Index formulas for ramified elliptic units. (English) Zbl 1045.11043

Oukhaba, Hassan

Compos. Math. 137, No. 1, 1-22 (2003).

Inspired by the work of W. Sinnott on cyclotomic units in abelian number fields, D. Kubert and S. Lang tried to develop an equivalent approach for elliptic units in their book [Modular units (1981; Zbl 0492.12002)], but their results were obtained only under some restrictive hypotheses. The aim of the present paper is to remove most of these restrictions.
There are two main results. In the first one there is a complicated formula for the unit index in the general case of a finite abelian extension \(F\) of the ground field \(K\) (imaginary quadratic). In the second theorem, \(F\) is a ray class field. Then, under a certain restriction, the index is equal to the class number of \(F\) times a factor depending only on the number of roots of unity in \(K\). The author uses ideas of L. Yin in the function field case [Compos. Math. 109, 49–66 (1997; Zbl 0902.11023) and J. Number Theory 63, 302–324 (1997; Zbl 0896.11023)].
Reviewer’s remark: If the normalized Klein form \(\varphi(t;\omega_1,\omega_2)\) is defined by means of the \(\sigma\)-function, then a factor \(2\pi\) is needed; see, e.g., the paper of R. Schertz [J. Théor. Nombres Bordx. 9, 383–394 (1997; Zbl 0902.11047)].

Reviewer: Veikko Ennola (Turku)

MSC:

11G16	Elliptic and modular units
11R27	Units and factorization
11R37	Class field theory

Keywords:

imaginary quadratic field; elliptic unit; class number; ray class field

Citations:

Zbl 0492.12002; Zbl 0902.11023; Zbl 0896.11023; Zbl 0902.11047