On the Galois structure of the class group of certain Kummer extensions.

Abstract: Let p ⩾ 5 and N be prime numbers 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 in Z / p Z, of certain units. Using the Gross–Koblitz formula and identities on the N‐adic Gamma function, we explicitly compute these logarithms. In particular, we give a new proof which does not use modular forms of a result of Calegari and Emerton. Using the same method, we prove a special case of a twisted form of a conjecture of Gross about the relation between some Stickelberger element and the Galois structure of the class group of the cyclotomic field Q ( ζ p , ζ N ). A special case of our formulae, for which we do not have an elementary proof, is the following: Assume there are some integers a, b such that N = ( a p + b p ) / ( a + b ). Then ( a + b ) · ( ∏ k = 1 ( N − 1 ) / 2 k 8 k ) is a pth power modulo N. [ABSTRACT FROM AUTHOR]