##
**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 \(S_ p(L)\).

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### References:

[1] | [B] Brumer, A.: On the Units of Algebraic Number Fields. Mathematika14 121-124, (1967) · Zbl 0171.01105 |

[2] | [C] Coleman, R.: Division Values in Local Fields. Invent. Math.53, 91-116 (1979) · Zbl 0429.12010 |

[3] | [Gi] Gillard, R.: Remarques sur les Unit?s Cyclotomiques et les Unit?s Elliptiques. J. Number Theory11, 21-48 (1979) · Zbl 0405.12008 |

[4] | [Gra] Gras, G.: Annulation du Groupe desl-classes G?n?ralis?es d’une Extension Ab?lienne R?elle de Degr? Premier ?l’. Ann. Inst. Fourier, Grenoble29(1), 15-32 (1979) |

[5] | [Gre] Greenberg, R.: Onp-AdicL-functions and Cyclotomic Fields II. Nagoya Math. J.67, 139-158 (1977) |

[6] | [M-Wi] Mazur, B., Wiles, A.: Class Fields of Abelian Extensions of ?. Invent. Math.76, 179-330 (1984) · Zbl 0545.12005 |

[7] | [O] Oriat, B.: Annulation de Groupes de Classes R?elles. Nagoya Math. J.81, 45-56 (1981) · Zbl 0495.12002 |

[8] | [Si] Sinnott, W.: On the Stickelberger Ideal and the Circular Units of an Abelian Field. Invent. Math.62 (no. 2), 181-234 (1980/81) · Zbl 0465.12001 |

[9] | [So] Solomon, D.: Galois Relations for Cyclotomic Numbers andp-Units. (to appear) |

[10] | [T] Thaine, F.: On the Ideal Class Groups of Real Abelian Number Fields. Ann. Math.128, p. 1-18 (1988) · Zbl 0665.12003 |

[11] | [Wa] Washington, L.: Introduction to Cyclotomic Fields Berlin Heidelberg New York: Springer 1982 · Zbl 0484.12001 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.