  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
$\sum_{k=0}^{p-1}\binom{2k}k^2\binom{3k}km^{-k}\mod {p^2}$.
