Editing Faltings's theorem

{{Short description|Curves of genus > 1 over the rationals have only finitely many rational points}}
{{Infobox mathematical statement
| name = Faltings's theorem
| image = Gerd Faltings MFO.jpg
| caption = Gerd Faltings
| field = [[Arithmetic geometry]]
| conjectured by = [[Louis Mordell]]
| conjecture date = 1922
| first proof by = [[Gerd Faltings]]
| first proof date = 1983
| open problem = 
| known cases =
| implied by =
| equivalent to = 
| generalizations = [[Bombieri–Lang conjecture]]<br/>[[Glossary of arithmetic and diophantine geometry#M|Mordell–Lang conjecture]]
| consequences = [[Siegel's theorem on integral points]]
}}

'''Faltings's theorem''' is a result in [[arithmetic geometry]], according to which a curve of [[Genus (mathematics)|genus]] greater than 1 over the field <math>\mathbb{Q}</math> of [[rational number]]s has only finitely many [[rational point]]s. This was conjectured in 1922 by [[Louis Mordell]],{{sfn|Mordell|1922}} and known as the '''Mordell conjecture''' until its 1983 proof by [[Gerd Faltings]].{{sfnm|1a1=Faltings|2a1=Faltings|1y=1983|2y=1984}} The conjecture was later generalized by replacing <math>\mathbb{Q}</math> by any [[number field]].

==Background==
Let <math>C</math> be a [[Singular point of a curve|non-singular]] algebraic curve of [[genus (mathematics)|genus]] <math>g</math> over <math>\mathbb{Q}</math>. Then the set of rational points on <math>C</math> may be determined as follows:
* When <math>g=0</math>, there are either no points or infinitely many. In such cases, <math>C</math> may be handled as a [[conic section]].
* When <math>g=1</math>, if there are any points, then <math>C</math> is an [[elliptic curve]] and its rational points form a [[finitely generated abelian group]]. (This is ''Mordell's Theorem'', later generalized to the [[Mordell–Weil theorem]].) Moreover, [[Mazur's torsion theorem]] restricts the structure of the torsion subgroup.
* When <math>g>1</math>, according to Faltings's theorem, <math>C</math> has only a finite number of rational points.

==Proofs==
[[Igor Shafarevich]] conjectured that there are only finitely many isomorphism classes of [[abelian variety|abelian varieties]] of fixed dimension and fixed [[Abelian variety#Polarisations|polarization]] degree over a fixed number field with [[good reduction]] outside a fixed finite set of [[place (mathematics)|place]]s.{{sfn|Shafarevich|1963}} [[Aleksei Parshin]] showed that Shafarevich's finiteness conjecture would imply the Mordell conjecture, using what is now called Parshin's trick.{{sfn|Parshin|1968}} 

[[Gerd Faltings]] proved Shafarevich's finiteness conjecture using a known reduction to a case of the [[Tate conjecture]], together with tools from [[algebraic geometry]], including the theory of [[Néron model]]s.{{sfn|Faltings|1983}} The main idea of Faltings's proof is the comparison of [[Height function#Faltings height|Faltings heights]] and [[Height function#Naive height|naive heights]] via [[Siegel modular variety|Siegel modular varieties]].{{efn|"Faltings relates the two notions of height by means of the Siegel moduli space.... It is the main idea of the proof." {{cite journal |last=Bloch |first=Spencer |s2cid=306251 |author-link=Spencer Bloch |page=44 |title=The Proof of the Mordell Conjecture |journal=The Mathematical Intelligencer |volume=6 |issue=2 |year=1984|doi=10.1007/BF03024155 }}}}

===Later proofs===
* [[Paul Vojta]] gave a proof based on [[Diophantine approximation]].{{sfn|Vojta|1991}} [[Enrico Bombieri]] found a more elementary variant of Vojta's proof.{{sfn|Bombieri|1990}}
*Brian Lawrence  and [[Akshay Venkatesh]] gave a proof based on [[p-adic Hodge theory|{{mvar|p}}-adic Hodge theory]], borrowing also some of the easier ingredients of Faltings's original proof.{{sfn|Lawrence|Venkatesh|2020}}

==Consequences==
Faltings's 1983 paper had as consequences a number of statements which had previously been conjectured:

* The ''Mordell conjecture'' that a curve of genus greater than 1 over a number field has only finitely many rational points;
* The ''Isogeny theorem'' that abelian varieties with isomorphic [[Tate module]]s (as <math>\mathbb{Q}_{\ell}</math>-modules with Galois action) are [[Isogeny|isogenous]].

