Turning Washington’s Heuristics in Favor of Vandiver’s Conjecture

Abstract

A famous conjecture bearing the name of Vandiver states that $p \nmid {h}_{p}^{+}$ in the p – cyclotomic extension of $\mathbb{Q}$. Heuristics arguments of Washington, which have been briefly exposed in Lang (Cyclotomic fields I and II, Springer, New York, 1978/1980, p 261) and Washington (Introduction to cyclotomic fields, Springer, New York/London, 1996, p 158) suggest that the Vandiver conjecture should be false if certain conditions of statistical independence are fulfilled. In this note, we assume that Greenberg’s conjecture is true for the p−th cyclotomic extensions and prove an elementary consequence of the assumption that Vandiver’s conjecture fails for a certain value of p: the result indicates that there are deep correlations between this fact and the defect ${\lambda }^{-} > i(p)$, where i(p) is like usual the irregularity index of p, i.e. the number of Bernoulli numbers ${B}_{2k} \equiv 0\mbox{ mod}p,1 < k < (p - 1)/2$. As a consequence, this result could turn Washington’s heuristic arguments, in a certain sense into an argument in favor of Vandiver’s conjecture.