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An effective result of André-Oort type II

Lars Kühne (2013)

Acta Arithmetica

We prove some new effective results of André-Oort type. In particular, we state certain uniform improvements of the main result in [L. Kühne, Ann. of Math. 176 (2012), 651-671]. We also show that the equation X + Y = 1 has no solution in singular moduli. As a by-product, we indicate a simple trick rendering André's proof of the André-Oort conjecture effective. A significantly new aspect is the usage of both the Siegel-Tatuzawa theorem and the weak effective lower bound on the class number of an...

Arithmetic of 0-cycles on varieties defined over number fields

Yongqi Liang (2013)

Annales scientifiques de l'École Normale Supérieure

Let X be a rationally connected algebraic variety, defined over a number field k . We find a relation between the arithmetic of rational points on  X and the arithmetic of zero-cycles. More precisely, we consider the following statements: (1) the Brauer-Manin obstruction is the only obstruction to weak approximation for  K -rational points on  X K for all finite extensions K / k ; (2) the Brauer-Manin obstruction is the only obstruction to weak approximation in some sense that we define for zero-cycles of degree...

Borne polynomiale pour le nombre de points rationnels des courbes

Gaël Rémond (2011)

Journal de Théorie des Nombres de Bordeaux

Soit F un polynôme en deux variables, de degré D et à coefficients entiers dans [ - M , M ] pour M ≥ 3 . Alors le nombre de zéros rationnels de F est soit infini soit plus petit que M 2 3 D 2 . Nous montrons aussi une version plus générale sur les corps de nombres.

Cohomology of integer matrices and local-global divisibility on the torus

Marco Illengo (2008)

Journal de Théorie des Nombres de Bordeaux

Let p ≠ 2 be a prime and let  G be a p -group of matrices in SL n ( ℤ ) , for some integer  n . In this paper we show that, when n < 3 ( p - 1 ) , a certain subgroup of the cohomology group H 1 ( G , 𝔽 p n ) is trivial. We also show that this statement can be false when n ≥ 3 ( p - 1 ) . Together with a result of Dvornicich and Zannier (see [2]), we obtain that any algebraic torus of dimension n < 3 ( p - 1 ) enjoys a local-global principle on divisibility by  p .

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