Class numbers and self-conjugate 7-cores

Abstract

We investigate sc7(n), the number of self-conjugate 7-core partitions of size n. It turns out that sc7(n)=0 for n≡7(mod8). For n≡1,3,5(mod8), with n≢5(mod7), we find that sc7(n) is essentially a Hurwitz class number. Using recent work of Gao and Qin, we show that sc7(n)=2−ε(n)−1⋅H(−Dn), where −Dn:=−4ε(n)(7n+14) and [Formula presented]. This fact implies several corollaries which are of interest. For example, if −Dn is a fundamental discriminant and p∉{2,7} is a prime with ordp(−Dn)≤1, then for every positive integer k we have [Formula presented] where [Formula presented] is the Legendre symbol. © 2021 Elsevier Inc.