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A '''height function''' is a [[function (mathematics)|function]] that quantifies the complexity of mathematical objects. In [[Diophantine geometry]], height functions quantify the size of solutions to [[Diophantine equations]] and are typically functions from a set of points on [[algebraic variety|algebraic varieties]] (or a set of algebraic varieties) to the [[real numbers]].<ref>{{harvs|txt|last=Lang|authorlink=Serge Lang|year=1997|loc1=pp. 43–67}}</ref>

For instance, the ''classical'' or ''naive height'' over the [[rational number]]s is typically defined to be the maximum of the numerators and denominators of the coordinates (e.g. 3{{math|7}} for the coordinates {{math|(3/97, 1/2)}}), but in a [[logarithmic scale]].

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==Significance==

Height functions allow mathematicians to count objects, such as [[rational point]]s, that are otherwise infinite in quantity. For instance, the set of rational numbers of naive height (the maximum of the numerator and denominator when [[Irreducible fraction|expressed in lowest terms]]) below any given constant is finite despite the set of rational numbers being infinite.<ref name="ReferenceA">{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|year=2006|loc1=pp. 15–21}}</ref> In this sense, height functions can be used to prove [[Asymptotic analysis|asymptotic results]] such as [[Baker's theorem]] in [[transcendental number theory]] which was proved by {{harvs|txt|authorlink=Alan Baker (mathematician)|first=Alan|last= Baker|year1=1966|year2=1967a|year3=1967b}}.

In other cases, height functions can distinguish some objects based on their complexity. For instance, the [[subspace theorem]] proved by {{harvs|txt|authorlink=Wolfgang M. Schmidt|first=Wolfgang M. |last=Schmidt|year= 1972}} demonstrates that points of small height (i.e. small complexity) in [[projective space]] lie in a finite number of [[hyperplane]]s and generalizes [[Siegel's theorem on integral points]] and solution of the [[S-unit equation]].<ref>{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|year=2006|loc1=pp. 176–230}}</ref>

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Height functions were crucial to the proofs of the [[Mordell–Weil theorem]] and [[Faltings's theorem]] by {{harvs|txt||last=Weil|authorlink=André Weil|year=1929}} and {{harvs|txt|last=Faltings|authorlink=Gerd Faltings|year=1983}} respectively. 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 equation]]s, [[arithmetic geometry]], and [[mathematical logic]].<ref>{{harvs|txt|last1=Vojta|authorlink=Paul Vojta|year=1987}}</ref><ref>{{harvs|txt|last1=Faltings|authorlink1=Gerd Faltings|year=1991}}</ref>

===History===

==Height functions in Diophantine geometry==

An early form of height function was proposed by [[Giambattista Benedetti]] (c. 1563), who argued that the [[consonance]] of a [[interval (music)|musical interval]] could be measured by the product of its numerator and denominator (in reduced form); see {{slink|Giambattista Benedetti|Music}}.{{cn|date=November 2022}}

===History===

Heights in Diophantine geometry were initially developed by [[André Weil]] and [[Douglas Northcott]] beginning in the 1920s.<ref>{{harvs|txt||last=Weil|authorlink=André Weil|year=1929}}</ref> 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 bundle]]s are in other parts of [[algebraic geometry]]. In the 1970s, [[Suren Arakelov]] developed Arakelov heights in [[Arakelov theory]].<ref>{{harvs|txt|last=Lang|authorlink=Serge Lang|year=1988}}</ref> In 1983, Faltings developed his theory of Faltings heights in his proof of Faltings's theorem.<ref>{{harvs|txt|last=Faltings|authorlink=Gerd Faltings|year=1983}}</ref>

==Height functions in Diophantine geometry==

===Naive height===

''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 [[bit]]s needed to store a point.<ref>{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|yearname=2006|loc1=pp. 15–21}}<"ReferenceA"/ref> 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.<ref>{{harvs|txt|last1=Baker | authorlink1=Alan Baker (mathematician)|last2= Wüstholz | authorlink2=Gisbert Wüstholz|year=2007|loc1=p. 3}}</ref>

