Some congruences for binomial coefficients. II

Abstract

Let $t \equiv 3 \mod 4$ ($t > 3$) be a prime and $\sigma_r\colon \zeta_t \rightarrow \zeta_t^r$ be a generator of $\operatorname{Gal}(\mathbf{Q}(\zeta_t) / \mathbf{Q}(\sqrt{-t}))$ for $r \in \{1,\dots,t-1\}$. If $p = tn + r$ is a prime, then $4p^h$ can be expressed as the form $4p^h = a^2 + tb^2$ where $h$ is the class number of $\mathbf{Q}(\sqrt{-t})$. Let $\alpha t$ be the sum of representatives of $\langle r \rangle $ in $(\mathbf{Z}/t\mathbf{Z})^{\times}$ and $\beta = \phi(t)/2 - \alpha$. If we choose the sign of $a$ then $a \equiv 2p^{\beta} \mod t$ and $a$ satisfies a certain congruence relation modulo $p$. We also treat the case of $t = 4k$ for a prime $k \equiv 1 \mod 4$.