Schanuel type conjectures and disjointness

Abstract

Given a subfield F of C, we study the linear disjointess of the field E generated by iterated exponentials of elements of F¯¯¯¯, and the field L generated by iterated logarithms, in the presence of Schanuel’s conjecture. We also obtain similar results replacing exp by the modular j-function, under an appropriate version of Schanuel’s conjecture, where linear disjointness is replaced by a notion coming from the action of GL2 on C . We also show that for certain choices of F we obtain unconditional versions of these statements.