Some congruences for binomial coefficients. II

Abstract

Let t = 3 mod 4 (t > 3) be a prime and sigma (r) : zetat --> zeta (r)(t) be a generator of Gal(Q(zetat)/Q(root -t)) for r is an element of {1,...,t - 1}. If p = tn + r is a prime, then 4p(h) can be expressed as the form 4p(h) = a(2) + tb(2) where h is the class number of Q(root -t). Let alphat be the sum of representatives of [r] in (Z/tZ)(X) and beta = phi>(*) over bar * (t)/2 - alpha. If we choose the sign of a then a = 2p(beta) mod t and a satisfies a certain congruence relation module p. We also treat the case of t = 4k for a prime k = 1 mod 4.