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

TL;DR

This paper estimates the p-rank of class groups of certain number fields using explicit calculations of p-adic logarithms of units, employing advanced formulas like Gross--Koblitz and N-adic Gamma functions.

Contribution

It introduces a method to compute the p-rank of class groups via explicit p-adic logarithms and provides a new proof of a related result without modular forms.

Findings

Explicit formulas for p-adic logarithms of units in Q(N^{1/p})

A special case relating N, a, b, and p-th powers modulo N

A new proof of a known result avoiding modular forms

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.