Class Numbers and Self-Conjugate 7-Cores

Abstract

We investigate $s{c}_7(n)$, the number of self-conjugate $7$-core partitions of size $n$. It turns out that $s{c}_7(n)=0$ for $n\equiv 7(\mathrm{mod}8)$. For $n\equiv 1,3,5(\mathrm{mod}8)$, with $n\not\equiv 5(\mathrm{mod}7),$ we find that $s{c}_7(n)$ is essentially a Hurwitz class number. Using recent work of Gao and Qin, we show that $$s{c}_7(n)=2^{-\epsilon (n)-1}\cdot H(-D_n),$$ where $-D_n:=-4^{\epsilon (n)}(7n+14)$ and $\epsilon (n):=\frac{1}{2}\cdot (1+(-1{)}^{\frac{n-1}{2}})$. This fact implies several corollaries which are of interest. For example, if $-D_n$ is a fundamental discriminant and $p\notin \{2,7\}$ is a prime with $or{d}_p(-D_n)\le 1$, then for every positive integer $k$ we have $$s{c}_7\left((n+2)p^{2k}-2\right)=s{c}_7(n)\cdot \left(1+\frac{p^{k+1}-p}{p-1}-\frac{p^k-1}{p-1}.\left(\frac{-D_n}{p}\right)\right),$$ where $\left(\frac{-D_n}{p}\right)$ is the Legendre symbol.