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SOLUTION TO THE INVERSE MORDELL-WEIL PROBLEM FOR ELLIPTIC CURVES

In [Ro76], M. Rosen showed that for any countable commutative group G, there is a field K, an elliptic curve E/K and a surjective group homomorphism E(K) → G. From this he deduced that any countable commutative group whatsoever is the ideal class group of an elliptic Dedekind domain – the ring of all functions on an elliptic curve which are regular away from some (fixed, possibly infinite) set S of closed points of E. Rosen left open the question of whether every uncountable commutative group can be achieved as the class group of an elliptic Dedekind domain. This was answered in the affirmative in [Cl09], by showing that for any free commutative group G, there is a field K and an elliptic curve E/K such that the Mordell-Weil group E(K) is isomorphic to G.