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.[lower-alpha 1]

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]