q-Congruences, with applications to supercongruences and the cyclic sieving phenomenon

We establish a supercongruence conjectured by Almkvist and Zudilin, by proving a corresponding q-supercongruence. Similar q-supercongruences are established for binomial coefficients and the Apéry numbers, by means of a general criterion involving higher derivatives at roots of unity. Our methods lead us to discover new examples of the cyclic sieving phenomenon, involving the q-Lucas numbers.