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.
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- 1.
One may encounter also the notation \({h}^{+} = \vert \mathrm{id}C({\mathbb{K}}^{+})\vert \), but we refer strictly to the p-part in this paper.
- 2.
Washington mentions explicitly that this is the critical point in the various heuristics of this kind.
References
P. Furtwängler, Über die Reziprozitätsgesetze zwischen l-ten Potenzresten in algebraischen Zahlkörpern, wenn l eine ungerade Primzahl bedeutet. Math. Ann. (German) 58, 1–50 (1904)
K. Iwasawa, A note on the group of units of an algebraic number fields. J. de Math. Pures et Appl. 35/121, 189–192 (1956)
S. Lang, Cyclotomic Fields I and II, 1st edn. (Springer, New York, 1978/1980)
P. Ribenboim, 13 Lectures on Fermat’s Last Theorem (Springer, New York, 1979)
L. Washington, Introduction to Cyclotomic Fields. Graduate Texts in Mathematics, vol. 83, 2nd edn. (Springer, New York/London, 1996)
Acknowledgements
I thank the anonymous referee for helpful questions and suggestions, which helped improve the clarity of the exposition.
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Dedicated to the 80th Anniversary of Professor Stephen Smale
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Mihăilescu, P. (2012). Turning Washington’s Heuristics in Favor of Vandiver’s Conjecture. In: Pardalos, P., Rassias, T. (eds) Essays in Mathematics and its Applications. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28821-0_12
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DOI: https://doi.org/10.1007/978-3-642-28821-0_12
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