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Faltings's theorem

Faltings's theorem

In number theory, the Mordell conjecture is the conjecture made by Mordell (1922) that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by Gerd Faltings (1983, 1984), and is now known as Faltings's theorem. The conjecture was later generalized by replacing Q by any number field.

Faltings's theorem
FieldAlgebraic geometry and number theory
Conjectured byLouis Mordell
Conjectured in1922
First proof byGerd Faltings
First proof in1983
GeneralizationsBombieri–Lang conjecture
Mordell–Lang conjecture


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

  • Case g = 0: no points or infinitely many; C is handled as a conic section.

  • Case g = 1: no points, or C is an elliptic curve and its rational points form a finitely generated abelian group (Mordell's Theorem, later generalized to the Mordell–Weil theorem). Moreover, Mazur's torsion theorem restricts the structure of the torsion subgroup.

  • Case g > 1: according to the Mordell conjecture, now Faltings's theorem, C has only a finite number of rational points.


Shafarevich (1963) posed a finiteness conjecture that asserted 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 given finite set of places. Parshin (1968) showed that the Mordell conjecture would hold if the Shafarevich finiteness conjecture was true using Parshin's trick, which gives an embedding of a curve into the Siegel modular variety.

Faltings (1983) proved the Shafarevich finiteness conjecture using a known reduction to a case of the Tate conjecture, and a number of tools from algebraic geometry, including the theory of Néron models. The main idea of Faltings' proof is the comparison of Faltings heights and naive heights via Siegel modular varieties.[1]

Later proofs

A proof based on diophantine approximation was given by Vojta (1991). A more elementary variant of Vojta's proof was given by Bombieri (1990).


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 Q-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 > 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 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 Γ of an abelian variety A. Generalizing by replacing C by an arbitrary subvariety of A and Γ by an arbitrary finite-rank subgroup of A leads to the Mordell–Lang conjecture, which was proved by Faltings (1991, 1994).

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 Manin (1963) and by Grauert (1965). In 1990, Coleman (1990) found and fixed a gap in Manin's proof.


Citation Linkpdfs.semanticscholar.org"Faltings relates the two notions of height by means of the Siegel moduli space.... It is the main idea of the proof." Bloch, Spencer (1984). "The Proof of the Mordell Conjecture" (PDF). The Mathematical Intelligencer. 6 (2): 44.
Sep 24, 2019, 5:54 AM
Citation Linkwww.numdam.org"The Mordell conjecture revisited"
Sep 24, 2019, 5:54 AM
Citation Link//www.ams.org/mathscinet-getitem?mr=10937121093712
Sep 24, 2019, 5:54 AM
Citation Linkweb.archive.org"Manin's proof of the Mordell conjecture over function fields"
Sep 24, 2019, 5:54 AM
Citation Link//www.worldcat.org/issn/0013-85840013-8584
Sep 24, 2019, 5:54 AM
Citation Link//www.ams.org/mathscinet-getitem?mr=10964261096426
Sep 24, 2019, 5:54 AM
Citation Linkretro.seals.chthe original
Sep 24, 2019, 5:54 AM
Citation Link//doi.org/10.1007%2F978-1-4613-8655-110.1007/978-1-4613-8655-1
Sep 24, 2019, 5:54 AM
Citation Link//www.ams.org/mathscinet-getitem?mr=08619690861969
Sep 24, 2019, 5:54 AM
Citation Link//doi.org/10.1007%2FBF0138843210.1007/BF01388432
Sep 24, 2019, 5:54 AM
Citation Link//www.ams.org/mathscinet-getitem?mr=07189350718935
Sep 24, 2019, 5:54 AM
Citation Link//doi.org/10.1007%2FBF0138857210.1007/BF01388572
Sep 24, 2019, 5:54 AM
Citation Link//www.ams.org/mathscinet-getitem?mr=07325540732554
Sep 24, 2019, 5:54 AM
Citation Link//doi.org/10.2307%2F294431910.2307/2944319
Sep 24, 2019, 5:54 AM
Citation Link//www.ams.org/mathscinet-getitem?mr=11093531109353
Sep 24, 2019, 5:54 AM
Citation Link//www.ams.org/mathscinet-getitem?mr=13073961307396
Sep 24, 2019, 5:54 AM
Citation Linkwww.numdam.org"Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper"
Sep 24, 2019, 5:54 AM
Citation Link//www.worldcat.org/issn/1618-19131618-1913
Sep 24, 2019, 5:54 AM
Citation Link//www.ams.org/mathscinet-getitem?mr=02220870222087
Sep 24, 2019, 5:54 AM
Citation Link//doi.org/10.1007%2F978-1-4612-1210-210.1007/978-1-4612-1210-2
Sep 24, 2019, 5:54 AM