Congruences involving $\binom{2k}k^2\binom{4k}{2k}m^{-k}$

preprint2011arXivOpen access

Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$ . In the paper, 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 $\sum_{k=0}^{p-1}\binom{2k}k^2\binom{4k}{2k}m^{-k}\mod {p^2}$ .

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Congruences involving $\binom{2k}k^2\binom{4k}{2k}m^{-k}$

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Congruences involving $\binom{2k}k^2\binom{4k}{2k}m^{-k}$