Unramified cohomology and Chow groups Alena Pirutka Universit - PowerPoint PPT Presentation

Unramified cohomology and Chow groups Alena Pirutka Université Paris-Sud, ENS Paris May 18, 2011 Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Let k be a field and G = Gal ( k s / k ) . Let X / k be a smooth projective geometrically integral variety. Z i ( X ) = � x ∈ X ( i ) Z CH i ( X ) = Z i ( X ) / ∼ rat We have a natural map φ i → CH i (¯ CH i ( X ) X ) G , where ¯ X = X × k k s . Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Examples X ) G is bijective; ◮ i = 0. φ 0 : CH 0 ( X ) → CH 0 (¯ Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Examples X ) G is bijective; ◮ i = 0. φ 0 : CH 0 ( X ) → CH 0 (¯ ◮ i = 1. Then CH 1 ( X ) ≃ Pic X and we have an exact sequence X ) G → Br k → Br X . φ 1 → ( Pic ¯ 0 → Pic X If Br k = 0 (example: k is a finite field), then φ 1 is surjective. Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Examples X ) G is bijective; ◮ i = 0. φ 0 : CH 0 ( X ) → CH 0 (¯ ◮ i = 1. Then CH 1 ( X ) ≃ Pic X and we have an exact sequence X ) G → Br k → Br X . φ 1 → ( Pic ¯ 0 → Pic X If Br k = 0 (example: k is a finite field), then φ 1 is surjective. ◮ i = dim X and k = F is a finite field. We have a surjection : CH 0 ( X ) ։ CH 0 (¯ X ) G . Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Examples X ) G is bijective; ◮ i = 0. φ 0 : CH 0 ( X ) → CH 0 (¯ ◮ i = 1. Then CH 1 ( X ) ≃ Pic X and we have an exact sequence X ) G → Br k → Br X . φ 1 → ( Pic ¯ 0 → Pic X If Br k = 0 (example: k is a finite field), then φ 1 is surjective. ◮ i = dim X and k = F is a finite field. We have a surjection : CH 0 ( X ) ։ CH 0 (¯ X ) G . This follows from : ◮ X has a zero-cycle of degree 1 (Lang-Weil estimates); ◮ A 0 ( X ) ։ Alb X ( F ) (Kato-Saito); ◮ A 0 ( ¯ → Alb X (¯ X ) ∼ F ) (Milne, Rojtman). Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Question (T. Geisser): For X / F , do we have a surjection X ) G ? φ 2 : CH 2 ( X ) → CH 2 (¯ Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Using techniques of K -theory, one shows: Theorem (Kahn, Colliot-Thélène and Kahn) Let F be a finite field, char F = p. Let X / F be a smooth projective geometrically rational variety . We have the following complex φ 2 X ) G → H 3 → CH 2 (¯ 0 → CH 2 ( X ) nr ( X , Q / Z ( 2 )) → 0 which is exact after tensorisation by Z [ 1 / p ] . Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Using techniques of K -theory, one shows: Theorem (Kahn, Colliot-Thélène and Kahn) Let F be a finite field, char F = p. Let X / F be a smooth projective geometrically rational variety . We have the following complex φ 2 X ) G → H 3 → CH 2 (¯ 0 → CH 2 ( X ) nr ( X , Q / Z ( 2 )) → 0 which is exact after tensorisation by Z [ 1 / p ] . Using a method of Colliot-Thélène and Ojanguren, we will produce, for infinitely many primes p , a geometrically rational variety X / F p with H 3 nr ( X , Z / 2 ) � = 0. Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Strategy First step. Write F = F p ( x , y ) and consider the quadric Q ⊂ P 4 F , p � = 2 defined by x 2 0 − ax 2 1 − fx 2 2 + afx 2 3 − g 1 g 2 x 2 4 = 0 with a ∈ F p ; f , g 1 , g 2 ∈ F . Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Strategy First step. Write F = F p ( x , y ) and consider the quadric Q ⊂ P 4 F , p � = 2 defined by x 2 0 − ax 2 1 − fx 2 2 + afx 2 3 − g 1 g 2 x 2 4 = 0 with a ∈ F p ; f , g 1 , g 2 ∈ F . We have : Q (¯ F p ( x , y )) � = ∅ ⇒ Q is ¯ F p ( x , y ) -rational . Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Strategy First step. Write F = F p ( x , y ) and consider the quadric Q ⊂ P 4 F , p � = 2 defined by x 2 0 − ax 2 1 − fx 2 2 + afx 2 3 − g 1 g 2 x 2 4 = 0 with a ∈ F p ; f , g 1 , g 2 ∈ F . We have : Q (¯ F p ( x , y )) � = ∅ ⇒ Q is ¯ F p ( x , y ) -rational . Theorem (Arason) ker [ H 3 ( F , Z / 2 ) → H 3 ( F ( Q ) , Z / 2 )] = Z / 2 ( a , f , g 1 g 2 ) . Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Second step. Find necessarily conditions on a , f , g 1 , g 2 such that, for Q defined by x 2 0 − ax 2 1 − fx 2 2 + afx 2 3 − g 1 g 2 x 2 4 = 0 , we have ◮ ( a , f , g 1 ) is a nonzero element of H 3 ( F ( Q ) , Z / 2 ) ; ◮ ( a , f , g 1 ) ∈ H 3 nr ( F ( Q ) , Z / 2 ) , Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Second step. Find necessarily conditions on a , f , g 1 , g 2 such that, for Q defined by x 2 0 − ax 2 1 − fx 2 2 + afx 2 3 − g 1 g 2 x 2 4 = 0 , we have ◮ ( a , f , g 1 ) is a nonzero element of H 3 ( F ( Q ) , Z / 2 ) ; ◮ ( a , f , g 1 ) ∈ H 3 nr ( F ( Q ) , Z / 2 ) , where ∂ A H 3 � Ker [ H 3 ( F ( Q ) , Z / 2 ) → H 2 ( k A , Z / 2 )] . nr ( F ( Q ) , Z / 2 ) = A dvr F ⊂ A Frac ( A )= F ( Q ) Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Third step. ◮ Find a ∈ Z and f , g 1 , g 2 ∈ Z ( x , y ) such that for infinitely many p their reductions modulo p a , ¯ ¯ f , ¯ g 1 , ¯ g 2 satisfy the conditions above. Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Third step. ◮ Find a ∈ Z and f , g 1 , g 2 ∈ Z ( x , y ) such that for infinitely many p their reductions modulo p a , ¯ ¯ f , ¯ g 1 , ¯ g 2 satisfy the conditions above. ◮ By Hironaka, find X / Q smooth and projective with a morphism X → P 2 Q whose generic fibre is defined by x 2 0 − ax 2 1 − fx 2 2 + afx 2 3 − g 1 g 2 x 2 4 = 0 . Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Third step. ◮ Find a ∈ Z and f , g 1 , g 2 ∈ Z ( x , y ) such that for infinitely many p their reductions modulo p a , ¯ ¯ f , ¯ g 1 , ¯ g 2 satisfy the conditions above. ◮ By Hironaka, find X / Q smooth and projective with a morphism X → P 2 Q whose generic fibre is defined by x 2 0 − ax 2 1 − fx 2 2 + afx 2 3 − g 1 g 2 x 2 4 = 0 . For infinitely many p , X has a reduction X p over F p which is smooth over F p and we have a nonzero element a , ¯ (¯ f , ¯ g 1 ) in H 3 nr ( X p , Z / 2 ) . Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups

Conclusion For p ≥ 13 the following choice works: p \ F ∗ 2 a ∈ F ∗ p f , g 1 , g 2 ∈ F p ( P 2 F p ) with homogeneous coordinates ( x : y : z ) : f = x y � j ( x + y + 2 z + h j ) g 1 = y 8 � j ( 3 x + 3 y + z + h j ) g 2 = z 8 where h j , j = 1 , . . . , 8, are the linear forms e x x + e y y + e z z with e x , e y , e z ∈ { 0 , 1 } . Alena Pirutka Université Paris-Sud, ENS Paris Unramified cohomology and Chow groups