Diophantine definability over non-finitely generated non-degenerate modules of algebraic extensions of ℚ

Abstract. We investigate the issues of Diophantine definability over the non-finitely generated version of non-degenerate modules contained in the infinite algebraic extensions of the rational numbers. In particular, we show the following. Let k be a number field and let Kinf be a normal algebraic, possibly infinite, extension of k such that k has a normal extension L linearly disjoint from Kinf over k. Assume L is totally real and Kinf is totally complex. Let Minf be a non-degenerate Ok-module, possibly non-finitely generated and contained in OKinf. Then Minf contains a submodule M¯inf such that Minf/M¯inf is torsion and Ok has a Diophantine definition over M¯inf.

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