去掉挠点后的某些阿⻉尔簇的强逼近性质.pdf

Strong approximation for Abelian varieties punctured at torsion points 去掉绕点后的某些阿贝尔簇的强逼近性质 梁 永祺 中国科学技术大学 2018年10月21日 中国数学会学术年会 － 贵阳 梁 永祺 Strong approx. for punctured Abelian varieties 1/19 Notation k : number field Ωk = Ωfk t ∞k set of places kv for v ∈ Ωk Ov ⊂ kv for v ∈ Ωfk Ak ring of adèles S ⊂ Ωk finite subset ASk adèles without S-components pr S : Ak → ASk natural projection X : smooth variety over k (variety = separated scheme of finite type, geometrically integral) Br(X ) = H2ét (X , Gm ) the cohomological Brauer group 梁 永祺 Strong approx. for punctured Abelian varieties 2/19 Weak approximation X (k) ,→ Q v ∈Ω X (kv ) diagonally Weak approximation holds if X (k) is dense w.r.t. product topology ∅= 6 U ⊂ X Zariski open weak approximation on X =⇒ weak approximation on U 梁 永祺 Strong approx. for punctured Abelian varieties 3/19 Weak approximation X (k) ,→ Q v ∈Ω X (kv ) diagonally Weak approximation holds if X (k) is dense w.r.t. product topology ∅= 6 U ⊂ X Zariski open weak approximation on X =⇒ weak approximation on U 梁 永祺 Strong approx. for punctured Abelian varieties 3/19 Weak approximation X (k) ,→ Q v ∈Ω X (kv ) diagonally Weak approximation holds if X (k) is dense w.r.t. product topology ∅= 6 U ⊂ X Zariski open weak approximation on X =⇒ weak approximation on U 梁 永祺 Strong approx. for punctured Abelian varieties 3/19 Weak approximation X (k) ,→ Q v ∈Ω X (kv ) diagonally Weak approximation holds if X (k) is dense w.r.t. product topology ∅= 6 U ⊂ X Zariski open weak approximation Q Qon X =⇒ weak approximation on U (uv )v ∈ S0 Mv × v ∈S / 0 U(kv ) with Mv ⊂ U(kv ) open, 梁 永祺 Strong approx. for punctured Abelian varieties 3/19 Weak approximation X (k) ,→ Q v ∈Ω X (kv ) diagonally Weak approximation holds if X (k) is dense w.r.t. product topology ∅= 6 U ⊂ X Zariski open weak approximation Q Qon X =⇒ weak approximation on U (uv )v ∈ QS0 Mv × Qv ∈S / 0 U(kv ) with Mv ⊂ U(kv ) open, (uv )v ∈ S0 Mv × v ∈S / 0 X (kv ), Mv ⊂ X (kv ) open 梁 永祺 Strong approx. for punctured Abelian varieties 3/19 Weak approximation X (k) ,→ Q v ∈Ω X (kv ) diagonally Weak approximation holds if X (k) is dense w.r.t. product topology ∅= 6 U ⊂ X Zariski open weak approximation Q Qon X =⇒ weak approximation on U (uv )v ∈ QS0 Mv × Qv ∈S / 0 U(kv ) with Mv ⊂ U(kv ) open, (uv )v ∈ S0 MQ X (kv ), Mv ⊂ X (kv ) open v × v ∈S / 0Q ∃x ∈ X (k) ∩ [ S0 Mv × v ∈S / 0 X (kv )] 梁 永祺 Strong approx. for punctured Abelian varieties 3/19 Weak approximation X (k) ,→ Q v ∈Ω X (kv ) diagonally Weak approximation holds if X (k) is dense w.r.t. product topology ∅= 6 U ⊂ X Zariski open weak approximation Q Qon X =⇒ weak approximation on U (uv )v ∈ QS0 Mv × Qv ∈S / 0 U(kv ) with Mv ⊂ U(kv ) open, (uv )v ∈ S0 MQ × X (kv ), Mv ⊂ X (kv ) open v v ∈S / 0Q ∃x ∈ X (k) ∩ [ S0Q Mv × v ∈S /Q0 X (kv )] =⇒ x ∈ U(k) ∩ [ S0 Mv × v ∈S / 0 U(kv )], don’t need to care about v ∈ / S0 梁 永祺 Strong approx. for punctured Abelian varieties 3/19 Strong approximation X (k) ,→ X (ASk ) diagonally Strong approximation off S holds if X (k) is dense w.r.t. adélic topology subtle difference between product topology and adélic topology: - strong approximation on X ; strong approximation on U Example: k = Q, S 6= ∅, X = A1 , U = A1 \ {0} = Gm X