Néron–Tate height

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.[1] Although the global Néron–Tate height is quadratic, the constituent local heights are not quite quadratic. Tate (unpublished) defined it globally by observing that the logarithmic height h L {\displaystyle h_{L}} associated to a symmetric invertible sheaf L {\displaystyle L} on an abelian variety A {\displaystyle A} is “almost quadratic,” and used this to show that the limit

h ^ L ( P ) = lim N → ∞ h L ( N P ) N 2 {\displaystyle {\hat {h}}_{L}(P)=\lim _{N\rightarrow \infty }{\frac {h_{L}(NP)}{N^{2}}}}

exists, defines a quadratic form on the Mordell–Weil group of rational points, and satisfies

h ^ L ( P ) = h L ( P ) + O ( 1 ) , {\displaystyle {\hat {h}}_{L}(P)=h_{L}(P)+O(1),}

where the implied O ( 1 ) {\displaystyle O(1)} constant is independent of P {\displaystyle P} .[2] If L {\displaystyle L} is anti-symmetric, that is [ − 1 ] ∗ L = L − 1 {\displaystyle [-1]^{*}L=L^{-1}} , then the analogous limit

h ^ L ( P ) = lim N → ∞ h L ( N P ) N {\displaystyle {\hat {h}}_{L}(P)=\lim _{N\rightarrow \infty }{\frac {h_{L}(NP)}{N}}}

converges and satisfies h ^ L ( P ) = h L ( P ) + O ( 1 ) {\displaystyle {\hat {h}}_{L}(P)=h_{L}(P)+O(1)} , but in this case h ^ L {\displaystyle {\hat {h}}_{L}} is a linear function on the Mordell-Weil group. For general invertible sheaves, one writes L ⊗ 2 = ( L ⊗ [ − 1 ] ∗ L ) ⊗ ( L ⊗ [ − 1 ] ∗ L − 1 ) {\displaystyle L^{\otimes 2}=(L\otimes [-1]^{*}L)\otimes (L\otimes [-1]^{*}L^{-1})} as a product of a symmetric sheaf and an anti-symmetric sheaf, and then

h ^ L ( P ) = 1 2 h ^ L ⊗ [ − 1 ] ∗ L ( P ) + 1 2 h ^ L ⊗ [ − 1 ] ∗ L − 1 ( P ) {\displaystyle {\hat {h}}_{L}(P)={\frac {1}{2}}{\hat {h}}_{L\otimes [-1]^{*}L}(P)+{\frac {1}{2}}{\hat {h}}_{L\otimes [-1]^{*}L^{-1}}(P)}

is the unique quadratic function satisfying

h ^ L ( P ) = h L ( P ) + O ( 1 ) and h ^ L ( 0 ) = 0. {\displaystyle {\hat {h}}_{L}(P)=h_{L}(P)+O(1)\quad {\mbox{and}}\quad {\hat {h}}_{L}(0)=0.}

The Néron–Tate height depends on the choice of an invertible sheaf on the abelian variety, although the associated bilinear form depends only on the image of L {\displaystyle L} in the Néron–Severi group of A {\displaystyle A} . If the abelian variety A {\displaystyle A} is defined over a number field K and the invertible sheaf is symmetric and 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 ) {\displaystyle A(K)} . More generally, h ^ L {\displaystyle {\hat {h}}_{L}} induces a positive definite quadratic form on the real vector space A ( K ) ⊗ R {\displaystyle A(K)\otimes \mathbb {R} } .

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 h ^ {\displaystyle {\hat {h}}} without reference to a particular line bundle. (However, the height that naturally appears in the statement of the Birch and 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, and the height used in the statement of the Birch–Swinnerton-Dyer conjecture is the Néron–Tate height associated to the Poincaré line bundle on A × A ^ {\displaystyle A\times {\hat {A}}} , the product of A {\displaystyle A} with its dual.

The elliptic and abelian regulators

The bilinear form associated to the canonical height h ^ {\displaystyle {\hat {h}}} on an elliptic curve E is

⟨ P , Q ⟩ = 1 2 ( h ^ ( P + Q ) − h ^ ( P ) − h ^ ( Q ) ) . {\displaystyle \langle P,Q\rangle ={\frac {1}{2}}{\bigl (}{\hat {h}}(P+Q)-{\hat {h}}(P)-{\hat {h}}(Q){\bigr )}.}

The elliptic regulator of E/K is

Reg ⁡ ( E / K ) = det ( ⟨ P i , P j ⟩ ) 1 ≤ i , j ≤ r , {\displaystyle \operatorname {Reg} (E/K)=\det {\bigl (}\langle P_{i},P_{j}\rangle {\bigr )}_{1\leq i,j\leq r},}

where P1,...,Pr 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 ≅ Pic0(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 Q1,...,Qr for the Mordell–Weil group A(K) modulo torsion and a basis η1,...,ηr for the Mordell–Weil group B(K) modulo torsion and setting

Reg ⁡ ( A / K ) = det ( ⟨ Q i , η j ⟩ P ) 1 ≤ i , j ≤ r . {\displaystyle \operatorname {Reg} (A/K)=\det {\bigl (}\langle Q_{i},\eta _{j}\rangle _{P}{\bigr )}_{1\leq i,j\leq r}.}

