Nagell–Lutz theorem
Encyclopedia
In mathematics
, the Nagell–Lutz theorem is a result in the diophantine geometry
of elliptic curve
s, which describes rational
torsion points on elliptic curves over the integers.
defines a non-singular cubic curve with integer coefficient
s a, b, c, and let D be the discriminant
of the cubic polynomial
on the right side:
general cubic equations.
For curves over the rationals, the
generalization says that, for a nonsingular cubic curve
whose Weierstrass form
has integer coefficients, any rational point P=(x,y) of finite
order must have integer coordinates, or else have order 2 and
coordinates of the form x=m/4, y=n/8, for m and n integers.
(1895–1988) who published it in 1935, and Élisabeth Lutz (1937).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Nagell–Lutz theorem is a result in the diophantine geometry
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
of elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s, which describes rational
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
torsion points on elliptic curves over the integers.
Definition of the terms
Suppose that the equationdefines a non-singular cubic curve with integer coefficient
Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression ; it is usually a number, but in any case does not involve any variables of the expression...
s a, b, c, and let D be the discriminant
Discriminant
In algebra, the discriminant of a polynomial is an expression which gives information about the nature of the polynomial's roots. For example, the discriminant of the quadratic polynomialax^2+bx+c\,is\Delta = \,b^2-4ac....
of the cubic polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
on the right side:
Statement of the theorem
If P = (x,y) is a rational point of finite order on C, for the elliptic curve group law, then:- 1) x and y are integers
- 2) either y = 0, in which case P has order two, or else y divides D, which immediately implies that y2 divides D.
Generalizations
The Nagell–Lutz theorem generalizes to arbitrary number fields and moregeneral cubic equations.
For curves over the rationals, the
generalization says that, for a nonsingular cubic curve
whose Weierstrass form
has integer coefficients, any rational point P=(x,y) of finite
order must have integer coordinates, or else have order 2 and
coordinates of the form x=m/4, y=n/8, for m and n integers.
History
The result is named for its two independent discoverers, the Norwegian Trygve NagellTrygve Nagell
Trygve Nagell was a Norwegian mathematician, known for his works on the Diophantine equations within number theory. He received his doctorate at the University of Oslo in 1926, and lectured at the University until 1931. He was a professor at the University of Uppsala from 1931 to 1962. Nagell was...
(1895–1988) who published it in 1935, and Élisabeth Lutz (1937).