$p$-adic analogues of hypergeometric identities and their applications

In this paper, we confirm several conjectures posed by Sun recently; for example, we prove that for any odd prime $p$ we have $$\sum _{k=0}^{p-1}{A}_k\equiv \{\begin{array}{ll}4x^2-2p(\mathrm{m}\mathrm{o}\mathrm{d}{p}^2)& \text{if }p=x^2+2y^2(x,y\in \mathbb{Z}),\\ 0(\mathrm{m}\mathrm{o}\mathrm{d}{p}^2)& \text{if }p\equiv 5,7(\mathrm{m}\mathrm{o}\mathrm{d}8),\end{array}$$ where $A_n:={\sum }_{k=0}^n{\left(\genfrac{}{}{0px}{}{n+k}{k}\right)}^2{\left(\genfrac{}{}{0px}{}{n}{k}\right)}^2$ are the Ap'{e}ry numbers.