satisfies strong approximation off S U does not satisfy strong approximation off S 梁 永祺 Strong approx. for punctured Abelian varieties 4/19 Strong approximation X (k) ,→ X (ASk ) diagonally Strong approximation off S holds if X (k) is dense w.r.t. adélic topology subtle difference between product topology and adélic topology: - strong approximation on X ; strong approximation on U Example: k = Q, S 6= ∅, X = A1 , U = A1 \ {0} = Gm X satisfies strong approximation off S U does not satisfy strong approximation off S 梁 永祺 Strong approx. for punctured Abelian varieties 4/19 Strong approximation X (k) ,→ X (ASk ) diagonally Strong approximation off S holds if X (k) is dense w.r.t. adélic topology subtle difference between product topology and adélic topology: - strong approximation on X ; strong approximation on U Example: k = Q, S 6= ∅, X = A1 , U = A1 \ {0} = Gm X satisfies strong approximation off S U does not satisfy strong approximation off S 梁 永祺 Strong approx. for punctured Abelian varieties 4/19 Strong approximation X (k) ,→ X (ASk ) diagonally Strong approximation off S holds if X (k) is dense w.r.t. adélic topology subtle difference between product topology and adélic topology: - strong Q approximation Q on X ; strong approximation on U (uv )v ∈ S0 Mv × v ∈S / 0 U(Ov ) with Mv ⊂ U(kv ) open Example: k = Q, S 6= ∅, X = A1 , U = A1 \ {0} = Gm X satisfies strong approximation off S U does not satisfy strong approximation off S 梁 永祺 Strong approx. for punctured Abelian varieties 4/19 Strong approximation X (k) ,→ X (ASk ) diagonally Strong approximation off S holds if X (k) is dense w.r.t. adélic topology subtle difference between product topology and adélic topology: - strong Q approximation Q on X ; strong approximation on U (uv )v ∈ QS0 Mv × Qv ∈S / 0 U(Ov ) with Mv ⊂ U(kv ) open (uv )v ∈ S0 Mv × v ∈S / 0 X (Ov ) with Mv ⊂ X (kv ) open Example: k = Q, S 6= ∅, X = A1 , U = A1 \ {0} = Gm X satisfies strong approximation off S U does not satisfy strong approximation off S 梁 永祺 Strong approx. for punctured Abelian varieties 4/19 Strong approximation X (k) ,→ X (ASk ) diagonally Strong approximation off S holds if X (k) is dense w.r.t. adélic topology subtle difference between product topology and adélic topology: - strong Q approximation Q on X ; strong approximation on U (uv )v ∈ QS0 Mv × Qv ∈S / 0 U(Ov ) with Mv ⊂ U(kv ) open (uv )v ∈ S0 MQ X (Ov ) with Mv ⊂ X (kv ) open v × v ∈S / 0Q ∃x ∈ X (k) ∩ [ S0 Mv × v ∈S / 0 X (Ov )] Example: k = Q, S 6= ∅, X = A1 , U = A1 \ {0} = Gm X satisfies strong approximation off S U does not satisfy strong approximation off S 梁 永祺 Strong approx. for punctured Abelian varieties 4/19 Strong approximation X (k) ,→ X (ASk ) diagonally Strong approximation off S holds if X (k) is dense w.r.t. adélic topology subtle difference between product topology and adélic topology: - strong Q approximation Q on X ; strong approximation on U (uv )v ∈ QS0 Mv × Qv ∈S / 0 U(Ov ) with Mv ⊂ U(kv ) open (uv )v ∈ S0 MQ X (Ov ) with Mv ⊂ X (kv ) open v × v ∈S / 0Q ∃x ∈ X (k) ∩ [ S0 Mv × v ∈S / 0 X (Ov )] =⇒ x ∈ U(k), but ; x ∈ U(Ov ) for v ∈ / S0 Example: k = Q, S 6= ∅, X = A1 , U = A1 \ {0} = Gm X satisfies strong approximation off S U does not satisfy strong approximation off S 梁 