Sur le p-rang du groupe des classes de Q(N^1/p)

Abstract

Let N and p be two prime numbers > 3 such that p divides N-1. We estimate the p-rank of the class group of Q(N^(1/p)) in terms of the discrete logarithm, with values un F_p, of certain units. Using the Gross--Koblitz formula and identities on the N-adic Gamma function, we explicitly compute these logarithms. A special case (for which we don't have an elementary proof) of our formula is the following: assume there are some integers $a$, $b$ such that N = (a^p+b^p)/(a+b). Then (a+b)*\prod_{k=1}^{(N-1)/2} k^{8k} is a p-th power modulo N. Furthermore we give a new proof which doesn't use modular forms of a result of Calegari and Emerton.