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

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

2010-11-29

arxiv.org/abs/1011.6283v1

Mathematics

Number Theory

Scientific paper

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 \cite{La}, p. 261 and \cite{Wa}, 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, {\em in a certain sense} into an argument in favor of Vandiver's conjecture.

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