A sample application of Faltings's theorem is to a weak form of [[Fermat's Last Theorem]]: for any fixed <math>n\ge 4</math> there are at most finitely many primitive integer solutions (pairwise [[coprime]] solutions) to <math>a^n+b^n=c^n</math>, since for such <math>n</math> the [[Fermat curve]] <math>x^n+y^n=1</math> has genus greater than 1.

==Generalizations==
Because of the [[Mordell–Weil theorem]], Faltings's theorem can be reformulated as a statement about the intersection of a curve <math>C</math> with a finitely generated subgroup <math>\Gamma</math> of an abelian variety <math>A</math>. Generalizing by replacing <math>A</math> by a [[semiabelian variety]], <math>C</math> by an arbitrary subvariety of <math>A</math>, and <math>\Gamma</math> by an arbitrary finite-rank subgroup of <math>A</math> leads to the [[Mordell–Lang conjecture]], which was proved in 1995 by [[Michael McQuillan (mathematician)|McQuillan]]{{sfn|McQuillan|1995}} following work of Laurent, [[Michel Raynaud|Raynaud]], Hindry, [[Paul Vojta|Vojta]], and [[Gerd Faltings|Faltings]].

Another higher-dimensional generalization of Faltings's theorem is the [[Bombieri–Lang conjecture]] that if <math>X</math> is a [[pseudo-canonical variety]] (i.e., a variety of general type) over a number field <math>k</math>, then <math>X(k)</math> is not [[Zariski topology|Zariski]] [[dense set|dense]] in <math>X</math>. Even more general conjectures have been put forth by [[Paul Vojta]].

The Mordell conjecture for function fields was proved by [[Yuri Ivanovich Manin]]{{sfn|Manin|1963}} and by [[Hans Grauert]].{{sfn|Grauert|1965}} In 1990, [[Robert F. Coleman]] found and fixed a gap in Manin's proof.{{sfn|Coleman|1990}}

