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Mordell conjecture

Idea

The Mordell conjecture or Falting’s theorem is a statement about the finiteness of rational points? on an algebraic curve over a number field of genus $g>1$.

Its statement motivated the introduction of anabelian geometry (Grothendieck).

The Mordell conjecture completed a classification of the behavior of rational points on curves over $\mathbb{Q}$. For genus 0 one has no points, or something isomorphic to ${\mathbb{P}}^1$ and hence infinitely many. For genus 1, the Mordell-Weil theorem? tells us it is either empty or a finitely generated abelian group (sometimes finite, sometimes infinite). To finish off this type of classification the Mordell conjecture shows that only the finite case can occur for higher genus.

This result also implies many non-trivial results. For example, fix a finite set of primes $S$, a dimension $n$, and a polarization degree? $d$. There are only finitely many abelian varieties of dimension $n$ and polarization degree $d$ with bad reduction? inside $S$. This result is so interesting that a whole industry popped up asking what other varieties this type of behavior occurs for. It seems to work for K3s.

Relation to other statements

The Mordell conjecture is implied by the abc conjecture. (See there.)

The Mordell conjecture implies Tate's isogeny theorem?.

See also Vojta's conjecture.

References

The Mordell comnjecture originates in

• Louis Mordell, On the rational solutions of the indeterminate equation of the third and fourth degrees, Proc. Cambridge Philos. Soc. 21: 179–192 (1922)

It was proven in

• Gerd Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Inventiones Mathematicae 73 (3): 349–366 (1983) doi:10.1007/BF01388432.

Reviews include

• Barry Mazur, Abelian varieties and the Mordell-Lang conjecture (pdf)

• Enrico Bombieri, The Mordell conjecture revisited, Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Sér. 4, 17 no. 4 (1990), p. 615-640 (Numdam)

Encyclopedia entries are in

• Encyclopedia of Mathematics, Mordell conjecture

• Wikipedia, Faltings’ theorem

The relation to anabelian geometry originates in

• Alexandre Grothendieck, letter to Faltings (1983) (pdf)