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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] It marked a breakthrough in the theory of transcendental numbers. Many longstanding open problems can be deduced as direct consequences.

Statement

If $G$ is a commutative algebraic group defined over an algebraic number field and $A$ is a Lie subgroup of $G$ with Lie algebra defined over the number field then $A$ does not contain any non-zero algebraic point of $G$ unless $A$ 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 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 for abelian varieties which Gerd Faltings 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.

Citations

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

References

Baker, Alan; Wüstholz, Gisbert (1993), "Logarithmic forms and group varieties", J. Reine Angew. Math., 1993 (442): 19–62, doi:10.1515/crll.1993.442.19, MR 1234835, S2CID 118335888
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.
Masser, David; Wüstholz, Gisbert (1993), "Isogeny estimates for abelian varieties and finiteness theorems", Annals of Mathematics, Second Series, 137 (3): 459–472, doi:10.2307/2946529, JSTOR 2946529, MR 1217345

[1] Wüstholz, Gisbert (1989). "Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen" [Algebraic points on analytic subgroups of algebraic groups]. Annals of Mathematics. Second Series (in German). 129 (3): 501–517. doi:10.2307/1971515. JSTOR 1971515. MR 0997311.

[2] Wüstholz, Gisbert (1989). "Multiplicity estimates on group varieties". Annals of Mathematics. Second Series. 129 (3): 471–500. doi:10.2307/1971514. JSTOR 1971514. MR 0997310.

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+{{Short description|Term used in transcendental number theory}}
-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.
+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.<ref>{{cite journal | last=Wüstholz | first=Gisbert | journal=[[Annals of Mathematics]] |series=Second Series | year=1989 | language=de | trans-title=Algebraic points on analytic subgroups of algebraic groups | title=Algebraische Punkte auf analytischen Untergruppen algebraischer Gruppen | volume=129 | number=3 | pages=501–517 | doi=10.2307/1971515 | jstor=1971515 | mr=997311}}</ref><ref>{{cite journal | last=Wüstholz | first=Gisbert | journal=[[Annals of Mathematics]] |series=Second Series | year=1989 | title=Multiplicity estimates on group varieties | volume=129 | number=3 | pages=471–500 | doi=10.2307/1971514 | jstor=1971514 | mr=997310}}</ref> It marked a breakthrough in the theory of transcendental numbers. Many longstanding open problems can be deduced as direct consequences.
-==History==
-The analytic subgroup theorem was proved in 2007 by [[Alan Baker (mathematician)|Alan Baker]] and [[Gisbert Wüstholz]].
 ==Statement==
-Let ''G'' be a [[Abelian group|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''.
+If <math>G</math> is a [[Abelian group|commutative]] [[algebraic group]] defined over an [[algebraic number field]] and <math>A</math> is a [[Lie subgroup]] of <math>G</math> with [[Lie algebra]] defined over the number field then <math>A</math> does not contain any non-zero algebraic point of <math>G</math> unless <math>A</math> contains a proper [[Algebraic group#Algebraic subgroup|algebraic subgroup]].
+One of the central new ingredients of the proof was the theory of multiplicity estimates of group varieties developed by [[David Masser]] 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]] for [[abelian variety|abelian varieties]] which [[Gerd Faltings]] 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 (mathematician)|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==
-*[[Abelian variety]]
 *[[Algebraic curve]]
+==Citations==
+{{reflist}}
 ==References==
+*{{Citation | last1=Baker | first1=Alan | last2=Wüstholz | first2=Gisbert | title=Logarithmic forms and group varieties | doi=10.1515/crll.1993.442.19 | mr=1234835 | year=1993 | journal=[[Crelle's Journal|J. Reine Angew. Math.]] | volume=1993 | issue=442 | pages=19–62| s2cid=118335888 }}
-* [[Alan Baker (mathematician)|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.
+* {{cite book|first1=Alan|last1=Baker|first2=Gisbert|last2=Wüstholz | title=Logarithmic Forms and Diophantine Geometry | series=New Mathematical Monographs | volume=9 | publisher=Cambridge University Press | year=2007 | isbn=978-0-521-88268-2 | mr=2382891 | location=Cambridge}}
+*{{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]] |series=Second Series | volume=137 | number=3 | pages=459–472| jstor=2946529 }}
-[[Category:Theorems in analytic number theory]]
+[[Category:Transcendental numbers]]