==Notes==
{{notelist}}

== Citations ==
{{reflist}}

==References==
{{refbegin|2}}
*{{cite journal | last=Bombieri | first=Enrico | author-link1=Enrico Bombieri |year=1990|title=The Mordell conjecture revisited| journal=Ann. Scuola Norm. Sup. Pisa Cl. Sci.|volume=17| issue=4| pages=615–640 | mr=1093712 | url=http://www.numdam.org/item?id=ASNSP_1990_4_17_4_615_0 }}
*{{Cite journal | last1=Coleman | first1=Robert F. | author-link1=Robert F. Coleman | year=1990 | title=Manin's proof of the Mordell conjecture over function fields | url=https://www.e-periodica.ch/digbib/view?pid=ens-001:1990:36#560 | journal=L'Enseignement Mathématique |series=2e Série | issn=0013-8584 | volume=36 | issue=3 | pages=393–427 | mr=1096426 }}
*{{cite book|title=Arithmetic geometry. Papers from the conference held at the University of Connecticut, Storrs, Connecticut, July 30 – August 10, 1984 |editor1-last=Cornell | editor1-first=Gary | editor2-link=Joseph Hillel Silverman| editor2-last=Silverman | editor2-first=Joseph H. |year=1986 |publisher=Springer-Verlag |location= New York |isbn=0-387-96311-1 | doi=10.1007/978-1-4613-8655-1 | mr=861969}} → Contains an English translation of {{harvtxt|Faltings|1983}}
*{{cite journal |author-link=Gerd Faltings| last=Faltings |first=Gerd |year=1983 |title=Endlichkeitssätze für abelsche Varietäten über Zahlkörpern |journal=[[Inventiones Mathematicae]] |volume=73 |issue=3 |pages=349–366 |doi=10.1007/BF01388432 | bibcode=1983InMat..73..349F | mr=0718935 | trans-title=Finiteness theorems for abelian varieties over number fields | language=de }}
*{{cite journal |last=Faltings |first=Gerd |year=1984 |title=Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern |journal=[[Inventiones Mathematicae]] |volume=75 |issue=2 |pages=381 |doi=10.1007/BF01388572 | mr=0732554 | language=de |doi-access=free }}
*{{cite journal | last=Faltings | first=Gerd | year=1991 | title=Diophantine approximation on abelian varieties | journal=[[Ann. of Math.]] | volume=133 | issue=3 | pages=549–576 | doi=10.2307/2944319 | jstor=2944319 | mr=1109353 }}
*{{cite encyclopedia | last=Faltings | first=Gerd | year=1994 | chapter=The general case of S. Lang's conjecture | title=Barsotti Symposium in Algebraic Geometry. Papers from the symposium held in Abano Terme, June 24–27, 1991. | editor1-first=Valentino | editor1-last=Cristante | editor2-first=William | editor2-last=Messing | isbn=0-12-197270-4 | series=Perspectives in Mathematics | publisher=Academic Press, Inc. | location=San Diego, CA | mr=1307396 }}
*{{Cite journal | author-link1=Hans Grauert|last1=Grauert | first1=Hans | title=Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper | url=http://www.numdam.org/item?id=PMIHES_1965__25__131_0 | year=1965 | journal=[[Publications Mathématiques de l'IHÉS]] |volume=25 | issn=1618-1913 | issue=25 | pages=131–149 |doi=10.1007/BF02684399 | mr=0222087 }}
*{{cite book | title=Diophantine geometry | first1=Marc | last1=Hindry |last2=Silverman | first2=Joseph H. | series= [[Graduate Texts in Mathematics]] | volume=201 | publisher=Springer-Verlag | year=2000 | isbn=0-387-98981-1 | mr=1745599 | doi=10.1007/978-1-4612-1210-2 | location=New York}} → Gives Vojta's proof of Faltings's Theorem.
*{{cite book | first=Serge | last=Lang | author-link=Serge Lang | title=Survey of Diophantine geometry | url=https://archive.org/details/surveydiophantin00lang_347 | url-access=limited | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | pages=[https://archive.org/details/surveydiophantin00lang_347/page/n114 101]–122 }}
*{{cite journal |last1=Lawrence |first1=Brian |last2=Venkatesh |first2=Akshay |title=Diophantine problems and {{mvar|p}}-adic period mappings |journal=Invent. Math. |date=2020 |volume=221 |issue=3 |pages=893–999 |doi=10.1007/s00222-020-00966-7|arxiv=1807.02721 |bibcode=2020InMat.221..893L }}
*{{Cite journal | author-link1=Yuri Ivanovich Manin | last1=Manin | first1=Ju. I. | title=Rational points on algebraic curves over function fields | url=http://mi.mathnet.ru/eng/izv3174 | year=1963 | language=ru | journal=Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya | issn=0373-2436 | volume=27 | pages=1395–1440 | mr=0157971 }} (Translation: {{Cite journal | last1=Manin | first1=Yu. | title=Rational points on algebraic curves over function fields | journal=American Mathematical Society Translations |series=Series 2 | issn= 0065-9290 | volume=59 | year=1966 | pages=189–234 | doi=10.1090/trans2/050/11 | isbn=9780821817506 }} )
*{{cite journal |last1=McQuillan |first1=Michael |title=Division points on semi-abelian varieties |journal=Invent. Math. |date=1995 |volume=120 |issue=1 |pages=143–159 |doi=10.1007/BF01241125|bibcode=1995InMat.120..143M }}
*{{Cite journal | author-link1=Louis J. Mordell | last1=Mordell | first1=Louis J. | title=On the rational solutions of the indeterminate equation of the third and fourth degrees | year=1922 | journal=Proc. Cambridge Philos. Soc. | volume=21 | pages=179–192 | url=https://archive.org/stream/proceedingscambr21camb#page/n0/mode/2up }}
*{{Cite conference | author1-link=Aleksei Nikolaevich Parshin | last1=Paršin | first1=A. N. | book-title=Actes du Congrès International des Mathématiciens | location=Nice | year=1970 | volume=Tome 1 | publisher=Gauthier-Villars | publication-date=1971 | title=Quelques conjectures de finitude en géométrie diophantienne | pages=467–471 | url=http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0467.0472.ocr.pdf | mr=0427323 | access-date=2016-06-11 | archive-url=https://web.archive.org/web/20160924235505/http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0467.0472.ocr.pdf | archive-date=2016-09-24 | url-status=dead }}
*{{eom|id=M/m064910|first=A. N. |last=Parshin |title=Mordell conjecture|mode=cs1}}
*{{cite journal|title=Algebraic curves over function fields I|last=Parshin|first=A. N.|author-link=Aleksei Parshin|journal=[[Izvestiya: Mathematics|Izv. Akad. Nauk SSSR Ser. Mat.]]|volume=32|year=1968|issue=5|pages=1191–1219|doi=10.1070/IM1968v002n05ABEH000723|bibcode=1968IzMat...2.1145P}}
*{{cite journal|title=Algebraic number fields|last=Shafarevich|first=I. R.|author-link=Igor Shafarevich|journal=Proceedings of the International Congress of Mathematicians|year=1963|pages=163–176}}
*{{cite journal | last=Vojta | first=Paul | author-link=Paul Vojta | title=Siegel's theorem in the compact case | journal=[[Ann. of Math.]] | year=1991 | volume=133 | issue=3 | pages=509–548 | mr=1109352 | doi=10.2307/2944318 | jstor=2944318 }}
{{refend}}

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