## On a construction of $$p$$-units in abelian fields.(English)Zbl 0772.11043

For a fixed prime $$p$$ and a number field $$K$$, let $$E_ p(K)$$ denote the group of $$p$$-units of $$K$$, i.e., elements of $$K$$ that are locally units at all finite places of $$K$$ outside $$S_ p(K)$$, the set of prime ideals above $$p$$. Let $$L$$ be an abelian field of conductor $$f$$ and assume that $$p$$ splits completely in $$L$$. An important $$p$$-unit of $$L$$ is obtained by means of the Gauss sum $$\tau(\chi^{-1})$$, where $$\chi$$ is the power residue character modulo $${\mathfrak P}\in S_ p(\mathbb{Q}(\zeta_ f))$$ ($$\zeta_ f$$ a primitive $$f$$th root of 1). In fact, $$\tau(\chi^{- 1})^ f$$ is an element of the splitting field of $$p$$ in $$\mathbb{Q}(\zeta_ f)$$ and its norm in $$L$$ is such a $$p$$-unit.
The author introduces “cyclotomic $$p$$-units” $$\kappa(L)$$ of $$L$$ whose role for the plus-part (even part) of the class group of $$L$$ is in some sense analogous to that played by the above $$p$$-unit for the minus-part. The construction of $$\kappa(L)$$ stems for a “wild” variant of a method of F. Thaine, V. A. Kolyvagin and others.
To give an idea of this construction, let $$p>2$$ and, for $$i\geq 0$$, let $$L_ i\subset L(\zeta_{p^{i+1}})$$ be the unique extension of $$L$$ which is of degree $$p^ i$$ and linearly disjoint with $$L(\zeta_ p)$$. Let $$\gamma_ i$$ be the generator of the Galois group of $$L_ i/L$$. Put $\alpha_ i=N_{\mathbb{Q}(\zeta_{fp^{i+1}})/L_ i} (1- \zeta_{fp^{i+1}}).$ Then $$N_{L_ i/L}(\alpha_ i)=1$$ and Hilbert’s theorem 90 yields an element $$\beta\in L_ i^ \times$$ such that $$\beta_ i^{\gamma_ i-1}=\alpha_ i$$. The element $$\kappa_ i=N_{L_ i/L} (\beta_ i)$$ is well-defined modulo $$p^ i$$th powers, and by letting $$i$$ vary one defines (after a normalisation) $$\kappa(L)$$ to be the projective limit of $$\gamma_ i$$. This $$p$$-unit $$\kappa(L)$$ is actually not an element of $$E_ p(L)$$ but, rather, of $$E_ p(L)\otimes_ \mathbb{Z} \mathbb{Z}_ p$$, where $$\mathbb{Z}_ p$$ denotes the ring of $$p$$-adic integers.
The author’s main result gives the $${\mathfrak p}$$-ordinals of $$\kappa(L)$$, for all $${\mathfrak p}\in S_ p(L)$$, in terms of a $$p$$-adic logarithm. The proof is a consequence of a more general result depending on the theory of Coleman power series. As applications the author proves a weak analogue of Stickelberger’s theorem for real fields (he conjectures a stronger analogue) and provides, again for real $$L$$, a construction of $$\text{Gal}(L/\mathbb{Q})$$-submodules of finite index in the group of ideals of $$L$$ supported on $$S_ p(L)$$.

### MSC:

 11R20 Other abelian and metabelian extensions 11R18 Cyclotomic extensions 11R29 Class numbers, class groups, discriminants
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### References:

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