Problems and results on the distribution of algebraic points on algebraic varieties

@article{Bombieri2009ProblemsAR,
  title={Problems and results on the distribution of algebraic points on algebraic varieties},
  author={Enrico Bombieri},
  journal={Journal de Theorie des Nombres de Bordeaux},
  year={2009},
  volume={21},
  pages={41-57}
}
  • E. Bombieri
  • Published 2009
  • Mathematics
  • Journal de Theorie des Nombres de Bordeaux
Cet article est un expose de plusieurs resultats sur la distribution des points algebriques sur les varietes algebriques. 

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References

SHOWING 1-10 OF 35 REFERENCES

A Lower Bound for the Height in Abelian Extensions

Abstract We produce an absolute lower bound for the height of the algebraic numbers (different from zero and from the roots of unity) lying in an abelian extension of the rationals. The proof rests

Number Theory and Polynomials: The Mahler measure of algebraic numbers: a survey

A survey of results for Mahler measure of algebraic numbers, and one-variable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (`house') of

Minorations des hauteurs normalisées des sous-variétés des tores

We extend to the case of torii the study of small points on subvarieties of abelian varieties initiated in a previous work, and which gave an effective proof of the generalized Bogomolov

On the local zeta function of a cubic threefold

L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions

Rational points of bounded height on Fano varieties

a prime pe7Z. Let V be an algebraic variety defined over F and lI~ a metrized line bundle on V, i.e., a system (L, ]'],) consisting of a line bundle L and a family of Banach v-adic metrics on L | F,,

Points de hauteur bornée et géométrie des variétés

Résumé. — Si V est une variété algébrique ayant une infinité de points rationnels sur un corps de nombres, il est naturel de munir V d’une hauteur et d’étudier de manière asymptotique les points

Le problème de Lehmer en dimension supérieure

we study a higher dimensional Lehmer problem, or alternatively the Lehmer problem for a power of the multiplicative group. More precisely, if α1, . . . , αn are multiplicatively independent algebraic

On a certain nonary cubic form and related equations

with xi, yi ∈ [1, P ] ∩ Z. Thus, in the case kj = j, estimates for Ss(P ;k), usually referred to as “Vinogradov’s Mean Value Theorem”, are central to the establishment of rather general estimates for

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 some topics connected with Waring's problem.

which was in fact the vehicle through which Hardy and Littlewood applied their hypothesis. So far, the best that has been achieved unconditionally in the direction »of (2) for small values of / are