永祺 Strong approx. for punctured Abelian varieties 4/19 Strong approximation X (k) ,→ X (ASk ) diagonally Strong approximation off S holds if X (k) is dense w.r.t. adélic topology subtle difference between product topology and adélic topology: - strong Q approximation Q on X ; strong approximation on U (uv )v ∈ QS0 Mv × Qv ∈S / 0 U(Ov ) with Mv ⊂ U(kv ) open (uv )v ∈ S0 MQ X (Ov ) with Mv ⊂ X (kv ) open v × v ∈S / 0Q ∃x ∈ X (k) ∩ [ S0 Mv × v ∈S / 0 X (Ov )] =⇒ x ∈ U(k), but ; x ∈ U(Ov ) for v ∈ / S0 Example: k = Q, S 6= ∅, X = A1 , U = A1 \ {0} = Gm X satisfies strong approximation off S U does not satisfy strong approximation off S 梁 永祺 Strong approx. for punctured Abelian varieties 4/19 Strong approximation Why? (X = A1 , U = Gm ) 1. étale fundamental groups 2. Brauer groups in such a case 梁 永祺 Strong approx. for punctured Abelian varieties 5/19 Strong approximation Why? (X = A1 , U = Gm ) Apart from the subtle adélic topology, two more reasons: 1. étale fundamental groups 2. Brauer groups in such a case 梁 永祺 Strong approx. for punctured Abelian varieties 5/19 Strong approximation Why? (X = A1 , U = Gm ) Apart from the subtle adélic topology, two more reasons: 1. étale fundamental groups π1ét (Xk̄ ) = 0 while π1ét (Uk̄ ) = Ẑ 2. Brauer groups in such a case 梁 永祺 Strong approx. for punctured Abelian varieties 5/19 Strong approximation Why? (X = A1 , U = Gm ) Apart from the subtle adélic topology, two more reasons: 1. étale fundamental groups π1ét (Xk̄ ) = 0 while π1ét (Uk̄ ) = Ẑ Theorem (Minchev) Let V be a variety defined over a number field k. If Vk̄ is not simply connected π1ét (Vk̄ ) 6= 0, then V can never satisfy strong approximation. 2. Brauer groups in such a case 梁 永祺 Strong approx. for punctured Abelian varieties 5/19 Strong approximation Why? (X = A1 , U = Gm ) Apart from the subtle adélic topology, two more reasons: 1. étale fundamental groups π1ét (Xk̄ ) = 0 while π1ét (Uk̄ ) = Ẑ Theorem (Minchev) Let V be a variety defined over a number field k. If Vk̄ is not simply connected π1ét (Vk̄ ) 6= 0, then V can never satisfy strong approximation. 2. Brauer groups in such a case 梁 永祺 Strong approx. for punctured Abelian varieties 5/19 Strong approximation Why? (X = A1 , U = Gm ) Apart from the subtle adélic topology, two more reasons: 1. étale fundamental groups π1ét (Xk̄ ) = 0 while π1ét (Uk̄ ) = Ẑ 2. Brauer groups Br(X )/Br(k) = 0 while Br1 (U)/Br(k) ' H1 (k, Q/Z) is infinite. in such a case 梁 永祺 Strong approx. for punctured Abelian varieties 5/19 Strong approximation Why? (X = A1 , U = Gm ) Apart from the subtle adélic topology, two more reasons: 1. étale fundamental groups π1ét (Xk̄ ) = 0 while π1ét (Uk̄ ) = Ẑ 2. Brauer groups Br(X )/Br(k) = 0 while Br1 (U)/Br(k) ' H1 (k, Q/Z) is infinite. In 1970s, Manin made use of the Brauer group to define an obstruction to approximation properties. in such a case 梁 永祺 Strong approx. for punctured Abelian varieties 5/19 Strong approximation Why? (X = A1 , U = Gm ) Apart from the subtle adélic topology, two more reasons: 1. étale fundamental groups π1ét (Xk̄ ) = 0 while π1ét (Uk̄ ) = Ẑ 2. Brauer groups Br(X )/Br(k) = 0 while Br1 (U)/Br(k) ' H1 (k, Q/Z) is infinite. In 1970s, Manin made use of the Brauer group to define an obstruction to approximation