Analytic subgroup theorem

In mathematics, the analytic subgroup theorem is a significant result in modern transcendental number theory. It may be seen as a generalisation of Baker's theorem on linear forms in logarithms. Gisbert Wüstholz proved it in the 1980s.

Statement

Let $G$ be a commutative algebraic group defined over ${\overline {\mathbb {Q} }}$ and let $B$ be an analytic subgroup of the complex points $G(\mathbb {C} )$ such that $Lie(B)$ is defined over ${\overline {\mathbb {Q} }}$ and positive-dimensional. Then, $B\cap G({\overline {\mathbb {Q} }})\neq 0$ if and only if there is a non-trivial algebraic subgroup $H$ of $G$ defined over ${\overline {\mathbb {Q} }}$ such that $H(\mathbb {C} )$ is contained in $B$ .

See also

• Abelian variety
• Algebraic curve

References

• Baker, Alan; Wüstholz, Gisbert (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs. Vol. 9. Cambridge: Cambridge University Press. ISBN 978-0-521-88268-2. MR 2382891.