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

* multiplicative height <math> H(p/q) = \max\{|p|,|q|\}</math><ref>[https://planetmath.org/heightfunction planetmath: height function]</ref>

* logarithmic height: <math> h(p/q) = \log H (p/q)</math><ref>[https://mathoverflow.net/q/203852 mathoverflow question: average-height-of-rational-points-on-a-curve]</ref>

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

The naive height ''H'' of an [[elliptic curve]] ''E'' given by {{math|''y² {{=}} x³ + Ax + B''}} is defined to be {{math|''H(E)'' {{=}} log max(4&#124;{{pipe}}''A''&#124;{{pipe}}³, 27&#124;{{pipe}}''B''&#124;{{pipe}}²)}}.<ref name="planetmath">{{PlanetMath |urlname=canonicalheightonanellipticcurve |title=Canonical height on an elliptic curve }}</ref>

===Néron–Tate height===

{{Main|Néron–Tate height}}

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,<ref>{{harvs|txt|last=Néron|authorlink=André Néron|year=1965}}</ref> and [[John Tate (mathematician)|John Tate]], who defined it globally in an unpublished work.<ref>{{harvs|txt|last=Lang|authorlink=Serge Lang|year=1997|page=72}}</ref>

===Weil height===

TheLet ''Weil heightX'' is defined onbe a [[projective variety]] ''X'' over a number field ''K''. equipped with a line bundleLet ''L'' on ''X''. Givenbe a [[Ample line bundle|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

One defines the ''Weil height'' on ''X'' with respect to ''L'' as follows.

<math>h_{L_0}(p) := h(\phi(p)).</math><ref name=Silverman/><ref name=Gubler/>

First, suppose that ''L'' is [[Ample line bundle|very ample]]. A choice of basis of the space <math>\Gamma(X,L)</math> of global sections defines a morphism ''ϕ'' from ''X'' to projective space, and for all points ''p'' on ''X'', one defines

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 [[Proj construction#The twisting sheaf of Serre|Serre's twisting sheaf]] ''O(1)'', so one may define the Weil height ''h_L'' on ''X'' with respect to ''L'' via

<math>h_L(p) := h(\phi(p))</math>, where ''h'' is the naive height on projective space.<ref name=Silverman/><ref name=Gubler/> For fixed ''X'' and ''L'', choosing a different basis of global sections changes <math>h_L</math>, but only by a bounded function of ''p''. Thus <math>h_L</math> is well-defined up to addition of a function that is ''O(1)''.

<math>h_{L} := h_{L_1} - h_{L_2},</math>

(up to ''O(1)'').<ref name=Silverman>{{harvs|txt|last=Silverman|authorlink=Joseph H. Silverman|year=1994|loc1=III.10}}</ref><ref name=Gubler>{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|year=2006|loc1=Sections 2.2–2.4}}</ref>

In general, one can write ''L'' as the difference of two very ample line bundles ''L₁'' and ''L₂'' on ''X'' and define <math>h_{L} := h_{L_1} - h_{L_2},</math>

(which again is well-defined up to ''O(1)'').<ref name=Silverman>{{harvs|txt|last=Silverman|authorlink=Joseph H. Silverman|year=1994|loc1=III.10}}</ref><ref name=Gubler>{{harvs|txt|last1=Bombieri|last2=Gubler|authorlink1=Enrico Bombieri|year=2006|loc1=Sections 2.2–2.4}}</ref>

====Arakelov height====

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==Height functions in algebra==

{{see also|Height (abelian group)|Height (ring theory)}}

===Height of a polynomial===

For a [[polynomial]] ''P'' of degree ''n'' given by

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==Height functions in automorphic forms==

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]].<ref>{{harvs|txt|last=Bump|authorlink=Daniel Bump|year=1998}}</ref>