properties. in such a case 梁 永祺 Strong approx. for punctured Abelian varieties 5/19 Strong approximation Why? (X = A1 , U = Gm ) Apart from the subtle adélic topology, two more reasons: 1. étale fundamental groups π1ét (Xk̄ ) = 0 while π1ét (Uk̄ ) = Ẑ 2. Brauer groups Br(X )/Br(k) = 0 while Br1 (U)/Br(k) ' H1 (k, Q/Z) is infinite. In 1970s, Manin made use of the Brauer group to define an obstruction to approximation properties. in such a case 梁 永祺 Strong approx. for punctured Abelian varieties 5/19 Strong approximation Why? (X = A1 , U = Gm ) Apart from the subtle adélic topology, two more reasons: 1. étale fundamental groups π1ét (Xk̄ ) = 0 while π1ét (Uk̄ ) = Ẑ 2. Brauer groups Br(X )/Br(k) = 0 while Br1 (U)/Br(k) ' H1 (k, Q/Z) is infinite. In 1970s, Manin made use of the Brauer group to define an obstruction to approximation properties. Question (Wittenberg 2014) What happens if Z = X \ U is of codimension ≥ 2 ? in such a case 梁 永祺 Strong approx. for punctured Abelian varieties 5/19 Strong approximation Why? (X = A1 , U = Gm ) Apart from the subtle adélic topology, two more reasons: 1. étale fundamental groups π1ét (Xk̄ ) = 0 while π1ét (Uk̄ ) = Ẑ 2. Brauer groups Br(X )/Br(k) = 0 while Br1 (U)/Br(k) ' H1 (k, Q/Z) is infinite. In 1970s, Manin made use of the Brauer group to define an obstruction to approximation properties. Question (Wittenberg 2014) What happens if Z = X \ U is of codimension ≥ 2 ? in such a case - Zariski-Nagata: π1ét (Xk̄ ) = π1ét (Uk̄ ) 梁 永祺 Strong approx. for punctured Abelian varieties 5/19 Strong approximation Why? (X = A1 , U = Gm ) Apart from the subtle adélic topology, two more reasons: 1. étale fundamental groups π1ét (Xk̄ ) = 0 while π1ét (Uk̄ ) = Ẑ 2. Brauer groups Br(X )/Br(k) = 0 while Br1 (U)/Br(k) ' H1 (k, Q/Z) is infinite. In 1970s, Manin made use of the Brauer group to define an obstruction to approximation properties. Question (Wittenberg 2014) What happens if Z = X \ U is of codimension ≥ 2 ? in such a case - Zariski-Nagata: π1ét (Xk̄ ) = π1ét (Uk̄ ) - purity for étale cohomology: Br(X ) = Br(U) 梁 永祺 Strong approx. for punctured Abelian varieties 5/19 First example: the affine space X = An satisfies strong approximation off S 6= ∅ Br(X \ Z ) Br(X ) = =0 In this case, Br(k) Br(k) In general, should take into account the Brauer-Manin obstruction 梁 永祺 Strong approx. for punctured Abelian varieties 6/19 First example: the affine space X = An satisfies strong approximation off S 6= ∅ Theorem (D. Wei; Y. Cao & F. Xu) Let Z be a Zariski closed subset of An such that codim(Z , An ) ≥ 2. Then An \ Z satisfies strong approximation off S 6= ∅. Br(X ) Br(X \ Z ) = =0 Br(k) Br(k) In general, should take into account the Brauer-Manin obstruction In this case, 梁 永祺 Strong approx. for punctured Abelian varieties 6/19 First example: the affine space X = An satisfies strong approximation off S 6= ∅ Theorem (D. Wei; Y. Cao & F. Xu) Let Z be a Zariski closed subset of An such that codim(Z , An ) ≥ 2. Then An \ Z satisfies strong approximation off S 6= ∅. Br(X ) Br(X \ Z ) = =0 Br(k) Br(k) In general, should take into account the Brauer-Manin obstruction In this case, 梁 永祺 Strong approx. for punctured Abelian varieties 6/19 First example: the affine space X = An