Abstract
Given a circle diffeomorphism f, we can construct a map taking each real number a to the rotation number of the diffeomorphism f +a. In 1978, V. I. Arnold suggested a complex analog To this map. Given a complex number z with Im z > 0, Arnold used the map f + z to construct an elliptic curve. The moduli map takes every number z to the modulus µ(z) of this elliptic curve.
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