Approximation forte en famille

Abstract

Abstract Soient k un corps de nombres et X une k -variété affine lisse intègre fibrée au-dessus de la droite affine <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi>𝔸</m:mi> <m:mi>k</m:mi> <m:mn>1</m:mn> </m:msubsup> </m:math> $\mathbb {A}^1_{k}$ . Supposons que toutes les fibres sont géométriquement intègres, et que la fibre générique est un espace homogène sous un groupe semisimple simplement connexe presque simple <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>k</m:mi> <m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> $G/k(t)$ , les stabilisateurs géométriques étant réductifs connexes. Soit v une place de k telle que la fibration <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:msubsup> <m:mi>𝔸</m:mi> <m:mi>k</m:mi> <m:mn>1</m:mn> </m:msubsup> </m:mrow> </m:math> $X \rightarrow \mathbb {A}^1_{k}$ admette une section rationnelle sur le complété k v . Supposons en outre que pour presque tout point <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>𝔸</m:mi> <m:mn>1</m:mn> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>k</m:mi> <m:mi>v</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> $x \in \mathbb {A}^1(k_{v})$ le k v -groupe G x est isotrope. Supposons enfin le groupe de Brauer de X réduit à celui de k . Alors l'approximation forte vaut pour X en dehors de la place v . Let k be a number field and X a smooth integral affine variety equipped with a surjective morphism <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>:</m:mo> <m:mi>X</m:mi> <m:mo>→</m:mo> <m:msubsup> <m:mi>𝔸</m:mi> <m:mi>k</m:mi> <m:mn>1</m:mn> </m:msubsup> </m:mrow> </m:math> $f: X \rightarrow \mathbb {A}^1_{k}$ to the affine line. Assume that all fibres of f are split, for instance that they are geometrically integral. Assume that the generic fibre of f is a homogeneous space of a simply connected, almost simple, semisimple group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>/</m:mo> <m:mi>k</m:mi> <m:mo>(</m:mo> <m:mi>t</m:mi> <m:mo>)</m:mo> </m:mrow> </m:math> $G/k(t)$ , and that the geometric stabilizers are connected reductive groups. Let v be a place of k such that the fibration f acquires a rational section over the completion k v at v . Assume moreover that at almost all points in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>𝔸</m:mi> <m:mn>1</m:mn> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>k</m:mi> <m:mi>v</m:mi> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> $x \in \mathbb {A}^1(k_{v})$ the specialized group G x is isotropic over k v . If the Brauer group of X is reduced to the Brauer group of k , then strong approximation holds for X away from the place v .

Locations

• arXiv (Cornell University) - View - PDF
• Journal für die reine und angewandte Mathematik (Crelles Journal) - View