satisfies strong approximation off S 6= ∅ Theorem (D. Wei; Y. Cao & F. Xu) Let Z be a Zariski closed subset of An such that codim(Z , An ) ≥ 2. Then An \ Z satisfies strong approximation off S 6= ∅. Br(X ) Br(X \ Z ) = =0 Br(k) Br(k) In general, should take into account the Brauer-Manin obstruction In this case, 梁 永祺 Strong approx. for punctured Abelian varieties 6/19 Brauer-Manin obstruction Manin’s pairing: Fact: X (k) ⊆ X (k) ⊆ X (Ak )Br ⊆ X (Ak ) S ⊆ Ωk finite subset pr S : X (Ak ) → X (ASk ) natural projections Similarly, we define Weak Approximation with Brauer-Manin obstruction using the product topology of X (kv ) instead of the adélic topology. 梁 永祺 Strong approx. for punctured Abelian varieties 7/19 Brauer-Manin obstruction Manin’s pairing: X (Ak ) × Br(X ) → Q/Z X ((xv )v ∈Ωk , b) 7→ invv (b(xv )), v ∈Ωk where invv : Br(kv ) → Q/Z comes from local class field theory Fact: X (k) ⊆ X (k) ⊆ X (Ak )Br ⊆ X (Ak ) S ⊆ Ωk finite subset pr S : X (Ak ) → X (ASk ) natural projections Similarly, we define Weak Approximation with Brauer-Manin obstruction using the product topology of X (kv ) instead of the adélic topology. 梁 永祺 Strong approx. for punctured Abelian varieties 7/19 Brauer-Manin obstruction Manin’s pairing: X (Ak ) × Br(X ) → Q/Z X ((xv )v ∈Ωk , b) 7→ invv (b(xv )), v ∈Ωk where invv : Br(kv ) → Q/Z comes from local class field theory Fact: X (k) ⊆ X (k) ⊆ X (Ak )Br ⊆ X (Ak ) S ⊆ Ωk finite subset pr S : X (Ak ) → X (ASk ) natural projections Similarly, we define Weak Approximation with Brauer-Manin obstruction using the product topology of X (kv ) instead of the adélic topology. 梁 永祺 Strong approx. for punctured Abelian varieties 7/19 Brauer-Manin obstruction Manin’s pairing: X (Ak ) × Br(X ) → Q/Z X ((xv )v ∈Ωk , b) 7→ invv (b(xv )), v ∈Ωk where invv : Br(kv ) → Q/Z comes from local class field theory Fact: X (k) ⊆ X (k) ⊆ X (Ak )Br ⊆ X (Ak ) S ⊆ Ωk finite subset pr S : X (Ak ) → X (ASk ) natural projections Similarly, we define Weak Approximation with Brauer-Manin obstruction using the product topology of X (kv ) instead of the adélic topology. 梁 永祺 Strong approx. for punctured Abelian varieties 7/19 Brauer-Manin obstruction Manin’s pairing: X (Ak ) × Br(X ) → Q/Z X ((xv )v ∈Ωk , b) 7→ invv (b(xv )), v ∈Ωk where invv : Br(kv ) → Q/Z comes from local class field theory Fact: X (k) ⊆ X (k) ⊆ X (Ak )Br ⊆ X (Ak ) S ⊆ Ωk finite subset pr S : X (Ak ) → X (ASk ) natural projections Definition We say that X satisfies strong approximation with Brauer-Manin obstruction off S if X (k) = pr S (X (Ak )Br ) ⊂ X (ASk ). Similarly, we define Weak Approximation with Brauer-Manin obstruction using the product topology of X (kv ) instead of the adélic topology. 梁 永祺 Strong approx. for punctured Abelian varieties 7/19 Brauer-Manin obstruction Manin’s pairing: X (Ak ) × Br(X ) → Q/Z X ((xv )v ∈Ωk , b) 7→ invv (b(xv )), v ∈Ωk where invv : Br(kv ) → Q/Z comes from local class field theory Fact: X (k) ⊆ X (k) ⊆ X (Ak )Br ⊆ X (Ak ) S ⊆ Ωk finite subset pr S : X (Ak ) → X (ASk ) natural projections Definition We say that X satisfies strong approximation with Brauer-Manin obstruction off S if X (k) = pr S (X (Ak )Br ) ⊂ X (ASk ). Similarly, we define Weak Approximation with Brauer-Manin obstruction using the product topology of X (kv ) instead of the adélic topology. 