Axiomatizing exponentiation

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.