Faltings's theorem

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

• When ${\displaystyle g=0}$ , there are either no points or infinitely many. In such cases, ${\displaystyle C}$  may be handled as a conic section.
• When ${\displaystyle g=1}$ , if there are any points, then ${\displaystyle C}$  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 ${\displaystyle g>1}$ , according to Faltings's theorem, ${\displaystyle C}$  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.^[3] Aleksei Parshin showed that Shafarevich's finiteness conjecture would imply the Mordell conjecture, using what is now called Parshin's trick.^[4]

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.^[5] The main idea of Faltings's proof is the comparison of Faltings heights and naive heights via Siegel modular varieties.^[a]

• Paul Vojta gave a proof based on Diophantine approximation.^[6] Enrico Bombieri found a more elementary variant of Vojta's proof.^[7]
• 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.^[8]

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 ${\displaystyle \mathbb {Q} _{\ell }}$ -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 ${\displaystyle n\geq 4}$  there are at most finitely many primitive integer solutions (pairwise coprime solutions) to ${\displaystyle a^{n}+b^{n}=c^{n}}$ , since for such ${\displaystyle n}$  the Fermat curve ${\displaystyle x^{n}+y^{n}=1}$  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 ${\displaystyle C}$  with a finitely generated subgroup ${\displaystyle \Gamma }$  of an abelian variety ${\displaystyle A}$ . Generalizing by replacing ${\displaystyle A}$  by a semiabelian variety, ${\displaystyle C}$  by an arbitrary subvariety of ${\displaystyle A}$ , and ${\displaystyle \Gamma }$  by an arbitrary finite-rank subgroup of ${\displaystyle A}$  leads to the Mordell–Lang conjecture, which was proved in 1995 by McQuillan^[9] following work of Laurent, Raynaud, Hindry, Vojta, and Faltings.

Another higher-dimensional generalization of Faltings's theorem is the Bombieri–Lang conjecture that if ${\displaystyle X}$  is a pseudo-canonical variety (i.e., a variety of general type) over a number field ${\displaystyle k}$ , then ${\displaystyle X(k)}$  is not Zariski dense in ${\displaystyle X}$ . Even more general conjectures have been put forth by Paul Vojta.

The Mordell conjecture for function fields was proved by Yuri Ivanovich Manin^[10] and by Hans Grauert.^[11] In 1990, Robert F. Coleman found and fixed a gap in Manin's proof.^[12]