In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Mathematics, a Diophantine equation is an indeterminate Polynomial Equation that allows the variables to be Integers only It was eventually proved by Gerd Faltings in 1983, about six decades after the conjecture was made; it is now known as Faltings' theorem. Gerd Faltings (born July 28, 1954 in Gelsenkirchen -Buer is a German Mathematician known for his work in arithmetic Algebraic
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Background
Suppose we are given an algebraic curve C defined over the rational numbers (that is, C is defined by polynomials with rational coefficients), and suppose further that C is non-singular (in this case that condition isn't a real restriction). In Algebraic geometry, an algebraic curve is an Algebraic variety of dimension one In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, a singular point of an Algebraic variety V is a point P that is 'special' (so singular in the geometric sense that V How many rational points (points with rational coefficients) are on C?
The answer depends upon the genus g of the curve. In Mathematics, genus has a few different but closely related meanings Topology Orientable surface As is common in number theory, there are three cases: g = 0, g = 1, and g greater than 1. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes The g = 0 case has been understood for a long time; Mordell solved the g = 1 case, and conjectured the result for the g greater than 1 case. Louis Joel Mordell ( 28 January 1888 - 12 March 1972) was a British mathematician known for pioneering research in Number theory.
Statement of results
The complete result is this:
Let C be a non-singular algebraic curve over the rationals of genus g. Then the number of rational points on C may be determined as follows:
- Case g = 0: no points or infinitely many; C is handled as a conic section. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface
- Case g = 1: no points, or C is an elliptic curve with a finite number of rational points forming an abelian group of quite restricted structure, or an infinite number of points forming a finitely generated abelian group (Mordell's Theorem, the initial result of the Mordell-Weil theorem). In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Algebraic geometry Mazur's torsion theorem, due to Barry Mazur, classifies the possible Torsion subgroups of the group of rational points on an Elliptic In Abstract algebra, an Abelian group ( G,+ is called finitely generated if there exist finitely many elements x 1 In Mathematics, the Mordell–Weil theorem states that for an Abelian variety A over a Number field K, the group A (
- Case g > 1: according to Mordell's conjecture, now Faltings' Theorem, only a finite number of points.
Proofs
Faltings' original proof used the known reduction to a case of the Tate conjecture, and a number of tools from algebraic geometry, including the theory of Néron models. In Mathematics, the Tate conjecture is a 1963 Conjecture of John Tate linking Algebraic geometry, and more specifically the identification of Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Algebraic geometry, a Néron model (or Néron minimal model, or minimal model)for an Abelian variety AK defined over A number of subsequent proofs have since been found, applying rather different methods.
References
- Cornell, Gary & Silverman, J. , eds. (1986), Arithmetic geometry, ISBN 0387963111 (Contains an English translation of the paper (Faltings 1983). )
- Faltings, G. (1983), “Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. (Finiteness theorems for abelian varieties over number fields)”, Invent. Math. 73 (3): 349--366, MR0718935, DOI 10. Mathematical Reviews is a journal and online database published by the American Mathematical Society that contains brief synopses (and occasionally evaluations of many 1007/BF01388432. “erratum”, Invent. Math. 75 (2): 381, 1984, DOI 10. 1007/BF01388572