en>P1ayer at 11:11, 26 July 2013

2013-07-26T11:11:05Z

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	~~I am Oscar~~ and ~~I completely dig~~ that ~~title~~. ~~South Dakota~~ is ~~exactly where me~~ and ~~my husband reside~~ and ~~my family enjoys~~ it. ~~One~~ of the ~~things she enjoys most~~ is to ~~do aerobics~~ and ~~now she~~ is ~~attempting~~ to ~~earn cash with it~~. ~~My day occupation~~ is a ~~meter reader~~.<br><br>~~Here~~ is ~~my web page~~ - ~~at home std testing~~ ([~~http~~://~~simple~~-~~crafts~~.~~com~~/~~groups~~/~~best~~-~~techniques~~-for-~~keeping~~-~~yeast~~-~~infections~~-~~under~~-~~control/ mouse click the next web site~~])		In [[number theory]], 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]] and [[John Tate]].

			==Definition and properties==
			Néron defined the Néron–Tate height as a sum of local heights.<ref>A. Néron, Quasi-fonctions et hauteurs sur les variétés abéliennes, ''Ann. of Math.'' 82 (1965), 249–331</ref> Although the global Néron–Tate height is quadratic, the local heights that it is the sum of are not quite quadratic. Tate (unpublished) defined it globally by observing that the [[Height_function#H\|logarithmic height]] ''h<SUB>L</SUB>'' associated to an [[invertible sheaf]] ''L'' on an [[abelian variety]] ''A'' is “almost quadratic,” and used this to show that the limit

			:<math>\hat h_L(P) = \lim_{N\rightarrow\infty}\frac{h_L(NP)}{N^2}</math>

			exists and defines a quadratic form on the Mordell-Weil group of rational points.<ref name=L72>Lang (1997) p.72</ref>

			The Néron–Tate height depends on the choice of an invertible sheaf (or an element of the [[Néron-Severi group]]) on the abelian variety. If the abelian variety ''A'' is defined over a number field ''K'' and the invertible sheaf is ample, then the Néron–Tate height is positive definite in the sense that it vanishes only on torsion elements of the Mordell-Weil group ''A''(''K''). More generally, <math>\hat h_L</math> induces a positive definite quadratic form on the real vector space <math>A(K)\otimes\mathbb{R}</math>.

			On an [[elliptic curve]], the Néron-Severi group is of rank one and has a unique ample generator, so this generator is often used to define the Néron–Tate height, which is denoted <math>\hat h</math> without reference to a particular line bundle. (However, the height that naturally appears in the statement of the Birch–Swinnerton-Dyer conjecture is twice this height.) On abelian varieties of higher dimension, there need not be a particular choice of smallest ample line bundle to be used in defining the Néron–Tate height.

			==The elliptic and abelian regulators==
			The bilinear form associated to the canonical height <math>\hat h</math> on an elliptic curve ''E'' is

			:<math> \langle P,Q\rangle = \frac{1}{2} \bigl( \hat h(P+Q) - \hat h(P) - \hat h(Q) \bigr) .</math>

			The '''elliptic regulator''' of ''E/K'' is

			:<math> \operatorname{Reg}(E/K) = \det\bigl( \langle P_i,P_j\rangle \bigr)_{1\le i,j\le r},</math>

			where ''P<sub>1</sub>,…,P<sub>r</sub>'' is a basis for the Mordell-Weil group ''E''(''K'') modulo torsion (cf. [[Gram determinant]]). The elliptic regulator does not depend on the choice of basis.

			More generally, let ''A/K'' be an abelian variety, let ''B'' ≅ Pic<sup>0</sup>(''A'') be the dual abelian variety to ''A'', and let ''P'' be the [[Poincaré line bundle]] on ''A'' × ''B''. Then the '''abelian regulator''' of ''A/K'' is defined by choosing a basis ''Q<sub>1</sub>,…,Q<sub>r</sub>'' for the Mordell-Weil group ''A''(''K'') modulo torsion and a basis η<sub>1</sub>,…,η<sub>''r''</sub> for the Mordell-Weil group ''B''(''K'') modulo torsion and setting

			:<math> \operatorname{Reg}(A/K) = \det\bigl( \langle P_i,\eta_j\rangle_{P} \bigr)_{1\le i,j\le r}.</math>

			(The definitions of elliptic and abelian regulator are not entirely consistent, since if ''A'' is an elliptic curve, then the latter is 2<sup>''r''</sup> times the former.)

			The elliptic and abelian regulators appear in the [[Birch–Swinnerton-Dyer conjecture]].

			==Lower bounds for the Néron–Tate height==
			There are two fundamental conjectures that give lower bounds for the Néron–Tate height. In the first, the field ''K'' is fixed and the elliptic curve ''E/K'' and point ''P ∈ E(K)'' vary, while in the second, the [[elliptic Lehmer conjecture]], the curve ''E/K'' is fixed while the field of definition of the point ''P'' varies.

