SOME REMARKS ON STRONG APPROXIMATION AND APPLICATIONS TO HOMOGENEOUS SPACES OF LINEAR ALGEBRAIC GROUPS - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... Year : 2020

SOME REMARKS ON STRONG APPROXIMATION AND APPLICATIONS TO HOMOGENEOUS SPACES OF LINEAR ALGEBRAIC GROUPS

Abstract

Let k be a number field and X a smooth, geometrically integral quasi-projective variety over k. For any linear algebraic group G over k and any G-torsor g : Z → X, we observe that if the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S for all twists of Z by elements in H^1(k, G), then the étale-Brauer obstruction is the only one for strong approximation off a finite set of places S for X. As an application, we show that any homogeneous space of the form G/H with G a connected linear algebraic group over k satisfies strong approximation off the infinite places with étale-Brauer obstruction, under some compactness assumptions when k is totally real. We also prove more refined strong approximation results for homogeneous spaces of the form G/H with G semisimple simply connected and H finite, using the theory of torsors and descent.
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hal-03013498 , version 1 (19-11-2020)

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  • HAL Id : hal-03013498 , version 1

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F Balestrieri. SOME REMARKS ON STRONG APPROXIMATION AND APPLICATIONS TO HOMOGENEOUS SPACES OF LINEAR ALGEBRAIC GROUPS. 2020. ⟨hal-03013498⟩
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