1). Let $Δ$ > 0 be the fundamental discriminant of a real quadratic field in which ${\ell}$ splits. We denote by h - ${\ell}$ ($Δ$) the order of the minus part (for the Galois action) of the ray class group of Q( $\sqrt$ $Δ$) of modulus ${\ell}$. In this paper, we study the indivisibility of h - ${\ell}$ ($Δ$) by p, and prove that under the assumption that this set is non-empty. This lower bound is made unconditional if ${\ell}$ = 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).