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Thursday 08 Dec 2016Cyclotomic Units and Greenberg's Conjecture

Filippo Majno di Capriglio - Universite de Saint-Etienne

H101 14:30-16:30

For an abelian number field F, already Kummer (in the case of cyclotomic fields), and

eventually Sinnott, defined the so-called cyclotomic units as a

subgroup of full rank of the usual units with the striking property that

their index in the full group of units coincides—up to known explicit constants—with

the class number of F. Suppose now that p is a prime not dividing [F : Q]: then Sinnott proved that these explicit constants stay bounded along the cy-

clotomic p-extension of F, and they are prime to p: in other words, at every

level of the cyclotomic extension, the p-part of the index of the cyclotomic

units inside the full group of units equals the order of the p-Sylow of the class group.

It is then natural to ask whether this equality comes from an isomorphism

(of abelian groups, say). Works by Kraft−Schoof and Kuz’min, later refined

by Ozaki and by Belliard−Nguyen-Quang-Do, shows that if p does not split

in F then a famous conjecture by Greenberg would indeed imply that the two aformentioned groups eventually

become isomorphic as Galois modules. In my talk I would like to discuss the

case when p splits in F, showing that Greenberg’s conjecture implies that the

natural map giving the isomorphism in the non-split case has non-trivial

kernels and cokernels whose order is controlled by a special value of the p-adic L-function.

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