==Other height functions==

The height of an irreducible [[rational number]] ''x'' = ''p''/''q'', ''q'' > 0 is <math>|p|+q</math> (this function is used for constructing a [[bijection]] between <math>\mathbb{N}</math> and <math>\mathbb{Q}</math>).<ref>{{harvs|txt|last1=Kolmogorov | authorlink1=Andrey Kolmogorov |last2= Fomin | authorlink2=Sergei Fomin|year=1957| loc1=p. 5}}</ref>

==See also==

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*[[Height zeta function]]

*[[Raynaud's isogeny theorem]]

*[[Tree height]]

==References==

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==Sources==

*{{cite journal| last1=Baker | first1=Alan | author-link = Alan Baker (mathematician) | title=Linear forms in the logarithms of algebraic numbers. I | doi=10.1112/S0025579300003971 | mr=0220680 | year=1966 | journal=[[Mathematika. A Journal of Pure and Applied Mathematics]] | issn=0025-5793 | volume=13 | issue=2 | pages=204–216 }}

*{{cite journal| last1=Baker | first1=Alan | title=Linear forms in the logarithms of algebraic numbers. II | doi=10.1112/S0025579300008068 | mr=0220680 | year=1967a | journal=[[Mathematika. A Journal of Pure and Applied Mathematics]] | issn=0025-5793 | volume=14 | pages=102–107 }}

*{{cite journal| last1=Baker | first1=Alan | title=Linear forms in the logarithms of algebraic numbers. III | doi=10.1112/S0025579300003843 | mr=0220680 | year=1967b | journal=[[Mathematika. A Journal of Pure and Applied Mathematics]] | issn=0025-5793 | volume=14 | issue=2 | pages=220–228 }}

*{{cite book | first1=Alan | last1=Baker | first2=Gisbert | last2= Wüstholz | author-link2=Gisbert Wüstholz | title=Logarithmic Forms and Diophantine Geometry | series=New Mathematical Monographs | volume=9 | publisher=[[Cambridge University Press]] | year=2007 | isbn=978-0-521-88268-2 | zbl=1145.11004 | page=3 }}

*{{cite book | first1=Enrico | last1=Bombieri | author-link1=Enrico Bombieri | first2=Walter | last2=Gubler | title=Heights in Diophantine Geometry | series=New Mathematical Monographs | volume=4 | publisher=[[Cambridge University Press]] | year=2006 | isbn=978-0-521-71229-3 | zbl=1130.11034 | doi=10.2277/0521846153 }}

*{{cite book | first=Peter | last=Borwein | author-link=Peter Borwein | title=Computational Excursions in Analysis and Number Theory | url=https://archive.org/details/computationalexc00borw | url-access=limited | series=CMS Books in Mathematics | publisher=[[Springer-Verlag]] | year=2002 | isbn=0-387-95444-9 | zbl=1020.12001 | pages=[https://archive.org/details/computationalexc00borw/page/n5 2], 3, 14148 }}

*{{cite book | first=Daniel | last=Bump| author-link1=Daniel Bump | title=Automorphic Forms and Representations | series=Cambridge Studies in Advanced Mathematics | volume=55 | publisher=Cambridge University Press | year=1998 | isbn=9780521658188 | page=300 }}

*{{cite book |title=Arithmetic geometry |last1=Cornell |first1=Gary |last2=Silverman | first2=Joseph H. |author-link2=Joseph H. Silverman |year=1986 |publisher=Springer |___location= New York |isbn=0387963111 }} → Contains an English translation of {{harvtxt|Faltings|1983}}

*{{cite journal |last=Faltings |first=Gerd |year=1983 |title=Endlichkeitssätze für abelsche Varietäten über Zahlkörpern |journal=Inventiones Mathematicae |volume=73 |issue=3 |pages=349&ndash;366 |doi=10.1007/BF01388432 |bibcode=1983InMat..73..349F | mr=0718935 |s2cid=121049418 | trans-title=Finiteness theorems for abelian varieties over number fields | language=de }}

