Six Exponentials Theorem - Generalization To Commutative Group Varieties

Generalization To Commutative Group Varieties

The exponential function uniformizes the exponential map of the multiplicative group . The six exponential theorem can be thus reformulated in a more abstract manner as follows:

Let be the field of complex numbers and let . Let → be a non-zero complex-analytic group homomorphism. If the group of numbers in, such that is an algebraic point of, can be only generated by more than two elements over the field of rational numbers, then the image is an algebraic subgroup of .

In this way the statement of the six exponentials theorem can be generalized to an arbitrary commutative group variety over the field of algebraic numbers. (Alternatively, one can replace by and "more than two elements" by "more than one element" to obtain another variant of the generalization.) This generalized six exponential conjecture, however, seems out of scope at the current state of transcendental number theory.

For the special, but interesting cases × and ×, with elliptic curves over the field of algebraic numbers, results towards the generalized six exponential conjecture could be established by Aleksander Momot. These results involve one exponential function and one Weierstrass function resp. two Weierstrass functions with algebraic invariants, instead of two exponential functions as in the classical statement (in the classical statement the numbers play the role of a generating set of, such that ). For an algebraic group × it is proved in, among others, that if is not isogenous to a curve over a real field and if is not an algebraic subgroup of, then can be either generated by two elements over, or a minimal generating set of over consists of three elements which are not all contained in a real line ( a non-zero complex number). A similar result is shown for × .