Height function: Difference between revisions

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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

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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==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> (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]]