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.[1][2]
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.
See also
Citations
- ^ Wüstholz, Gisbert (1989). "Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen" [Algebraic points on analytic subgroups of algebraic groups]. Annals of Mathematics (in German). 129 (3): 501–517. doi:10.2307/1971515. MR 0997311.
- ^ Wüstholz, Gisbert (1989). "Multiplicity estimates on group varieties". Annals of Mathematics. 129 (3): 471–500. doi:10.2307/1971514. MR 0997310.
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.