# Faltings' product theorem

In arithmetic geometry, **Faltings' product theorem** gives sufficient conditions for a subvariety of a product of projective spaces to be a product of varieties in the projective spaces. It was introduced by Faltings (1991) in his proof of Lang's conjecture that subvarieties of an abelian variety containing no translates of non-trivial abelian subvarieties have only finitely many rational points.

(Evertse 1995) and (Ferretti 1996) gave explicit versions of Faltings' product theorem.

## References

- Evertse, Jan-Hendrik (1995), "An explicit version of Faltings' product theorem and an improvement of Roth's lemma",
*Acta Arithmetica***73**(3): 215–248, doi:10.4064/aa-73-3-215-248, ISSN 0065-1036, http://matwbn.icm.edu.pl/ksiazki/aa/aa73/aa7332.pdf - Faltings, Gerd (1991), "Diophantine approximation on abelian varieties",
*Annals of Mathematics*, Second Series**133**(3): 549–576, doi:10.2307/2944319, ISSN 0003-486X - Ferretti, Roberto (1996), "An effective version of Faltings' product theorem",
*Forum Mathematicum***8**(4): 401–427, doi:10.1515/form.1996.8.401, ISSN 0933-7741

Original source: https://en.wikipedia.org/wiki/ Faltings' product theorem.
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