The Brauer–Manin–Scharaschkin obstruction for subvarieties of a semi-abelian variety and its dynamical analog

Sun, Chia-Liang

doi:10.1016/j.jnt.2014.07.015

Cited by 4 publications

(2 citation statements)

References 14 publications

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“…element in H generates a subgroup of O × S with rank at most one, Bartolome, Bilu and Luca (Theorem 1.2 in [2]) prove the implication (L * f,H ) ⇒ (G * f,H ) in the number field case. Using an essentially different argument, the second author (Corollary 2 in [16]) generalizes this result to any global field.…”

Section: Introductionmentioning

confidence: 90%

Multiplicative subgroups avoiding linear relations in finite fields and a local-global principle

Pasten

Sun

2015

Proc. Amer. Math. Soc.

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We study a local-global principle for polynomial equations with coefficients in a finite field and solutions restricted in a rank-one multiplicative subgroup in a function field over this finite field. We prove such a local-global principle for all sufficiently large characteristics, and we show that the result should hold in full generality under certain reasonable hypothesis related to the existence of large multiplicative subgroups of finite fields avoiding linear relations. We give a method for verifying the latter hypothesis in specific cases, and we show that it is a consequence of the classical Artin primitive root conjecture. In particular, this function field local-global principle is a consequence of GRH. We also discuss the relation of these problems with a finite field version of the Manin-Mumford conjecture.

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Section: Introductionmentioning

confidence: 90%

Multiplicative subgroups avoiding linear relations in finite fields and a local-global principle

Pasten

Sun

2015

Proc. Amer. Math. Soc.

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“…(2) The analogue of [1] for global function fields, i.e. k is a finite field, was proved by Chia-Liang Sun recently in [7] with a different approach.…”

Section: Remarkmentioning

confidence: 99%

On the exponential local-global principle for meromorphic functions and algebraic functions

2014

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Very strong approximation for certain algebraic varieties

Liu

2015

Math. Ann.

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Let F be a global field. In this work, we show that the Brauer-Manin condition on adelic points for subvarieties of a torus T over F cuts out exactly the rational points, if either F is a function field or, if F is the field of rational numbers and T is split. As an application, we prove a conjecture of Harari-Voloch over global function fields which states, roughly speaking, that on any rational hyperbolic curve, the local integral points with the Brauer-Manin condition are the global integral points. Finally we prove for tori over number fields a theorem of Stoll on adelic points of zero-dimensional subvarieties in abelian varieties.Comment: 25 pages. Minor corrections. To appear in Math. An

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The Brauer–Manin–Scharaschkin obstruction for subvarieties of a semi-abelian variety and its dynamical analog

Cited by 4 publications

References 14 publications

Multiplicative subgroups avoiding linear relations in finite fields and a local-global principle

Multiplicative subgroups avoiding linear relations in finite fields and a local-global principle

On the exponential local-global principle for meromorphic functions and algebraic functions

Very strong approximation for certain algebraic varieties

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