Height function: Difference between revisions

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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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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===

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}}.

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===

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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===

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===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

<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)''.

OneIn maygeneral, writeone ancan arbitrary line bundlewrite ''L'' on ''X'' as the difference of two very ample line bundles ''L₁'' and ''L₂'' on ''X'', upand todefine [[Proj<math>h_{L} construction#The:= twistingh_{L_1} sheaf- of Serre|Serre's twisting sheaf]] ''O(1)''h_{L_2}, so one may define the Weil height ''h<sub>L</submath>'' on ''X'' with respect to ''L'' via

(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>

<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>

====Arakelov height====

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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>

==Some otherOther 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> (this function is used for constructing a [[bijection]] between <math>\mathbb{N}</math> and <math>\mathbb{Q}</math>).

==See also==

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*[[Height zeta function]]

*[[Raynaud's isogeny theorem]]

*[[Tree height]]

==References==

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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}}

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[[Category:Diophantine geometry]]

[[Category:Algebraic number theory]]

[[Category:AlgebraAbstract algebra]]