Turning Washington's heuristics in favor of Vandiver's conjecture

Abstract

A famous conjecture bearing the name of Vandiver states that in the p - cyclotomic extension of. 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 i(p)$$ ]], where i(p) is like usual the irregularity index of p, i.e. the number of Bernoulli numbers. As a consequence, this result could turn Washington's heuristic arguments, in a certain sense into an argument in favor of Vandiver's conjecture.