梁 永祺 Strong approx. for punctured Abelian varieties 7/19 Arithmetic purity Recall Question (We call it arithmetic purity) Suppose that X satisfies approximation properties, what about X \ Z for a closed subvariety Z of codimension ≥ 2? What about strong approximation with Brauer-Manin obstruction. 梁 永祺 Strong approx. for punctured Abelian varieties 8/19 Arithmetic purity Recall Question (We call it arithmetic purity) Suppose that X satisfies approximation properties, what about X \ Z for a closed subvariety Z of codimension ≥ 2? Theorem (well known) Arithmetic purity holds for weak approximation with Brauer-Manin obstruction. What about strong approximation with Brauer-Manin obstruction. 梁 永祺 Strong approx. for punctured Abelian varieties 8/19 Arithmetic purity Recall Question (We call it arithmetic purity) Suppose that X satisfies approximation properties, what about X \ Z for a closed subvariety Z of codimension ≥ 2? Theorem (well known) Arithmetic purity holds for weak approximation with Brauer-Manin obstruction. What about strong approximation with Brauer-Manin obstruction. 梁 永祺 Strong approx. for punctured Abelian varieties 8/19 Answers Positive answers Negative answers 梁 永祺 Strong approx. for punctured Abelian varieties 9/19 Positive answers Positive answers 梁 永祺 Strong approx. for punctured Abelian varieties 10/19 Positive answers Generalising examples An and Pn 梁 永祺 Strong approx. for punctured Abelian varieties 11/19 Positive answers Generalising examples An and Pn First result: Theorem (D. Wei 2014) Let X be a smooth toric variety such that k̄[X ]× = k̄ × . Then X verifies arithmetic purity for str. approx. with BM obs. off S 6= ∅. 梁 永祺 Strong approx. for punctured Abelian varieties 11/19 Positive answers joint work with Y. Cao and F. Xu Theorem (Cao-L.-Xu 2017) Let G be a semi-simple simply connected linear algebraic group defined over a number field. Suppose that G is quasi-split (a Borel subgroup is defined over k). Then G verifies arithmetic purity for strong approximation off S 6= ∅. Example: SLn For any Zariski closed subset Z such that codim(Z , SLn ) ≥ 2, SLn \ Z satisfies strong approximation off S 6= ∅. Open problem: remove the quasi-splitness condition. 梁 永祺 Strong approx. for punctured Abelian varieties 12/19 Positive answers joint work with Y. Cao and F. Xu Theorem (Cao-L.-Xu 2017) Let G be a semi-simple simply connected linear algebraic group defined over a number field. Suppose that G is quasi-split (a Borel subgroup is defined over k). Then G verifies arithmetic purity for strong approximation off S 6= ∅. (In this case Br(G )/Br(k) = 0.) Example: SLn For any Zariski closed subset Z such that codim(Z , SLn ) ≥ 2, SLn \ Z satisfies strong approximation off S 6= ∅. Open problem: remove the quasi-splitness condition. 梁 永祺 Strong approx. for punctured Abelian varieties 12/19 Positive answers joint work with Y. Cao and F. Xu Theorem (Cao-L.-Xu 2017) Let G be a semi-simple simply connected linear algebraic group defined over a number field. Suppose that G is quasi-split (a Borel subgroup is defined over k). Then G verifies arithmetic purity for strong approximation off S 6= ∅. (In this case Br(G )/Br(k) = 0.) Example: SLn For any Zariski closed subset Z such that codim(Z , SLn ) ≥ 2, SLn \ Z satisfies strong approximation off S 6= ∅. Open problem: remove the quasi-splitness condition. 