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.
It marked a breakthrough in the theory of transcendental numbers. Many longstanding open problems can be deduced as direct consequences.
Statement
If
is a
commutative algebraic group defined over an
algebraic number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
and
is a
Lie subgroup
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the add ...
of
with
Lie algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
defined over the number field then
does not contain any non-zero algebraic point of
unless
contains a proper
algebraic subgroup.
One of the central new ingredients of the proof was the theory of multiplicity estimates of group varieties developed by
David Masser
David William Masser (born 8 November 1948) is Professor Emeritus in the Department of Mathematics and Computer Science at the University of Basel. He is known for his work in transcendental number theory, Diophantine approximation, and Diophanti ...
and
Gisbert Wüstholz in special cases and established by Wüstholz in the general case which was necessary for the proof of the analytic subgroup theorem.
Consequences
One of the spectacular consequences of the analytic subgroup theorem was the Isogeny Theorem published by Masser and Wüstholz. A direct consequence is the
Tate conjecture
In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The conje ...
for
abelian varieties which
Gerd Faltings
Gerd Faltings (; born 28 July 1954) is a German mathematician known for his work in arithmetic geometry.
Education
From 1972 to 1978, Faltings studied mathematics and physics at the University of Münster. In 1978 he received his PhD in mathema ...
had proved with totally different methods which has many applications in modern arithmetic geometry.
Using the multiplicity estimates for group varieties Wüstholz succeeded to get the final expected form for lower bound for linear forms in logarithms. This was put into an effective form in a joint work of him with
Alan Baker which marks the current state of art. Besides the multiplicity estimates a further new ingredient was a very sophisticated use of geometry of numbers to obtain very sharp lower bounds.
See also
*
Algebraic curve
Citations
References
*
*
*{{Citation , last1=Masser , first1=David , last2=Wüstholz , first2=Gisbert , title=Isogeny estimates for abelian varieties and finiteness theorems , doi=10.2307/2946529 , mr=1217345 , year=1993 , journal=
Annals of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the ...
, series=Second Series , volume=137 , number=3 , pages=459–472, jstor=2946529
Transcendental numbers