Height function

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. This may be used to define height on a point in projective space over Q, or of a polynomial, regarded as a vector of coefficients, or of an algebraic number, from the height of its minimal polynomial.[9]

The naive height of a rational number x = p/q (in lowest terms) is

• multiplicative height ${\displaystyle H(p/q)=\max\{|p|,|q|\}}$
• logarithmic height: ${\displaystyle h(p/q)=\log H(p/q)}$[10]

Therefore, the naive multiplicative and logarithmic heights of 4/10 are 5 and log(5), for example.

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).

The Néron–Tate height, or canonical height, is a quadratic form on the Mordell–Weil group of rational points of an abelian variety defined over a global field. It is named after André Néron, who first defined it as a sum of local heights,[11] and John Tate, who defined it globally in an unpublished work.[12]

Let X be a projective variety over a number field K. Let L be a line bundle on X. One defines the Weil height on X with respect to L as follows.

First, suppose that L is very ample. A choice of basis of the space ${\displaystyle \Gamma (X,L)}$ of global sections defines a morphism ϕ from X to projective space, and for all points p on X, one defines ${\displaystyle h_{L}(p):=h(\phi (p))}$, where h is the naive height on projective space.[13][14] For fixed X and L, choosing a different basis of global sections changes ${\displaystyle h_{L}}$, but only by a bounded function of p. Thus ${\displaystyle h_{L}}$ is well-defined up to addition of a function that is O(1).

In general, one can write L as the difference of two very ample line bundles L₁ and L₂ on X and define ${\displaystyle h_{L}:=h_{L_{1}}-h_{L_{2}},}$ which again is well-defined up to O(1).[13][14]

The Arakelov height on a projective space over the field of algebraic numbers is a global height function with local contributions coming from Fubini–Study metrics on the Archimedean fields and the usual metric on the non-Archimedean fields.[15][16] It is the usual Weil height equipped with a different metric.[17]

The Faltings height of an abelian variety defined over a number field is a measure of its arithmetic complexity. It is defined in terms of the height of a metrized line bundle. It was introduced by Faltings (1983) in his proof of the Mordell conjecture.

For a polynomial P of degree n given by

${\displaystyle P=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},}$

the height H(P) is defined to be the maximum of the magnitudes of its coefficients:[18]

${\displaystyle H(P)={\underset {i}{\max }}\,|a_{i}|.}$

One could similarly define the length L(P) as the sum of the magnitudes of the coefficients:

${\displaystyle L(P)=\sum _{i=0}^{n}|a_{i}|.}$

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

${\displaystyle {\binom {n}{\lfloor n/2\rfloor }}^{-1}H(P)\leq M(P)\leq H(P){\sqrt {n+1}};}$

${\displaystyle L(p)\leq 2^{n}M(p)\leq 2^{n}L(p);}$

${\displaystyle H(p)\leq L(p)\leq (n+1)H(p)}$

where ${\displaystyle \scriptstyle {\binom {n}{\lfloor n/2\rfloor }}}$ is the binomial coefficient.