梁 永祺 Strong approx. for punctured Abelian varieties 12/19 Positive answers joint work with Y. Cao and F. Xu Theorem (Cao-L.-Xu 2017) Let G be a semi-simple simply connected linear algebraic group defined over a number field. Suppose that G is quasi-split (a Borel subgroup is defined over k). Then G verifies arithmetic purity for strong approximation off S 6= ∅. (In this case Br(G )/Br(k) = 0.) Example: SLn For any Zariski closed subset Z such that codim(Z , SLn ) ≥ 2, SLn \ Z satisfies strong approximation off S 6= ∅. Open problem: remove the quasi-splitness condition. 梁 永祺 Strong approx. for punctured Abelian varieties 12/19 GLn Proposition (1) GLn verifies arithmetic purity (codim 2) for str. approx. with BM obs. off ∞k if and only if the number field k is neither Q nor an imaginary quadratic field. The additional 1(= 3 − 2) dimension comes from Gm = GLn /SLn and Dirichlet’s unit theorem. 梁 永祺 Strong approx. for punctured Abelian varieties 13/19 GLn Proposition (1) GLn verifies arithmetic purity (codim 2) for str. approx. with BM obs. off ∞k if and only if the number field k is neither Q nor an imaginary quadratic field. (2)Over any number field k, GLn verifies 3-codimensional arithmetic purity for str. approx. with BM obs. off ∞k . The additional 1(= 3 − 2) dimension comes from Gm = GLn /SLn and Dirichlet’s unit theorem. 梁 永祺 Strong approx. for punctured Abelian varieties 13/19 GLn Proposition (1) GLn verifies arithmetic purity (codim 2) for str. approx. with BM obs. off ∞k if and only if the number field k is neither Q nor an imaginary quadratic field. (2)Over any number field k, GLn verifies 3-codimensional arithmetic purity for str. approx. with BM obs. off ∞k . The additional 1(= 3 − 2) dimension comes from Gm = GLn /SLn and Dirichlet’s unit theorem. 梁 永祺 Strong approx. for punctured Abelian varieties 13/19 Linear algebraic groups GLn most general setting G connected linear algebraic group G red = G /G u , G ss = [G red , G red ], G tor = G red /G ss , G sc → G ss 梁 永祺 Strong approx. for punctured Abelian varieties 14/19 Linear algebraic groups GLn most general setting G connected linear algebraic group G red = G /G u , G ss = [G red , G red ], G tor = G red /G ss , G sc → G ss 梁 永祺 Strong approx. for punctured Abelian varieties 14/19 Linear algebraic groups GLn most general setting G connected linear algebraic group G red = G /G u , G ss = [G red , G red ], G tor = G red /G ss , G sc → G ss Theorem Suppose that G sc verifies arithmetic purity for str. approx. off ∞k (in particular when it is quasi-split). G verifies arithmetic purity of codimension (2 + dim G tor ) for str. approx. with BM obs. off ∞k . 