Height function

Heights in Diophantine geometry were initially developed by André Weil and Douglas Northcott beginning in the 1920s.[6] Innovations in 1960s were the Néron–Tate height and the realization that heights were linked to projective representations in much the same way that ample line bundles are in other parts of algebraic geometry. In the 1970s, Suren Arakelov developed Arakelov heights in Arakelov theory.[7] In 1983, Faltings developed his theory of Faltings heights in his proof of Faltings's theorem.[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.[9] 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.[10]

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

• multiplicative height ${\displaystyle H(p/q)=\max\{|p|,|q|\}}$[11]
• logarithmic height: ${\displaystyle h(p/q)=\log H(p/q)}$[12]

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).[13]

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after André Néron, who first defined it as a sum of local heights,[14] and John Tate, who defined it globally in an unpublished work.[15]

The Weil height is defined on a projective variety X over a number field K equipped with a line bundle L on X. Given a very ample line bundle L₀ on X, one may define a height function using the naive height function h. Since L₀' is very ample, its complete linear system gives a map ϕ from X to projective space. Then for all points p on X, define ${\displaystyle h_{L_{0}}(p):=h(\phi (p)).}$[16][17]

One may write an arbitrary line bundle L on X as the difference of two very ample line bundles L₁ and L₂ on X, up to Serre's twisting sheaf O(1), so one may define the Weil height h_L on X with respect to L via ${\displaystyle h_{L}:=h_{L_{1}}-h_{L_{2}},}$ (up to O(1)).[16][17]

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.[18][19] It is the usual Weil height equipped with a different metric.[20]

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:[21]

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

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.[22] 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.