Congruences involving $\binom{2k}k^2\binom{3k}km^{-k}$

Zhi-Hong Sun

Published 2011-04-14, updated 2011-04-28Version 3

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