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Covers of the multiplicative group of an algebraically closed field of characteristic zero

B. Zilber
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Abstract

We study the universal cover of the complex one-dimensional torus as a model-theoretic structure in a natural language. We consider also abstract covers of one-dimensional tori over algebraically closed fields of characteristic zero. The main result states that the structures can be described uniquely up to isomorphism by a simple (non-first order) sentence, given a fixed uncountable cardinality of the underlying field. The proof reduces to the Kummer theory of cyclic Galois extensions over some, not necessarily finitely generated fields.

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B. Zilber. 2004-01-22. Covers of the multiplicative group of an algebraically closed field of characteristic zero. https://arxiv.org/abs/math/0401301

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