Class numbers of real cyclotomic fields of prime conductor

Abstract:

The class numbers $h_{l}^{+}$ of the real cyclotomic fields $\mathbf {Q}(\zeta _{l}^{}+\zeta _{l}^{-1})$ are notoriously hard to compute. Indeed, the number $h_{l}^{+}$ is not known for a single prime $l\ge 71$. In this paper we present a table of the orders of certain subgroups of the class groups of the real cyclotomic fields $\mathbf {Q}(\zeta _{l}^{}+\zeta _{l}^{-1})$ for the primes $l<10,000$. It is quite likely that these subgroups are in fact equal to the class groups themselves, but there is at present no hope of proving this rigorously. In the last section of the paper we argue —on the basis of the Cohen-Lenstra heuristics— that the probability that our table is actually a table of class numbers $h_{l}^{+}$, is at least $98\%$.