Zilber gave non-first order axioms for an exponential field B with good model-theoretic properties which is conjecturally isomorphic to complex exponentiation. I will explain how the diophantine conjecture CIT allows us to axiomatize the complete first-order theory Th(B), and how the existence of an axiomatization of Th(B) along similar lines implies CIT. This is joint work with Boris Zilber.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/4066