Combinatorial Congruences and $\psi$-Operators

The $\psi$-operator for $(\varphi ,\mathrm{\Gamma })$-modules plays an important role in the study of Iwasawa theory via Fontaine’s big rings. In this note, we prove several sharp estimates for the $\psi$-operator in the cyclotomic case. These estimates immediately imply a number of sharp $p$-adic combinatorial congruences, one of which extends the classical congruences of Fleck (1913) and Weisman (1977).