Néron–Tate height
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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.
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[edit] Definition and properties
Néron defined the Néron–Tate height as a sum of local heights. Tate defined it globally, by observing that the logarithmic height h associated to an invertible sheaf on an abelian variety is "almost quadratic", and used this to show that the Néron–Tate height
of a rational point P is a quadratic form on the Mordell-Weil group of rational points.
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 invertible sheaf is ample, the Néron–Tate height is positive definite (so it vanishes only on torsion elements of the Mordell-Weil group). On an elliptic curve, the Néron-Severi group is of rank one and has a unique ample generator, so this is usually used to define the Néron–Tate height. On an abelian variety of higher dimension, there need not be a canonical choice of line bundle for defining the Néron–Tate height.
[edit] The elliptic regulator
The elliptic regulator of an elliptic curve is the absolute value of determinant of the inner product matrix of a basis of the Mordell-Weil group (modulo torsion) (cf. Gram determinant). It does not depend on the choice of a basis. It appears in the Birch–Swinnerton-Dyer conjecture.
[edit] References
- Enrico Bombieri, Walter Gubler Heights in Diophantine Geometry ISBN 0521846153 DOI 10.2277/0521846153
- A. Néron, Quasi-fonctions et hauteurs sur les variétés abéliennes Ann. of Math. (2) , 82 (1965) pp. 249–331
- A.N. Parshin (2001), “Height, in Diophantine geometry”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
- J. H .Silverman, The arithmetic of elliptic curves, ISBN 0-387-96203-4
