Encyclopedia > Mordell conjecture

  Article Content

Mordell conjecture

The conjecture, eventually proved by Gerd Faltings[?] after about six decades, states a basic result on rational number solutions to Diophantine equations[?].

From the point of view of number theory, the classification of algebraic curves[?] that matters is into three classes, according to their genus g. Suppose given such a curve C defined over the rational numbers (that is, by polynomials with rational coefficients) and non-singular[?] (in this case that condition isn't a real restriction). Then there are three cases, according to how many rational points (points with rational coefficients) are on C

  • Case g = 0 : no points or infinitely many, C is handled as a conic section.
  • Case g = 1: no points, or an elliptic curve with a finite number forming an abelian group of quite restricted structure, or an infinite number forming a finitely generated abelian group (Mordell's Theorem, the initial result of the Mordell-Weil Theorem[?]).
  • Case g = 2: according to Mordell's conjecture, now Faltings' Theorem, only a finite number of points.

Therefore the conjecture took its natural place in the overall picture.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Monaco Grand Prix

... what would become the famous British racing green color. As a street race held on the streets of Monte Carlo and La Condamine[?], it has many elevation shifts, tight ...

 
 
 
This page was created in 26.8 ms