Elliptic Dedekind Domains Revisited

We give an affirmative answer to a 1976 question of M. Rosen: every abelian group is isomorphic to the class group of an elliptic Dedekind domain R. We can choose R to be the integral closure of a PID in a separable quadratic field extension. In particular, this yields new and – we feel – simpler proofs of theorems of L. Claborn and C.R. Leedham-Green. Luther Claborn received his PhD from U. Michigan in 1963 and died in a car accident in August of 1967. In between he wrote 11 papers, including [3], which shows that every abelian group is the class group of a Dedekind domain. Evidently this result remains of interest to this day, more than 40 years after his untimely death. This paper is dedicated to him. Terminology: For a scheme X , PicX denotes the Picard group H1(X,O X), whose elements correspond to isomorphism classes of line bundles on X . When X = SpecR is affine, we abbreviate Pic(SpecR) to Pic(R). When R is a Dedekind domain, Pic(R) is the ideal class group of R. An overring of an integral domain R is a ring intermediate between R and its fraction field. A Dedekind domain R will be called affine if it is the coordinate ring of a nonsingular, geometrically integral affine curve C defined over some field k. A geometric Dedekind domain is an overring of an affine domain. An elliptic curve over a field k means, as usual, a complete, nonsingular, geometrically integral genus one curve E/k with a distinguished k-rational point O. To an elliptic curve we associate its function field k(E) and its standard affine ring k[E], the ring of all functions on E which are regular away from O. An elliptic domain is an overring of the standard affine ring of some elliptic curve. A quadratic domain is a Dedekind domain which can be obtained by taking the integral closure of a PID in a quadratic field extension. For a cardinal κ, FA(κ) := ⊕ x∈κ Z denotes the free abelian group of rank κ.