Analytic subgroup theorem

The analytic subgroup theorem is a significant result in modern transcendence theory. It may be seen as a generalisation of Baker's theorem on linear forms in logarithms.

Statement

Let G be a commutative algebraic group defined over a number field K and let B be a subgroup of the complex points G(C) defined over K. There are points of B defined over the field of algebraic numbers if and only if there is a non-trivial analytic subgroup H of G defined over a number field such that H(C) is contained in B.

References

• Alan Baker and Gisbert Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs 9, Cambridge University Press, 2007, ISBN 978-0-521-88268-2. Chapter 6, pp.109-146.