Congruences for odd class numbers of quadratic fields with odd discriminant

Abstract

For any distinct two primes p1≡ p2≡ 3 (mod4), let h(- p1) , h(- p2) and h(p1p2) be the class numbers of the quadratic fields Q(-p1), Q(-p2) and Q(p1p2), respectively. Let ωp1p2:=(1+p1p2)/2 and let Ψ(ωp1p2) be the Hirzebruch sum of ωp1p2. We show that h(-p1)h(-p2)≡h(p1p2)Ψ(ωp1p2)/n(mod8), where n= 6 (respectively, n= 2) if minp1 , p2 > 3 (respectively, otherwise). We also consider the real quadratic order with conductor 2 in Q(p1p2). © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.