nLab Mordell conjecture

Idea

Il s’agit là d’une question qu’un arithméticien ne peut guère manquer de se poser; on n’aperçoit d’ailleurs aucun motif sérieux de parier pour ou contre. Peut-être dira-t-on que l’existence d’une infinité de solutions rationnelles pour une équation $f(x,y)=0$, en l’absence d’une raison algébrique qui la justifie, est infiniment peu probable. Mais ce n’est pas un argument…. (André Weil, Commentaire 1979)

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 $g \gt 1$.

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.

Sketch of Faltings’ proof

We follow the seminar notes BhattSnowden.

The proof proceeds along the following steps:

1. Using Parshin’s trick and passing to the Jacobian variety to reduce the Mordell conjecture to the Shafarevich conjecture that there are only finitely many abelian varieties of dimension $g$ and polarization degree $d$ with good reduction outside a fixed set of places $S$.

2. Using the semisimplicity of the Tate module and the isogeny theorem

   $$Hom_{K}(A,B)\otimes\mathbb{Z}_{\ell}\cong Hom_{G_{K}}(T_{\ell}(A),T_{\ell}(B))$$

   for abelian varieties $A$ and $B$ over $K$, to prove the Shafarevich conjecture. The idea is that there are only finitely many possibilities for the Tate module, which shows finiteness up to isogeny; a height argument then shows finiteness up to isomorphism.

3. Showing that the semisimplicity of the Tate module and the isogeny theorem will follow if there are only finitely many abelian varieties (up to isomorphism) isogenous to a fixed abelian variety $A$ (follows a strategy originally used by Tate to prove the analogous result over finite fields).

4. Showing that there are only finitely many abelian varieties (up to isomorphism) isogenous to a fixed abelian variety $A$ by relating the Faltings height to the moduli-theoretic height (for which there are only finitely many points of bounded height) using Arakelov theory then studying how the Faltings height changes under isogeny. This makes use of the Hodge-Tate decomposition from p-adic Hodge theory.

Other proofs

Other proofs of the Mordell conjecture using different methods have been given by Paul Vojta (Vojta91) and Enrico Bombieri (Bombieri90). Recently another proof involving period mappings together with o-minimality methods was given by Brian Lawrence and Akshay Venkatesh (LawrenceVenkatesh18).

Effective Mordell conjecture

Faltings’ proof of the Mordell conjecture determines the finiteness of the rational points but it does not find the set of rational points. The latter problem is also known as the effective Mordell conjecture. An approach towards this has been proposed by Minhyong Kim and is related to his proposed arithmetic gauge theory (see also Balakrishnan19 for a survey).

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 conjecture 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 (1983) 349-366 [doi:10.1007/BF01388432]

Seminar notes on Faltings’ proof:

• Bhargav Bhatt, Andrew Snowden (org.): Faltings’ proof of the Mordell conjecture (seminar notes, 2016) [pdf, pdf]

Other proofs after Faltings:

• Paul Vojta, 1991, Siegel’s theorem in the compact case, Ann. of Math. 133 (3): 509–548.

• 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:ASNSP_1990_4_17_4_615_0]

• Brian Lawrence, Akshay Venkatesh: Diophantine problems and $p$-adic period mappings, Invent. math. 221 (2020) 893–999 [arxiv:1807.02721, doi:10.1007/s00222-020-00966-7]

Reviews:

• Spencer Bloch, The Proof of the Mordell Conjecture, Math. Int. 6 2 (1984) 41-47 [pdf, doi:10.1007/BF03024155]

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

A survey of two approaches to the effective Mordell conjecture:

• Jennifer Balakrishnan, A. J. Best, F. Bianchi, Brian Lawrence, J. S. Müller, N. Triantafillou, J. Vonk: Two recent $p$-adic approaches to the (effective) Mordell conjecture, in: Arithmetic L-Functions and Differential Geometric Methods Progress in Mathematics 338 Birkhäuser (2021) [arxiv:1910.12755, doi:10.1007/978-3-030-65203-6_2]

Encyclopedia entries:

• Encyclopedia of Mathematics, Mordell conjecture

• Wikipedia, Faltings’ theorem

The relation to anabelian geometry originates in

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

The Weil quote stems from

• André Weil, Œuvres Scientifiques - Collected Papers Volume III, Springer Heidelberg 1979. (p.454)