Combinatorial Congruences and ψ-Operators

The 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). 1 Combinatorial Congruences Let p be a prime, n ∈ Z>0. Throughout this paper, let [x] denote the integer part of x if x ≥ 0 and [x] = 0 if x < 0. In the author’s course lectures [4] on Fontaine’s theory and p-adic L-functions given at UC Irvine (spring 2005) and at the Morningside Center of Mathematics (summer 2005), the following two congruences were discovered. Theorem 1.1. For integers r ∈ Z, j ≥ 0, we have