Covers of the Multiplicative Group of an Algebraically Closed Field of Characteristic Zero

By B. Zilber. 2004

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We consider the classical universal covering exp: C → C* of the complex torus as an algebraic structure. The exponentiation is seen here as an abstract homomorphism from a divisible torsion‐free group onto the multiplicative group of an algebraically closed field of characteristic zero, with cyclic kernel. We prove that any structure satisfying this description is isomorphic to the classical one provided that the cardinality of the underlying field is equal to that of C. This can also be seen as a model‐theoretic statement on the categoricity of a corresponding Lω1,ω‐sentence. The proof is a combination of arithmetic and model‐theoretic methods.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/389f5ccb72ad060e3a140428d5e0205111dbee2d