梁 永祺 Strong approx. for punctured Abelian varieties 14/19 Negative answers Negative answers 梁 永祺 Strong approx. for punctured Abelian varieties 15/19 Negative answers Example (Y. Cao & F. Xu 2013): k = Q or an imaginary quadratic field - X = Gm × A1 satisfies str. approx. with BM obs. off ∞k (Harari 2008, arithmetic duality theorems) X fails arithmetic purity 梁 永祺 Strong approx. for punctured Abelian varieties 16/19 Negative answers Example (Y. Cao & F. Xu 2013): k = Q or an imaginary quadratic field - X = Gm × A1 satisfies str. approx. with BM obs. off ∞k (Harari 2008, arithmetic duality theorems) - U = X \ {one rational point} does not satisfy str. approx. with BM obs. off ∞k X fails arithmetic purity 梁 永祺 Strong approx. for punctured Abelian varieties 16/19 Negative answers Example (Y. Cao & F. Xu 2013): k = Q or an imaginary quadratic field - X = Gm × A1 satisfies str. approx. with BM obs. off ∞k (Harari 2008, arithmetic duality theorems) - U = X \ {one rational point} does not satisfy str. approx. with BM obs. off ∞k X fails arithmetic purity 梁 永祺 Strong approx. for punctured Abelian varieties 16/19 Punctured Abelian varieties Theorem (L. 2018) E : elliptic curve, rank(E (k)) > 0 A : Abelian variety, rank(A(k)) = 0, dimA 6= 0 T ⊂ E × A a finite set of torsion points X = (E × A) \ T If prA (T ) contains a k-rational point, then X does not satisfy Str. Approx. with BM obs. off ∞k . The converse is also true if X(E × A, k) < ∞. Remark: the case where T = {O} was known in [Cao-Liang-Xu 2017]. 梁 永祺 Strong approx. for punctured Abelian varieties 17/19 Punctured Abelian varieties Theorem (L. 2018) E : elliptic curve, rank(E (k)) > 0 A : Abelian variety, rank(A(k)) = 0, dimA 6= 0 T ⊂ E × A a finite set of torsion points X = (E × A) \ T If prA (T ) contains a k-rational point, then X does not satisfy Str. Approx. with BM obs. off ∞k . The converse is also true if X(E × A, k) < ∞. Remark: the case where T = {O} was known in [Cao-Liang-Xu 2017]. 梁 永祺 Strong approx. for punctured Abelian varieties 17/19 Idea of proof The proof use fibration methods with the following lemma. Lemma E : elliptic curve, rank(E (k)) > 0. T ⊂ E finite set of torsion points. U = E \ T. Then U does not satisfy str. approx. with BM obs. w.r.t Br(E ) off ∞k . This lemma was known to [Harari-Voloch 2010] only in the very special case: k = Q, E : y 2 = x 3 + 3, rank(E (Q)) = 1 and T = {O}. Final remark: As a consequence, E \ O does not satisfy str. approx. with BM obs.off ∞k 梁 永祺 Strong approx. for punctured Abelian varieties 18/19 Idea of proof The proof use fibration methods with the following lemma. Lemma E : elliptic curve, rank(E (k)) > 0. T ⊂ E finite set of torsion points. U = E \ T. Then U does not satisfy str. approx. with BM obs. w.r.t Br(E ) off ∞k . This lemma was known to [Harari-Voloch 2010] only in the very special case: k = Q, E : y 2 = x 3 + 3, rank(E (Q)) = 1 and T = {O}. Final remark: As a consequence, E \ O does not satisfy str. approx. with BM obs.off ∞k 梁 永祺 Strong approx. for punctured Abelian varieties 18/19 Idea of proof The proof use fibration methods with the following lemma. Lemma E : elliptic curve, rank(E (k)) > 0. T ⊂ E finite set of torsion points. U = E \ T. Then U does not satisfy str. approx. with BM obs. w.r.t Br(E ) off ∞k . This lemma was known to [Harari-Voloch 2010] only in the very special case: k = Q, E : y 2 = x 3 + 3, rank(E (Q)) = 1 and T = {O}. Final remark: As a consequence, E \ O does not satisfy str. approx. with BM obs.off ∞k 梁 永祺 Strong approx. for punctured Abelian varieties 18/19 Thank you for your attention ! 谢谢大家！ 梁 永祺 Strong approx. for punctured Abelian varieties 19/19