Some results are obtained on the group of rational points on elliptic curves over infinite algebraic number fields. A certain naturally defined class of Dedekind domains, elliptic Dedekind domains, are described and it is shown that every countable abelian group can be realized as the class group of an elliptic Dedekind domain. Introduction. Let E be an elliptic curve defined over a field K. Le...

For a given family (Gi)i∈N of finitely generated abelian groups, we construct Dedekind domain D having the following properties. Pic(D)≅⨁i∈NGi. each i∈N, there exists submonoid Si⊆D• with Pic(DSi)≅Gi. Each class Pic(D) and all Pic(DSi) contains infinitely many prime ideals.

We study the relation of the ideal class group of a Dedekind domain A to that of As, where S is a multiplicatively closed subset of A. We construct examples of (a) a Dedekind domain with no principal prime ideal and (b) a Dedekind domain which is not the integral closure of a principal ideal domain. We also obtain some qualitative information on the number of non-principal prime ideals in an ar...

In this article we study the problem of finding such finite groups that the modular forms associated with all elements of these groups by means of a certain faithful representation belong to a special class of modular forms (so-called multiplicative η−products). This problem is open. We find metacyclic groups with such property and describe the Sylow p-subgroups, p 6= 2, for such groups. We als...