• Corpus ID: 118546075

Heights, ranks and regulators of abelian varieties

@article{Pazuki2015HeightsRA,
  title={Heights, ranks and regulators of abelian varieties},
  author={Fabien Pazuki},
  journal={arXiv: Number Theory},
  year={2015}
}
  • 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… 

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References

SHOWING 1-10 OF 40 REFERENCES

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

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

Good reduction of abelian varieties

We study the relation between abelian varieties having good reduction at a discrete valuation and the action of the inertia group on torsion points of the abelian variety. For onedimensional abelian

Neron models for semiabelian varieties: Congruence and change of base field

Let O be a henselian discrete valuation ring with perfect residue field. Denote by K the fraction field of O = OK , and by p = pK the maximal ideal of O. Then every abelian variety A over K has a

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

On infinite unramified extensions

Let k be a number field. A natural question is: Does k admit an infinite unramified extension? The answer is no, if the root discriminant of k is less than Odlyzko’s bounds. The answer is yes, if k