Faltings's theorem

Let $C$ be a non-singular algebraic curve of genus $g$ over $\backslash mathbb\{Q\}$. Then the set of rational points on $C$ may be determined as follows:

• When $g=0$, there are either no points or infinitely many. In such cases, $C$ may be handled as a conic section.
• When $g=1$, if there are any points, then $C$ 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 $g>1$, according to Faltings's theorem, $C$ has only a finite number of rational points.

Igor Shafarevich conjectured that there are only finitely many isomorphism classes of abelian varieties of fixed dimension and fixed polarization degree over a fixed number field with good reduction outside a fixed finite set of places.{{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 models.{{sfn|Faltings|1983}} The main idea of Faltings's proof is the comparison of Faltings heights and naive heights via 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 }}}}

• 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, borrowing also some of the easier ingredients of Faltings's original proof.{{sfn|Lawrence|Venkatesh|2020}}

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 modules (as $\backslash mathbb\{Q\}\_\{\backslash ell\}$-modules with Galois action) are isogenous.

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

Because of the Mordell–Weil theorem, Faltings's theorem can be reformulated as a statement about the intersection of a curve $C$ with a finitely generated subgroup $\backslash Gamma$ of an abelian variety $A$. Generalizing by replacing $A$ by a semiabelian variety, $C$ by an arbitrary subvariety of $A$, and $\backslash Gamma$ by an arbitrary finite-rank subgroup of $A$ leads to the Mordell–Lang conjecture, which was proved in 1995 by McQuillan{{sfn|McQuillan|1995}} following work of Laurent, Raynaud, Hindry, Vojta, and Faltings.

Another higher-dimensional generalization of Faltings's theorem is the Bombieri–Lang conjecture that if $X$ is a pseudo-canonical variety (i.e., a variety of general type) over a number field $k$, then $X(k)$ is not Zariski dense in $X$. 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}}

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• {{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=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 }}

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