*{{cite journal |last1=Faltings | first1=Gerd | author1-link=Gerd Faltings | title=Diophantine approximation on abelian varieties | journal=Annals of Mathematics | mr=1109353| year=1991 | volume=123 | pages=549–576 | doi=10.2307/2944319 | issue=3 | jstor=2944319 }}

*{{cite journal|title=Energy integrals and small points for the Arakelov height|journal=Archiv der Mathematik|last1=Fili|first1=Paul|last2=Petsche|first2=Clayton|last3=Pritsker|first3=Igor|volume=109|issue=5|year=2017|pages=441–454 |doi=10.1007/s00013-017-1080-x|arxiv=1507.01900|s2cid=119161942}}

*{{cite journal | first=K. | last=Mahler | author-link=Kurt Mahler | title=On two extremum properties of polynomials | journal=[[Illinois J.Journal Math.of Mathematics]] | volume=7 | pages=681–701 | year= 1963 | issue=4 | zbl=0117.04003 | doi=10.1215/ijm/1255645104| doi-access=free }}

*{{cite journal | first=André | last=Néron | author-link=André Néron | title=Quasi-fonctions et hauteurs sur les variétés abéliennes | journal=[[Ann.Annals of Math.Mathematics]] | volume=82 | year=1965 | issue=2 | pages=249–331 | doi=10.2307/1970644 | jstor=1970644 | mr=0179173 | language=fr }}

*{{cite book | last=Schinzel | first=Andrzej | author-link=Andrzej Schinzel | title=Polynomials with special regard to reducibility | zbl=0956.12001 | series=Encyclopedia of Mathematics and Its Applications | volume=77 | ___location=Cambridge | publisher=[[Cambridge University Press]] | year=2000 | isbn=0-521-66225-7 | page=[https://archive.org/details/polynomialswiths0000schi/page/212 212] | url=https://archive.org/details/polynomialswiths0000schi/page/212 }}

*{{cite journal | last1=Schmidt | first1=Wolfgang M. | author-link=Wolfgang M. Schmidt | title=Norm form equations | mr=0314761 | year=1972 | journal=[[Annals of Mathematics]] |series=Second Series | volume=96 | pages=526–551 | issue=3 | doi=10.2307/1970824 | jstor=1970824 }}

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*{{cite journal|last=Weil|first=André|author-link=André Weil|title=L'arithmétique sur les courbes algébriques|journal=[[Acta Mathematica]]|volume=52|year=1929|pages=281–315|issue=1|doi=10.1007/BF02592688|mr=1555278 |doi-access=free}}

*{{cite book |title=Advanced Topics in the Arithmetic of Elliptic Curves |last=Silverman |first=Joseph H. |author-link=Joseph H. Silverman |year=1994|publisher=Springer |___location= New York |isbn=978-1-4612-0851-8 }}

*{{cite book | last1=Vojta | first1=Paul | author1-link=Paul Vojta | title=Diophantine approximationsApproximations and valueValue distributionDistribution theoryTheory | publisher=[[Springer-Verlag]] | ___location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-17551-3 | doi=10.1007/BFb0072989 | zbl=0609.14011 | mr=883451 | year=1987 | volume=1239 }}

*{{cite book | first1=Andrey | last1=Kolmogorov | author-link1=Andrey Kolmogorov | first2=Sergei | last2= Fomin | author-link2=Sergei Fomin | title=Elements of the Theory of Functions and Functional Analysis |___location= New York | publisher=Graylock Press | year=1957}}

==External links==

* [http://mathworld.wolfram.com/PolynomialHeight.html Polynomial height at Mathworld]

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[[Category:Diophantine geometry]]

[[Category:Algebraic number theory]]

[[Category:AlgebraAbstract algebra]]