Combinatorial congruences and -operators

Abstract: The -operator for -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 -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 [Some congruences for binomial coefficients, Michigan Math. J. 24 (1977) 141–151]. [Copyright &y& Elsevier]