Schanuel type conjectures and disjointness

Abstract

AbstractGiven a subfield F of $${\mathbb {C}}$$ C , we study the linear disjointess of the field E generated by iterated exponentials of elements of $${\overline{F}}$$ F ¯ , and the field L generated by iterated logarithms, in the presence of Schanuel’s conjecture. We also obtain similar results replacing $$\exp $$ 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 $$\textrm{GL}_2$$ GL 2 on $${\mathbb {C}}$$ C . We also show that for certain choices of F we obtain unconditional versions of these statements.