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

Preda Mihăilescu ()
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Preda Mihăilescu: Mathematisches Institut der Universität Göttingen

A chapter in Essays in Mathematics and its Applications, 2012, pp 287-294 from Springer

Abstract: 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

Keywords: Galois Group; Principal Ideal; Galois Extension; Bernoulli Number; Heuristic Argument (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1007/978-3-642-28821-0_12

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