### Abstract

The combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those points x of a given subvariety of G _{M}^{N}(Q̄)=(̄Q*)N, that with respect to the height are very close to a given subgroup of finite rank of _{M} ^{N}(Q̄). Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form. In this paper we prove, for certain special classes of varieties , effective versions of the Lang-Bogomolov conjecture, giving explicit upper bounds for the heights and degrees of the points x under consideration. The main feature of our results is that the points we consider do not have to lie in a prescribed number field. Our main tools are Baker-type logarithmic forms estimates and Bogomolov-type estimates for the number of points on the variety with very small height.

Original language | English |
---|---|

Pages (from-to) | 69-94 |

Number of pages | 26 |

Journal | Mathematical Proceedings of the Cambridge Philosophical Society |

Volume | 147 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jul 2009 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematical Proceedings of the Cambridge Philosophical Society*,

*147*(1), 69-94. https://doi.org/10.1017/S030500410900231X

**Effective results for points on certain subvarieties of tori.** / Bérczes, Attila; Györy, K.; Evertse, Jan Hendrik; Pontreau, Corentin.

Research output: Contribution to journal › Article

*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 147, no. 1, pp. 69-94. https://doi.org/10.1017/S030500410900231X

}

TY - JOUR

T1 - Effective results for points on certain subvarieties of tori

AU - Bérczes, Attila

AU - Györy, K.

AU - Evertse, Jan Hendrik

AU - Pontreau, Corentin

PY - 2009/7

Y1 - 2009/7

N2 - The combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those points x of a given subvariety of G MN(Q̄)=(̄Q*)N, that with respect to the height are very close to a given subgroup of finite rank of M N(Q̄). Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form. In this paper we prove, for certain special classes of varieties , effective versions of the Lang-Bogomolov conjecture, giving explicit upper bounds for the heights and degrees of the points x under consideration. The main feature of our results is that the points we consider do not have to lie in a prescribed number field. Our main tools are Baker-type logarithmic forms estimates and Bogomolov-type estimates for the number of points on the variety with very small height.

AB - The combined conjecture of Lang-Bogomolov for tori gives an accurate description of the set of those points x of a given subvariety of G MN(Q̄)=(̄Q*)N, that with respect to the height are very close to a given subgroup of finite rank of M N(Q̄). Thanks to work of Laurent, Poonen and Bogomolov, this conjecture has been proved in a more precise form. In this paper we prove, for certain special classes of varieties , effective versions of the Lang-Bogomolov conjecture, giving explicit upper bounds for the heights and degrees of the points x under consideration. The main feature of our results is that the points we consider do not have to lie in a prescribed number field. Our main tools are Baker-type logarithmic forms estimates and Bogomolov-type estimates for the number of points on the variety with very small height.

UR - http://www.scopus.com/inward/record.url?scp=68349135372&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=68349135372&partnerID=8YFLogxK

U2 - 10.1017/S030500410900231X

DO - 10.1017/S030500410900231X

M3 - Article

VL - 147

SP - 69

EP - 94

JO - Mathematical Proceedings of the Cambridge Philosophical Society

JF - Mathematical Proceedings of the Cambridge Philosophical Society

SN - 0305-0041

IS - 1

ER -