Class Numbers and Self-Conjugate 7-Cores

Abstract

We investigate $sc_7(n)$, the number of self-conjugate $7$-core partitions of size $n$. It turns out that $sc_7(n)=0$ for $n\equiv 7\pmod 8$. For $n\equiv 1, 3, 5\pmod 8$, with $n\not \equiv 5\pmod 7,$ we find that $sc_7(n)$ is essentially a Hurwitz class number. Using recent work of Gao and Qin, we show that $$ sc_7(n) = 2^{-\varepsilon(n)-1}\cdot H(-D_n), $$ where $-D_n:=-4^{\varepsilon(n)}(7n+14)$ and $\varepsilon(n):=\frac{1}{2}\cdot(1+(-1)^{\frac{n-1}{2}})$. This fact implies several corollaries which are of interest. For example, if $-D_n$ is a fundamental discriminant and $p\not \in \{2, 7\}$ is a prime with $ord_p(-D_n)\leq 1$, then for every positive integer $k$ we have $$ sc_7\left((n+2)p^{2k}-2\right)=sc_7(n)\cdot \left(1+\frac{p^{k+1}-p}{p-1}-\frac{p^k-1}{p-1}.\left(\frac{-D_n}{p}\right)\right), $$ where $\left(\frac{-D_n}{p}\right)$ is the Legendre symbol.