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

TL;DR

This paper explores the relationship between Washington's heuristics and Vandiver's conjecture, suggesting that deep correlations with irregularity indices could support Vandiver's conjecture under certain assumptions.

Contribution

It demonstrates that assuming Greenberg's conjecture, failure of Vandiver's conjecture implies correlations with the irregularity index, potentially turning heuristics into supporting evidence.

Findings

Failure of Vandiver's conjecture correlates with the defect λ^- > i(p).

Assuming Greenberg's conjecture, deep links are found between Vandiver's conjecture failure and Bernoulli number irregularities.

Results suggest Washington's heuristics could support Vandiver's conjecture under specific conditions.

Abstract

A famous conjecture bearing the name of Vandiver states that $p \nmid h_p^+$ in the $p$ - cyclotomic extension of $\Q$. Heuristics arguments of Washington, which have been briefly exposed in Lang (1978), p. 261 and Washington (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 \nth{p} 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 $B_{2k} \equiv 0 \bmod p, 1 < k < (p-1)/2$. As a consequence, this result could turn Washington's heuristic arguments, in a certain sense into an argument in favor of Vandiver's conjecture.