Elliptic Curves and Dedekind Domains

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. Let S be a set of K rational points on E and Rs(E) the ring of K rational functions on E having all their poles in S. Then Rs(E) is a Dedekind domain. We call such a ring an elliptic Dedekind domain. What abelian groups arise as class groups of such rings? In [6] it is shown that every finitely generated abelian group arises in this way. In this paper we extend this result as follows. THEOREMA.Every countable abelian group can be realized as the class group of an elliptic Dedekind domain. The proof of Theorem A is dependent on the following theorem which is of independent interest. THEOREMB. Let E be an elliptic curve defined over an algebraic number field k . Let K/k be a solvable algebraic extension, possibly infinite. Assume E has no complex multiplications. Then the torsion subgroup of E(K) is finite and modulo torsion E(K) is a free abelian group. The proof of Theorem B is dependent on deep results of J.-P. Serre on division points of elliptic curves (see [7] and [8]). It is related to, and in fact inspired by, results of B. Mazur [4]. L. Claborn has shown [ l ] that any abelian group can be realized as the class group of a Dedekind domain. The same result may be true for elliptic Dedekind domains. The main stumbling block for our methods is that Proposition 1 of this paper is not true if the hypothesis of countability is removed. Hopefully an adequate substitute can be found. 1 . A result on abelian groups. We need a proposition on abelian groups which is essentially due to L. Pontryagin [5]. By the rank of an abelian group we mean the maximal number of linearly Received by the editors September 29, 1975. AMS (MOS) subject classifcations (1970). Primary 14K15, 13F05.