𝔖 Bobbio Scriptorium
✦   LIBER   ✦

On a Quotient of the Unramified Iwasawa Module over an Abelian Number Field

✍ Scribed by Humio Ichimura

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
154 KB
Volume
88
Category
Article
ISSN
0022-314X

No coin nor oath required. For personal study only.

✦ Synopsis

Let p be a fixed odd prime number and k an imaginary abelian field containing a primitive p th root `p of unity. Let k Γ‚k be the cyclotomic Z p -extension and LΓ‚k the maximal unramified pro-p abelian extension. We put

where E is the group of units of k . Let X=Gal(LΓ‚k ) and Y=Gal(L & NΓ‚k ), and let X & , Y & be the odd parts of the respective Galois groups. It is well known that X & is (finitely generated and) torsion free over Z p (cf. Washington [19, Corollary 13.29]). In the previous paper [6] on a power integral basis problem over cyclotomic Z p -extensions, one of the crucial points was the question: Is the quotient Y & of X & also torsion free over Z p ? In this paper, we continue the study on this quotient.

Let A be the ideal class group of k , and A + its even part. It is conjectured by Greenberg [4]

and hence Y & is also torsion free over Z p (see, for example, the formula (18) in Section 6). However, at present, the conjecture is far from being settled in general, while no counterexample is known.

Assume, for simplicity, that the exponent of 2=Gal(kΓ‚Q) equals p&1. Let be a nontrivial even Q p -valued character of 2 (of degree 1) and *

πŸ“œ SIMILAR VOLUMES

On Number Fields with an Unramified Abel
On Number Fields with an Unramified Abelian Extension of Degree 2n+2
✍ Y.Z. Lan πŸ“‚ Article πŸ“… 1993 πŸ› Elsevier Science 🌐 English βš– 315 KB

Assume that \(K\) is either a totally real or a totally imaginary number field. Let \(F\) be the maximal unramified elementary abelian 2-extension of \(K\) and \([F: K]=2^{n}\). The purpose of this paper is to describe a family of cubic cyclic extension of \(K\). We have constructed an unramified ab

On the Number of Solutions of Diagonal E
On the Number of Solutions of Diagonal Equations over a Finite Field
✍ Qi Sun; Ping-Zhi Yuan πŸ“‚ Article πŸ“… 1996 πŸ› Elsevier Science 🌐 English βš– 181 KB

In this paper, we give a reduction theorem for the number of solutions of any diagonal equation over a finite field. Using this reduction theorem and the theory of quadratic equations over a finite field, we also get an explicit formula for the number of solutions of a diagonal equation over a finit

CORRIGENDUM: Volume 73, Number 2 (1998),
CORRIGENDUM: Volume 73, Number 2 (1998), in the article β€œReal Polynomials with All Roots on the Unit Circle and Abelian Varieties over Finite Fields,” by Stephen A. DiPippo and Everett W. Howe, pages 426–450
πŸ“‚ Article πŸ“… 2000 πŸ› Elsevier Science 🌐 English βš– 27 KB

The expression q n(n&1)Γ‚4 should be replaced with the expression q (n+2)(n&1)Γ‚4 in the first displayed equation in the statement of Theorem 1.2 (page 427), as well as in the first displayed equation in the statement of Proposition 3.2.1 (page 445) and in the displayed equation at the bottom of page

Number of Points on the Projective Curve
Number of Points on the Projective Curves aYl=bXl+cZl and aY2l=bX2l+cZ2l Defined over Finite Fields, l an Odd Prime
✍ N. Anuradha; S.A. Katre πŸ“‚ Article πŸ“… 1999 πŸ› Elsevier Science 🌐 English βš– 201 KB

The number of points on the curve aY e =bX e +c (abc{0) defined over a finite field F q , q#1 (mod e), is known to be obtainable in terms of Jacobi sums and cyclotomic numbers of order e with respect to this field. In this paper, we obtain explicitly the Jacobi sums and cyclotomic numbers of order e