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

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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

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Let k be a field and G = Gal(ks/k). Let X/k be a smooth projective geometrically integral variety. Z i(X) =

x∈X (i) Z

CHi(X) = Z i(X)/ ∼rat We have a natural map CHi(X)

φi

→ CHi(¯ X)G, where ¯ X = X ×k ks.

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

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Examples

◮ i = 0. φ0 : CH0(X) → CH0(¯

X)G is bijective;

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

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Examples

◮ i = 0. φ0 : CH0(X) → CH0(¯

X)G is bijective;

◮ i = 1. Then CH1(X) ≃ Pic X and we have an exact sequence

0 → Pic X

φ1

→ (Pic ¯ X)G → Br k → Br X. If Br k = 0 (example: k is a finite field), then φ1 is surjective.

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

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Examples

◮ i = 0. φ0 : CH0(X) → CH0(¯

X)G is bijective;

◮ i = 1. Then CH1(X) ≃ Pic X and we have an exact sequence

0 → Pic X

φ1

→ (Pic ¯ X)G → Br k → Br X. If Br k = 0 (example: k is a finite field), then φ1 is surjective.

◮ i = dim X and k = F is a finite field.

We have a surjection : CH0(X) ։ CH0(¯ X)G.

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

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Examples

◮ i = 0. φ0 : CH0(X) → CH0(¯

X)G is bijective;

◮ i = 1. Then CH1(X) ≃ Pic X and we have an exact sequence

0 → Pic X

φ1

→ (Pic ¯ X)G → Br k → Br X. If Br k = 0 (example: k is a finite field), then φ1 is surjective.

◮ i = dim X and k = F is a finite field.

We have a surjection : CH0(X) ։ CH0(¯ X)G. This follows from :

◮ X has a zero-cycle of degree 1 (Lang-Weil estimates); ◮ A0(X) ։ AlbX(F) (Kato-Saito); ◮ A0( ¯

X)

∼

→ AlbX(¯ F) (Milne, Rojtman).

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

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Question (T. Geisser): For X/F, do we have a surjection φ2 : CH2(X) → CH2(¯ X)G ?

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

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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 0 → CH2(X)

φ2

→ CH2(¯ X)G → H3

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

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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 0 → CH2(X)

φ2

→ CH2(¯ X)G → H3

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/Fp with H3

nr(X, Z/2) = 0.

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

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Strategy

First step. Write F = Fp(x, y) and consider the quadric Q ⊂ P4

F, p = 2

defined by x2

0 − ax2 1 − fx2 2 + afx2 3 − g1g2x2 4 = 0

with a ∈ Fp; f , g1, g2 ∈ F.

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

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Strategy

First step. Write F = Fp(x, y) and consider the quadric Q ⊂ P4

F, p = 2

defined by x2

0 − ax2 1 − fx2 2 + afx2 3 − g1g2x2 4 = 0

with a ∈ Fp; f , g1, g2 ∈ F. We have : Q(¯ Fp(x, y)) = ∅ ⇒ Q is ¯ Fp(x, y)-rational.

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

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Strategy

First step. Write F = Fp(x, y) and consider the quadric Q ⊂ P4

F, p = 2

defined by x2

0 − ax2 1 − fx2 2 + afx2 3 − g1g2x2 4 = 0

with a ∈ Fp; f , g1, g2 ∈ F. We have : Q(¯ Fp(x, y)) = ∅ ⇒ Q is ¯ Fp(x, y)-rational.

Theorem (Arason)

ker[H3(F, Z/2) → H3(F(Q), Z/2)] = Z/2(a, f , g1g2).

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

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Second step. Find necessarily conditions on a, f , g1, g2 such that, for Q defined by x2

0 − ax2 1 − fx2 2 + afx2 3 − g1g2x2 4 = 0,

we have

◮ (a, f , g1) is a nonzero element of H3(F(Q), Z/2); ◮ (a, f , g1) ∈ H3 nr(F(Q), Z/2),

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

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Second step. Find necessarily conditions on a, f , g1, g2 such that, for Q defined by x2

0 − ax2 1 − fx2 2 + afx2 3 − g1g2x2 4 = 0,

we have

◮ (a, f , g1) is a nonzero element of H3(F(Q), Z/2); ◮ (a, f , g1) ∈ H3 nr(F(Q), Z/2), where

H3

nr(F(Q), Z/2) =

A dvr

F⊂A Frac(A)=F(Q)

Ker[H3(F(Q), Z/2)

∂A

→ H2(kA, Z/2)].

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

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Third step.

◮ Find a ∈ Z and f , g1, g2 ∈ Z(x, y) such that for infinitely

many p their reductions modulo p ¯ a, ¯ f , ¯ g1, ¯ g2 satisfy the conditions above.

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

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Third step.

◮ Find a ∈ Z and f , g1, g2 ∈ Z(x, y) such that for infinitely

many p their reductions modulo p ¯ a, ¯ f , ¯ g1, ¯ g2 satisfy the conditions above.

◮ By Hironaka, find X/Q smooth and projective with a

morphism X → P2

Q whose generic fibre is defined by

x2

0 − ax2 1 − fx2 2 + afx2 3 − g1g2x2 4 = 0.

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

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Third step.

◮ Find a ∈ Z and f , g1, g2 ∈ Z(x, y) such that for infinitely

many p their reductions modulo p ¯ a, ¯ f , ¯ g1, ¯ g2 satisfy the conditions above.

◮ By Hironaka, find X/Q smooth and projective with a

morphism X → P2

Q whose generic fibre is defined by

x2

0 − ax2 1 − fx2 2 + afx2 3 − g1g2x2 4 = 0.

For infinitely many p, X has a reduction Xp over Fp which is smooth over Fp and we have a nonzero element (¯ a, ¯ f , ¯ g1) in H3

nr(Xp, Z/2).

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

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Conclusion

For p ≥ 13 the following choice works: a ∈ F∗

p \ F∗2 p

f , g1, g2 ∈ Fp(P2

Fp) with homogeneous coordinates (x : y : z) :

f = x y g1 =

j(x + y + 2z + hj)

y8 g2 =

j(3x + 3y + z + hj)

z8 where hj, j = 1, . . . , 8, are the linear forms exx + eyy + ezz with ex, ey, ez ∈ {0, 1}.

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