Arithmetic height functions over finitely generated fields

@article{Moriwaki1998ArithmeticHF,
  title={Arithmetic height functions over finitely generated fields},
  author={Atsushi Moriwaki},
  journal={Inventiones mathematicae},
  year={1998},
  volume={140},
  pages={101-142}
}
  • A. Moriwaki
  • Published 4 September 1998
  • Mathematics
  • Inventiones mathematicae
Abstract.In this paper, we propose a new height function for a variety defined over a finitely generated field over ℚ. For this height function, we prove Northcott’s theorem and Bogomolov’s conjecture, so that we can recover the original Raynaud’s theorem (Manin-Mumford’s conjecture). 
Height of varieties over finitely generated fields
We show that the height of a variety over a finitely generated field of characteristic zero can be written as an integral of local heights over the set of places of the field. This allows us to apply
Heights and arithmetic dynamics over finitely generated fields.
We develop a theory of vector-valued heights and intersections defined relative to finitely generated extensions K/k. These generalize both number field and geometric heights. When k is Q or F_p, or
On the Specialization Theorem for Abelian Varieties
In this paper, we apply Moriwaki's arithmetic height functions to obtain an analogue of Silverman's specialization theorem for families of Abelian varieties over K, where K is any field finitely
Heights over finitely generated fields
This is an expository account about height functions and Arakelov theory in arithmetic geometry. We recall Conrad’s description of generalized global fields in order to describe heights over function
A note on polarizations of finitely generated fields
In our previous paper, we established Northcott's theorem for height functions over finitely generated fields. Unfortunately, Northcott's theorem on finitely generated fields does not hold in
Points of Small Height on Varieties Defined over a Function Field
  • D. Ghioca
  • Mathematics
    Canadian Mathematical Bulletin
  • 2009
Abstract We obtain a Bogomolov type of result for the affine space defined over the algebraic closure of a function field of transcendence degree 1 over a finite field.
A Note on the Specialization Theorem for Families of Abelian Varieties
In this note, we apply Moriwaki's arithmetic height functions to obtain an analogue of Silverman's Specialization Theorem for families of Abelian varieties over $K$, where $K$ is any field finitely
The number of algebraic cycles with bounded degree
Let X be a projective scheme over a finite field. In this paper, we consider the asymptotic behavior of the number of effective cycles on X with bounded degree as it goes to the infinity. By this
Roth’s Theorem over arithmetic function fields
Roth's theorem is extended to finitely generated field extensions of $\Bbb Q$, using Moriwaki's framework for heights.
Big line bundles over arithmetic varieties
We prove a Hilbert-Samuel type result of arithmetic big line bundles in Arakelov geometry, which is an analogue of a classical theorem of Siu. An application of this result gives equidistribution of
...
...

References

SHOWING 1-10 OF 16 REFERENCES
Inequalities for semistable families of arithmetic varieties
In this paper, we will consider a generalization of Bogomolov's inequality and Cornalba-Harris-Bost's inequality to semistable families of arithmetic varieties under the idea that geometric
The size function of abelian varieties
The size function is defined for points in projective space over any field K, finitely generated field over Q, generalizing the height function for number fields. We prove that the size function on
Positivite et discretion des points algebriques des courbes
We prove the discreteness of algebraic points (with respect to the Neron-Tate height) on a curve of genus greater than one embedded in his jacobian. This result was conjectured by Bogomolov. We also
POSITIVE LINE BUNDLES ON ARITHMETIC VARIETIES
p 5~~~~~~~~~~~~~~~~~~u integers is defined to count the volume of the lattice I(L?n) of integral sections in the space J7(Lon) of real sections with supremum norm. We want to prove that the leading
Relative Bogomolov’s inequality and the cone of positive divisors on the moduli space of stable curves
An interesting point of the above theorem is that even if the weak positivity of disx/y (E) at y is a global property on Y, it can be derived from the local assumption "the goodness of Xg and the
Sous-variétés d’une variété abélienne et points de torsion
Soient A une variete abelienne definie sur le corps des nombres complexes, T le sous-groupe de torsion de A et X un sous-schema ferme integre de A.
Arithmetic intersection theory
© Publications mathématiques de l’I.H.É.S., 1990, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://
Séminaire sur les pinceaux arithmétiques : la conjecture de Mordell
© Société mathématique de France, 1985, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les
Courbes sur une variété abélienne et points de torsion
...
...