			* (Lang)<ref name=L734>Lang (1997) pp.73–74</ref> <math> \hat h(P) \ge c(K) \log(\operatorname{Norm}_{K/\mathbb{Q}}\operatorname{Disc}(E/K))\quad</math> for all <math>E/K</math> and all nontorsion <math>P\in E(K).</math>
			* (Lehmer)<ref name=L243>Lang (1997) pp.243</ref> <math>\hat h(P) \ge \frac{c(E/K)}{[K(P):K]}</math> for all <math>P\in E(\bar K).</math>

			In both conjectures, the constants are positive and depend only on the indicated quantities. It is known that the [[Abc conjecture\|''abc'' conjecture]] implies Lang's conjecture.<ref name=L734/><ref>{{cite journal \| first1=M. \|last1=Hindry \|first2=J.H. \|last2=Silverman \| authr2-link=Joseph H. Silverman\| title=The canonical height and integral points on elliptic curves \| journal=Invent. Math. \| volume=93 \|year=1988 \| pages=419–450 \| zbl=0657.14018 }}</ref> The best general result on Lehmer's conjecture is the weaker estimate <math>\hat h(P)\ge c(E/K)/[K(P):K]^{3+\epsilon}</math> due to [[David Masser\|Masser]].<ref>D. Masser, Counting points of small height on elliptic curves, ''Bull. Soc. Math. France'' 117 (1989), 247-265</ref> When the elliptic curve has [[complex multiplication]], this has been improved to <math>\hat h(P)\ge c(E/K)/[K(P):K]^{1+\epsilon}</math> by Laurent.<ref>M. Laurent, Minoration de la hauteur de Néron-Tate, Séminaire de Théorie des Nombres (Paris 1981-1982), Progress in Mathematics, Birkhäuser 1983, 137-151</ref>

			==Generalizations==
			A polarized [[arithmetic dynamics\|algebraic dynamical system]] is a triple (''V'',φ,''L'') consisting of a (smooth projective) algebraic variety ''V'', a self-morphism φ : V → V, and a line bundle ''L'' on ''V'' with the property that <math>\phi^*L = L^{\otimes d}</math> for some integer ''d'' > 1. The associated canonical height is given by the Tate limit<ref>G. Call and J.H. Silverman, Canonical heights on varieties with morphisms, ''Compositio Math.'' 89 (1993), 163-205</ref>

			:<math> \hat h_{V,\phi,L}(P) = \lim_{n\to\infty} \frac{h_{V,L}(\phi^{(n)}(P))}{d^n}, </math>

			where φ<sup>(''n'')</sup> = φ o φ o … o φ is the ''n''-fold iteration of φ. For example, any morphism φ : '''P'''<sup>''N''</sup> → '''P'''<sup>''N''</sup> of degree ''d'' > 1 yields a canonical height associated to the line bundle relation φ*''O''(1) = ''O''(''d''). If ''V'' is defined over a number field and ''L'' is ample, then the canonical height is non-negative, and

			:<math> \hat h_{V,\phi,L}(P) = 0 ~~ \Longleftrightarrow ~~ P~{\rm is~preperiodic~for~}\phi.</math>

			(''P'' is preperiodic if its forward orbit ''P'', φ(''P''), φ<sup>2</sup>(''P''), φ<sup>3</sup>(''P''),… contains only finitely many distinct points.)

			==References==
			{{reflist}}

			General references for the theory of canonical heights
			* {{cite book \| first1=Enrico \| last1=Bombieri \| authorlink1=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 \| first1=Marc \| last1=Hindry \| first2=Joseph H. \| last2=Silverman \| authorlink2=Joseph H. Silverman \| title=Diophantine Geometry: An Introduction \| series=[[Graduate Texts in Mathematics]] \| volume=201 \| year=2000 \| isbn=0-387-98981-1 \| zbl=0948.11023 }}
			* {{cite book \| first=Serge \| last=Lang \| authorlink=Serge Lang \| title=Survey of Diophantine Geometry \| publisher=[[Springer-Verlag]] \| year=1997 \| isbn=3-540-61223-8 \| zbl=0869.11051 }}
			*J.H. Silverman, ''The Arithmetic of Elliptic Curves'', ISBN 0-387-96203-4

			==External links==
			* {{planetmath reference\|id=8534\|title=Canonical height on an elliptic curve}}

			{{DEFAULTSORT:Neron-Tate height}}
			[[Category:Number theory]]
			[[Category:Algebraic geometry]]

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I am Oscar and I completely dig that title. South Dakota is exactly where me and my husband reside and my family enjoys it. One of the things she enjoys most is to do aerobics and now she is attempting to earn cash with it. My day occupation is a meter reader.<br><br>Here is my web page - at home std testing ([http://simple-crafts.com/groups/best-techniques-for-keeping-yeast-infections-under-control/ mouse click the next web site])

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