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

Solomon, David

Invent. Math. 109, No. 2, 329-350 (1992).

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 \(S_ p(L)\).

Reviewer: Tauno Metsänkylä (Turku)

MSC:

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

Keywords:

group of \(p\)-units; Gauss sum; cyclotomic \(p\)-units; cyclotomic fields; class groups; Stickelberger’s theorem

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