• Corpus ID: 118546075

Heights, ranks and regulators of abelian varieties

  title={Heights, ranks and regulators of abelian varieties},
  author={Fabien Pazuki},
  journal={arXiv: Number Theory},
  • F. Pazuki
  • Published 16 June 2015
  • Mathematics
  • arXiv: Number Theory
We lower bound the Faltings height of an abelian variety over a number field by the sum of its injectivity diameter and the norm of its bad reduction primes. It leads to an unconditional bound on the rank of Mordell-Weil groups. Assuming the height conjecture of Lang and Silverman, we then obtain a Northcott property for the regulator on the set of simple abelian varieties defined over a fixed number field, of fixed dimension $g$, bounded rank and with dense rational points over a number field… 
Canonical Kähler metrics and arithmetics: Generalizing Faltings heights
  • Y. Odaka
  • Mathematics
    Kyoto Journal of Mathematics
  • 2018
We extend the Faltings modular heights of abelian varieties to general arithmetic varieties and show direct relations with the Kahler-Einstein geometry, the Minimal Model Program, heights of Bost and
Bertini and Northcott
We prove a new Bertini-type Theorem with explicit control of the genus, degree, height, and the field of definition of the constructed curve. As a consequence we provide a general strategy to reduce
Bornes sur le nombre de points rationnels des courbes : en quête d’uniformité
The aim of this paper is to show how a conjectural lower bound on the canonical height function in the spirit of Lang and Silverman leads to an explicit uniform bound on the number of rational points
Explicit open image theorems for abelian varieties with trivial endomorphism ring
Let $K$ be a number field and $A/K$ be an abelian variety of dimension $g$ with $\operatorname{End}_{\overline{K}}(A)=\mathbb{Z}$. We provide a semi-effective bound $\ell_0(A/K)$ such that the
Homoth\'eties explicites des repr\'esentations $\ell$-adiques.
We present classical and new results on the size of the subgroup of homotheties of $\ell$-adic representations associated to the torsion of an abelian variety. From these estimates, we derive uniform
Split Jacobians and Lower Bounds on Heights
Cette these concerne des proprietes des varietes jacobiennes de courbes de genre 2 qui couvrent des courbes elliptiques. Soit E une courbe plane, donnee par une equation y^2=F(x), ou
Représentations galoisiennes et groupe de Mumford-Tate associé à une variété abélienne
Soient $K$ un corps de nombres et $A$ une variete abelienne sur $K$ dont nous notons $g$ la dimension. Pour tout premier $ell$, le module de Tate $ell$-adique de $A$ nous fournit une representation


Heights and regulators of number fields and elliptic curves
We compare general inequalities between invariants of number fields and invariants of abelian varieties over number fields. On the number field side, we remark that there is only a finite number of
Décompositions en hauteurs locales
  • F. Pazuki
  • Mathematics
    Arithmetic Geometry: Computation and Applications
  • 2019
Let A be an abelian variety defined over a number field k. We provide in this paper different decomposition formulas for the N\'eron-Tate height of k-rational points on A. We deduce a decomposition
Theta height and Faltings height
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
A note on generators of number fields
We establish upper bounds for the smallest height of a generator of a number field $k$ over the rational field $\Q$. Our first bound applies to all number fields $k$ having at least one real
Décompte dans une conjecture de Lang
Abstract.Faltings has proven the following conjecture of Lang: if A is an abelian variety over a number field and X any subvariety then all rational points of X lie on a finite number N of
Polynomial bounds for Arakelov invariants of Belyi curves
We explicitly bound the Faltings height of a curve over Q polynomially in its Belyi degree. Similar bounds are proven for three other Arakelov invariants: the discriminant, Faltings’ delta invariant
Remarques sur une conjecture de Lang
The aim of this paper is to study a conjecture predicting a lower bound on the canonical height on abelian varieties, formulated by S. Lang and generalized by J. H. Silverman. We give here an
An Analogue of the Brauer–Siegel Theorem for Abelian Varieties in Positive Characteristic
Consider a family of abelian varieties Ai of fixed dimension defined over the function field of a curve over a finite field. We assume finiteness of the Shafarevic-Tate group of Ai. We ask then when
Nombre de points rationnels des courbes
Let F be a polynomial in two variables with integer coefficients, let D be its degree and let M ⩾ 3 be an upper bound for the absolute value of its coefficients. Then the number of rational zeroes of
Heights in Diophantine Geometry
I. Heights II. Weil heights III. Linear tori IV. Small points V. The unit equation VI. Roth's theorem VII. The subspace theorem VIII. Abelian varieties IX. Neron-Tate heights X. The Mordell-Weil