In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties (or a set of algebraic varieties) to the real numbers.
Height functions allow mathematicians to count objects, such as rational points, that are otherwise infinite in quantity.
For instance, the set of rational numbers of naive height (the maximum of the numerator and denominator when expressed in lowest terms) below any given constant is finite despite the set of rational numbers being infinite.
In other cases, height functions can distinguish some objects based on their complexity.
For instance, the subspace theorem proved by Wolfgang M. Schmidt (1972) demonstrates that points of small height (i.e. small complexity) in projective space lie in a finite number of hyperplanes and generalizes Siegel's theorem on integral points and solution of the S-unit equation.
Several outstanding unsolved problems about the heights of rational points on algebraic varieties, such as the Manin conjecture and Vojta's conjecture, have far-reaching implications for problems in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic.
[4][5] An early form of height function was proposed by Giambattista Benedetti (c. 1563), who argued that the consonance of a musical interval could be measured by the product of its numerator and denominator (in reduced form); see Giambattista Benedetti § Music.
[citation needed] Heights in Diophantine geometry were initially developed by André Weil and Douglas Northcott beginning in the 1920s.
[8] Classical or naive height is defined in terms of ordinary absolute value on homogeneous coordinates.
It is typically a logarithmic scale and therefore can be viewed as being proportional to the "algebraic complexity" or number of bits needed to store a point.
[2] It is typically defined to be the logarithm of the maximum absolute value of the vector of coprime integers obtained by multiplying through by a lowest common denominator.
The naive height H of an elliptic curve E given by y2 = x3 + Ax + B is defined to be H(E) = log max(4|A|3, 27|B|2).
In general, one can write L as the difference of two very ample line bundles L1 and L2 on X and define
[17] The Faltings height of an abelian variety defined over a number field is a measure of its arithmetic complexity.
For a polynomial P of degree n given by the height H(P) is defined to be the maximum of the magnitudes of its coefficients:[18] One could similarly define the length L(P) as the sum of the magnitudes of the coefficients: The Mahler measure M(P) of P is also a measure of the complexity of P.[19] The three functions H(P), L(P) and M(P) are related by the inequalities where
One of the conditions in the definition of an automorphic form on the general linear group of an adelic algebraic group is moderate growth, which is an asymptotic condition on the growth of a height function on the general linear group viewed as an affine variety.