Glossary of Arithmetic and Diophantine Geometry - N

N

Naive height

The naive or classical height of a vector of rational numbers is the maximum absoulte value of the vector of coprime integers obtained by multiplying through by a lowest common denominator. This may be used to define height on a point in projective space over Q, or of a polynomial, regarded as a vector of coefficients, or of an algebraic number, from the height of its minimal polynomial.

Néron symbol

The Néron symbol is a bimultiplicative pairing between divisors and algebraic cycles on an Abelian variety used in Néron's formulation of the Néron–Tate height as a sum of local contributions. The global Néron symbol, which is the sum of the local symbols, is just the negative of the height pairing.

Néron–Tate height

The Néron–Tate height (also often referred to as the canonical height) on an abelian variety A is a height function (q.v.) that is essentially intrinsic, and an exact quadratic form, rather than approximately quadratic with respect to the addition on A as provided by the general theory of heights. It can be defined from a general height by a limiting process; there are also formulae, in the sense that it is a sum of local contributions.

Nevanlinna invariant

The Nevanlinna invariant of an ample divisor D on a normal projective variety X is a real number which describes the rate of growth of the number of rational points on the variety with respect to the embedding defined by the divisor. It has similar formal properties to the abscissa of convergence of the height zeta function and it is conjectured that they are essentially the same.