Faltings's theorem

Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field <math>\mathbb{Q}</math> of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell,Template:Sfn and known as the Mordell conjecture until its 1983 proof by Gerd Faltings.Template:Sfnm The conjecture was later generalized by replacing <math>\mathbb{Q}</math> by any number field.

Background

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

• When <math>g=0</math>, there are either no points or infinitely many. In such cases, <math>C</math> may be handled as a conic section.
• When <math>g=1</math>, if there are any points, then <math>C</math> 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 <math>g>1</math>, according to Faltings's theorem, <math>C</math> has only a finite number of rational points.

Proofs

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.Template:Sfn Aleksei Parshin showed that Shafarevich's finiteness conjecture would imply the Mordell conjecture, using what is now called Parshin's trick.Template:Sfn

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.Template:Sfn The main idea of Faltings's proof is the comparison of Faltings heights and naive heights via Siegel modular varieties.Template:Efn

Later proofs

• Paul Vojta gave a proof based on Diophantine approximation.Template:Sfn Enrico Bombieri found a more elementary variant of Vojta's proof.Template:Sfn
• Brian Lawrence and Akshay Venkatesh gave a proof based on [[p-adic Hodge theory|Template:Mvar-adic Hodge theory]], borrowing also some of the easier ingredients of Faltings's original proof.Template:Sfn

Consequences

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 <math>\mathbb{Q}_{\ell}</math>-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 <math>n\ge 4</math> there are at most finitely many primitive integer solutions (pairwise coprime solutions) to <math>a^n+b^n=c^n</math>, since for such <math>n</math> the Fermat curve <math>x^n+y^n=1</math> has genus greater than 1.

Generalizations

Because of the Mordell–Weil theorem, Faltings's theorem can be reformulated as a statement about the intersection of a curve <math>C</math> with a finitely generated subgroup <math>\Gamma</math> of an abelian variety <math>A</math>. Generalizing by replacing <math>A</math> by a semiabelian variety, <math>C</math> by an arbitrary subvariety of <math>A</math>, and <math>\Gamma</math> by an arbitrary finite-rank subgroup of <math>A</math> leads to the Mordell–Lang conjecture, which was proved in 1995 by McQuillanTemplate:Sfn following work of Laurent, Raynaud, Hindry, Vojta, and Faltings.

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

The Mordell conjecture for function fields was proved by Yuri Ivanovich ManinTemplate:Sfn and by Hans Grauert.Template:Sfn In 1990, Robert F. Coleman found and fixed a gap in Manin's proof.Template:Sfn

References

• Template:Cite journal
• Template:Cite journal
• Template:Cite book → Contains an English translation of Template:Harvtxt
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal
• Template:Cite encyclopedia
• Template:Cite journal
• Template:Cite book → Gives Vojta's proof of Faltings's Theorem.
• Template:Cite book
• Template:Cite journal
• Template:Cite journal (Translation: Template:Cite journal )
• Template:Cite journal
• Template:Cite journal
• Template:Cite conference
• Template:Eom
• Template:Cite journal
• Template:Cite journal
• Template:Cite journal