  Let $p$ be an odd prime. In the paper, by using the properties of Legendre
polynomials we prove some congruences for
$\sum_{k=0}^{\frac{p-1}2}\binom{2k}k^2m^{-k}\mod {p^2}$. In particular, we
confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on
supercongruences.
