Congruences involving ${\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{3k}{k}\right)m^{-k}$

Abstract

Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper, based on the work of Brillhart and Morton, by using the work of Ishii and Deuring's theorem for elliptic curves with complex multiplication we solve some conjectures of Zhi-Wei Sun concerning ${\sum }_{k=0}^{p-1}{\left(\genfrac{}{}{0ex}{}{2k}{k}\right)}^2\left(\genfrac{}{}{0ex}{}{3k}{k}\right)m^{-k}\mathrm{mod}{p}^2$.

28 pages