Recent Progress on Congruences Involving Binomial Coefficients

In 1913 A. Fleck proved that if p is a prime, and n > 0 and r are integers then ∑ k≡r (mod p) (n k ) (−1) ≡ 0 ( mod pb(n−1)/(p−1)c ) . Only recently the significance of Fleck’s congruence was realized. It plays a fundamental role in Colmez’ and Wan’s investigation of the ψ-operator related to Fontaine’s theory and p-adic Langlands correspondence. In this talk we give a survey of the recent developments of Fleck’s congruence and its various extensions, as well as some important applications to Stirling numbers of the second kind and homotopy exponents of special unitary groups given by Davis and the speaker. Both number-theoretic and combinatorial approaches will be introduced. 1. Lucas’ and Wolstenholeme’s congruences Let (x)0 = 1 and (x)k = x(x−1) · · · (x−k+1) for k ∈ Z = {1, 2, 3, . . . }. For k ∈ N = {0, 1, 2, . . . }, define the binomial coefficient ( x k ) = (x)k k! 1