Title: Quasiminimality of a correspondence between two elliptic curves
Abstract: One of the well-known accomplishments of model theory is the study of the field of complex numbers. Its complete theory possesses numerous nice properties, including quantifier elimination. Moreover, using quantifier elimination, one can see that any definable subset is finite or cofinite, i.e. the theory is strongly minimal. However, adding the exponential map to the structure makes it possible to define the ring of integers, preventing minimality and many other properties. Nevertheless, there is still hope that the theory is somewhat well-behaved. For instance, Zilber’s quasiminimality conjecture states that the complex exponential field is quasiminimal, i.e. every definable subset is countable or co-countable. Analogous conjectures were made, replacing the exponential map with other interesting analytical functions. In this talk we will focus on a reduct of one of these conjectures, which involves a correspondence between two elliptic curves, and discuss the quasiminimality of the structure in question.