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The arithmetic Grothendieck-Riemann-Roch theorem for general projective morphisms

José Ignacio Burgos Gil, Gerard Freixas i Montplet, Răzvan Liţcanu (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

In this paper we extend the arithmetic Grothendieck-Riemann-Roch Theorem to projective morphisms between arithmetic varieties that are not necessarily smooth over the complex numbers. The main ingredient of this extension is the theory of generalized holomorphic analytic torsion classes previously developed by the authors.

The modified diagonal cycle on the triple product of a pointed curve

Benedict H. Gross, Chad Schoen (1995)

Annales de l'institut Fourier

Let X be a curve over a field k with a rational point e . We define a canonical cycle Δ e Z 2 ( X 3 ) hom . Suppose that k is a number field and that X has semi-stable reduction over the integers of k with fiber components non-singular. We construct a regular model of X 3 and show that the height pairing τ * ( Δ e ) , τ * ' ( Δ e ) is well defined where τ and τ ' are correspondences. The paper ends with a brief discussion of heights and L -functions in the case that X is a modular curve.

Théorème de Hilbert-Samuel «arithmétique»

Ahmed Abbes, Thierry Bouche (1995)

Annales de l'institut Fourier

On donne une nouvelle démonstration directe du théorème de Hilbert-Samuel arithmétique et on déduit un critère numérique pour l’existence de sections d’un fibré en droite sur une variété arithmétique de norme sup inférieure à un.

Theta height and Faltings height

Fabien Pazuki (2012)

Bulletin de la Société Mathématique de France

Using original ideas from J.-B. Bost and S. David, we provide an explicit comparison between the Theta height and the stable Faltings height of a principally polarized Abelian variety. We also give as an application an explicit upper bound on the number of K -rational points of a curve of genus g 2 under a conjecture of S. Lang and J. Silverman. We complete the study with a comparison between differential lattice structures.

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