Analytical subgroup set

In mathematics, the analytical subgroup set is an important result of modern transcendence theory. It can be seen as a generalization of Baker 's theorem about linear forms in logarithms. Gisbert Wüstholz proved it in the 1980s. It marked a breakthrough in the theory of transcendent numbers . Many longstanding problems can be deduced as direct consequences.

statement

If is a commutative algebraic group that is defined over an algebraic number field and is a Lie subgroup of with Lie algebra defined over the number field , then does not contain a non-trivial algebraic point of unless contains a real algebraic subgroup. ${\displaystyle G}$ ${\displaystyle A}$ ${\displaystyle G}$ ${\displaystyle A}$ ${\displaystyle G}$ ${\displaystyle A}$

One of the central new components of the proof was the theory of multiplicity estimates on group varieties, developed by David Masser and Gisbert Wüstholz in special cases and founded by Wüstholz in the general case, which plays a central role in the proof of the analytical subgroup theorem.

The consequences

One of the spectacular consequences of the analytic subgroup theorem was the isogeny estimates proved by Masser and Wüstholz . A direct consequence is the Tate conjecture for Abelian varieties , which Gerd Faltings has proven with completely different methods and which has many applications in modern arithmetic geometry.

With the help of multiplicity estimates on group varieties, Wüstholz succeeded in obtaining the final form of Baker's theorem on linear forms in logarithms. This was done effectively in a joint work with Alan Baker . It reflects the current state of art. In addition to the multiplicity estimates, another new component was a very sophisticated use of the geometry numbers to get a very sharp lower bound.

Individual evidence

1. ↑ Gisbert Wüstholz: Algebraic points on analytic subgroups of algebraic groups [Algebraic points on analytic subgroups of algebraic groups] , Annals of Mathematics , Series 2, Volume 129, No. 3, 1989, pp. 501-517. doi: 10.2307 / 1971515
2. ↑ Gisbert Wüstholz: Multiplicity estimates on group varieties , Annals of Mathematics , Series 2, Volume 129, No. 3, 1989, pp. 471-500. doi: 10.2307 / 1971514

literature

• Alan Baker, Gisbert Wüstholz: Logarithmic forms and group varieties , J. Reine. Angew. Math. , Vol. 442, 1993, pp. 19-62. doi: 10.1515 / crll.1993.442.19 .
• Alan Baker, Gisbert Wüstholz: Logarithmic Forms and Diophantine Geometry , New Mathematical Monographs, Volume 9, Cambridge University Press, Cambridge, 2007, ISBN 978-0-521-88268-2 .
• David Masser, Gisbert Wüstholz: Isogeny estimates for abelian varieties and finiteness theorems , Annals of Mathematics , Series 2, Volume 137, No. 3, 1993, pp. 459-472. doi: 10.2307 / 2946529 .