(The definitions of elliptic and abelian regulator are not entirely consistent, since if A is an elliptic curve, then the latter is 2r 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)[3]      h ^ ( P ) ≥ c ( K ) log ⁡ max { Norm K / Q ⁡ Disc ⁡ ( E / K ) , h ( j ( E ) ) } {\displaystyle {\hat {h}}(P)\geq c(K)\log \max {\bigl \{}\operatorname {Norm} _{K/\mathbb {Q} }\operatorname {Disc} (E/K),h(j(E)){\bigr \}}\quad } for all E / K {\displaystyle E/K} and all nontorsion P ∈ E ( K ) . {\displaystyle P\in E(K).}
  • (Lehmer)[4]     h ^ ( P ) ≥ c ( E / K ) [ K ( P ) : K ] {\displaystyle {\hat {h}}(P)\geq {\frac {c(E/K)}{[K(P):K]}}} for all nontorsion P ∈ E ( K ¯ ) . {\displaystyle P\in E({\bar {K}}).}

In both conjectures, the constants are positive and depend only on the indicated quantities. (A stronger form of Lang's conjecture asserts that c {\displaystyle c} depends only on the degree [ K : Q ] {\displaystyle [K:\mathbb {Q} ]} .) It is known that the abc conjecture implies Lang's conjecture, and that the analogue of Lang's conjecture over one dimensional characteristic 0 function fields is unconditionally true.[3][5] The best general result on Lehmer's conjecture is the weaker estimate h ^ ( P ) ≥ c ( E / K ) / [ K ( P ) : K ] 3 + ε {\displaystyle {\hat {h}}(P)\geq c(E/K)/[K(P):K]^{3+\varepsilon }} due to Masser.[6] When the elliptic curve has complex multiplication, this has been improved to h ^ ( P ) ≥ c ( E / K ) / [ K ( P ) : K ] 1 + ε {\displaystyle {\hat {h}}(P)\geq c(E/K)/[K(P):K]^{1+\varepsilon }} by Laurent.[7] There are analogous conjectures for abelian varieties, with the nontorsion condition replaced by the condition that the multiples of P {\displaystyle P} form a Zariski dense subset of A {\displaystyle A} , and the lower bound in Lang's conjecture replaced by h ^ ( P ) ≥ c ( K ) h ( A / K ) {\displaystyle {\hat {h}}(P)\geq c(K)h(A/K)} , where h ( A / K ) {\displaystyle h(A/K)} is the Faltings height of A / K {\displaystyle A/K} .

Generalizations

A polarized algebraic dynamical system is a triple ( V , φ , L ) {\displaystyle (V,\varphi ,L)} consisting of a (smooth projective) algebraic variety V {\displaystyle V} , an endomorphism φ : V → V {\displaystyle \varphi :V\to V} , and a line bundle L → V {\displaystyle L\to V} with the property that φ ∗ L = L ⊗ d {\displaystyle \varphi ^{*}L=L^{\otimes d}} for some integer d > 1 {\displaystyle d>1} . The associated canonical height is given by the Tate limit[8]

h ^ V , φ , L ( P ) = lim n → ∞ h V , L ( φ ( n ) ( P ) ) d n , {\displaystyle {\hat {h}}_{V,\varphi ,L}(P)=\lim _{n\to \infty }{\frac {h_{V,L}(\varphi ^{(n)}(P))}{d^{n}}},}

where φ ( n ) = φ ∘ ⋯ ∘ φ {\displaystyle \varphi ^{(n)}=\varphi \circ \cdots \circ \varphi } is the n-fold iteration of φ {\displaystyle \varphi } . For example, any morphism φ : P n → P n {\displaystyle \varphi :\mathbb {P} ^{n}\to \mathbb {P} ^{n}} of degree d > 1 {\displaystyle d>1} yields a canonical height associated to the line bundle relation φ ∗ O ( 1 ) = O ( n ) {\displaystyle \varphi ^{*}{\mathcal {O}}(1)={\mathcal {O}}(n)} . If V {\displaystyle V} is defined over a number field and L {\displaystyle L} is ample, then the canonical height is non-negative, and

h ^ V , φ , L ( P ) = 0     ⟺     P  is preperiodic for  φ . {\displaystyle {\hat {h}}_{V,\varphi ,L}(P)=0~~\Longleftrightarrow ~~P{\text{ is preperiodic for }}\varphi .}

( P {\displaystyle P} is preperiodic if its forward orbit P , φ ( P ) , φ 2 ( P ) , φ 3 ( P ) , … {\displaystyle P,\varphi (P),\varphi ^{2}(P),\varphi ^{3}(P),\ldots } contains only finitely many distinct points.)

References

  1. ^ Néron, André (1965). "Quasi-fonctions et hauteurs sur les variétés abéliennes". Ann. of Math. (in French). 82 (2): 249–331. doi:10.2307/1970644. JSTOR 1970644. MR 0179173.
  2. ^ Lang (1997) p.72
  3. ^ a b Lang (1997) pp.73–74
  4. ^ Lang (1997) pp.243
  5. ^ Hindry, Marc; Silverman, Joseph H. (1988). "The canonical height and integral points on elliptic curves". Invent. Math. 93 (2): 419–450. doi:10.1007/bf01394340. MR 0948108. S2CID 121520625. Zbl 0657.14018.
  6. ^ Masser, David W. (1989). "Counting points of small height on elliptic curves". Bull. Soc. Math. France. 117 (2): 247–265. doi:10.24033/bsmf.2120. MR 1015810.
  7. ^ Laurent, Michel (1983). "Minoration de la hauteur de Néron–Tate" [Lower bounds of the Nerón-Tate height]. In Bertin, Marie-José (ed.). Séminaire de théorie des nombres, Paris 1981–82 [Seminar on number theory, Paris 1981–82]. Progress in Mathematics (in French). Birkhäuser. pp. 137–151. ISBN 0-8176-3155-0. MR 0729165.
  8. ^ Call, Gregory S.; Silverman, Joseph H. (1993). "Canonical heights on varieties with morphisms". Compositio Mathematica. 89 (2): 163–205. MR 1255693.

General references for the theory of canonical heights

External links

  • "Canonical height on an elliptic curve". PlanetMath.