2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75999The $ψ$-operator for $(ϕ, Γ)$-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 $ψ$-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).11 pagesNumber TheoryCombinatoricstext