One of the recent questions, in fact the answer to it, reminded me about the binomial sequence $$ a_n=\sum_{k=0}^n{\binom{n}{k}}^2{\binom{n+k}{k}}^2, \qquad n=0,1,2,\dots, $$ of the Apéry numbers. The numbers $a_n$ come as denominators of rational approximations to $\zeta(3)$ in Apéry's famous proof of the irrationality of the number. There are many nice properties of the sequence, one of these is the observation that for primes $p\ge5$, $$ a_{pn}\equiv a_n\pmod{p^3}, \qquad n=0,1,2,\dots. $$ The congruence was conjectured for $n=1$ by S. Chowla, J. Cowles and M. Cowles and proved in the full generality by I. Gessel (1982).

There is probably nothing strange in the congruence
(which belongs, by the way, to the class of *supercongruences* as it happens
to hold for a power of prime higher than one). But already the classical
binomials behave very similarly: for primes $p\ge5$,
$$
\binom{pm}{pn}\equiv\binom mn\pmod{p^3},
$$
the result due to G.S. Kazandzidis (1968). There are many other examples
of modulo $p^3$ congruences, most of them explicitly or implicitly related
to some modular objects, but that is a different story. My question is:
what are the grounds for the above (very simple) congruence for binomial
coefficients to hold modulo $p^3$? Not modulo $p$ or $p^2$ but $p^3$.
I do not ask you to prove the supercongruence but to indicate a general
mechanism which provides some kind of evidence for it and can be used
in other similar problems.

My motivation rests upon my own research on supercongruences; most of them are just miracles coming from nowhere...