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===

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| last1=Baker | first1=Alan | title=Linear forms in the logarithms of algebraic numbers. III | doi=10.1112/S0025579300003843 | mr=0220680 | year=1967b | journal=[[Mathematika]] | 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}}

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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]]