Involutions on Zilber fields

Abstract

In this paper, we briefly outline the definition of Zilber field, which is a structure analogue to the complex field with the exponential function. An open conjecture, including Schanuel’s Conjecture, is whether the complex field is itself one of these structure.

In view of this conjecture, a natural question raised by Zilber, Kirby, Macintyre and others is whether they have an automorphism of order two akin to complex conjugation.

We announce, without proof, the positive answer: for cardinality up to the continuum there exists an involution of the field commuting with the exponential function. Moreover, in the case of cardinality of the continuum, the automorphism can be taken such that its fixed field is exactly $R$, and the kernel of the exponential function is $2\pi i\mathbb{Z}$.