Abstract

We survey the history of, and recent developments on, two major conjectures originating in Zilber’s model-theoretic work on complex exponentiation: existential closedness and Zilber–Pink. The main focus is on the modular versions of these conjectures and specifically on novel variants incorporating the derivatives of modular functions. The functional analogues of all the conjectures are already theorems, which we also present. The paper also contains some new results and conjectures.

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Mathematical Subject Classification

Milestones

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