Objects of interest

Algebraic cycles on varieties over finite fields

Alena Pirutka

CNRS, École Polytechnique

May 18, 2015, CIRM, Luminy

Arithmétique, Géométrie, Cryptographie et Théorie des Codes

Objects of interest

Fa finite field, X ⊂Pⁿ_F a smooth projective variety, d =dim(X).

(defined by homogeneous polynomials with coefficients inF).

Examples :

I X =E is an elliptic curve; Higher dimensions:

I X =Ais an abelian variety;

I X ⊂Pⁿ_F is a cubic hypersurface f(x₀, . . .x_n) =0 withf

homogeneous of degree 3.

Question: what objects one can associate to X?

Objects of interest

Fa finite field, X ⊂Pⁿ_F a smooth projective variety, d =dim(X).

(defined by homogeneous polynomials with coefficients inF).

Examples :

I X =E is an elliptic curve; Higher dimensions:

I X =Ais an abelian variety;

I X ⊂Pⁿ_F is a cubic hypersurface f(x₀, . . .x_n) =0 withf

homogeneous of degree 3.

Question: what objects one can associate to X?

Objects of interest

Fa finite field, X ⊂Pⁿ_F a smooth projective variety, d =dim(X).

(defined by homogeneous polynomials with coefficients inF).

Examples :

I X =E is an elliptic curve;

Higher dimensions:

I X =Ais an abelian variety;

I X ⊂Pⁿ_F is a cubic hypersurface f(x₀, . . .x_n) =0 withf

homogeneous of degree 3.

Question: what objects one can associate to X?

Objects of interest

Fa finite field, X ⊂Pⁿ_F a smooth projective variety, d =dim(X).

(defined by homogeneous polynomials with coefficients inF).

Examples :

I X =E is an elliptic curve;

Higher dimensions:

I X =Ais an abelian variety;

I X ⊂Pⁿ_F is a cubic hypersurface f(x₀, . . .x_n) =0 withf

homogeneous of degree 3.

Question: what objects one can associate to X?

Objects of interest

Fa finite field, X ⊂Pⁿ_F a smooth projective variety, d =dim(X).

(defined by homogeneous polynomials with coefficients inF).

Examples :

I X =E is an elliptic curve;

Higher dimensions:

I X =Ais an abelian variety;

I X ⊂Pⁿ_F is a cubic hypersurface f(x₀, . . .x_n) =0 withf

homogeneous of degree 3.

Question: what objects one can associate to X?

Objects of interest

Fa finite field, X ⊂Pⁿ_F a smooth projective variety, d =dim(X).

(defined by homogeneous polynomials with coefficients inF).

Examples :

I X =E is an elliptic curve;

Higher dimensions:

I X =Ais an abelian variety;

I X ⊂Pⁿ_F is a cubic hypersurface f(x₀, . . .x_n) =0 withf

homogeneous of degree 3.

Question: what objects one can associate to X?

Objects of interest

Fa finite field, X ⊂Pⁿ_F a smooth projective variety, d =dim(X).

(defined by homogeneous polynomials with coefficients inF).

Examples :

I X =E is an elliptic curve;

Higher dimensions:

I X =Ais an abelian variety;

I X ⊂Pⁿ_F is a cubic hypersurface f(x₀, . . .x_n) =0 withf

homogeneous of degree 3.

Question: what objects one can associate to X?

Subvarieties of smaller dimension

Look at all Y ⊂X irreducibles of dimensiond −i. Acycleis a formal linear combination of suchY’s: Thegroup of cyclesof

codimensioni is Zⁱ(X) =⊕ZY too huge!

Equivalence relations:

I ForX =C a curve andi =1 define : XajPj ∼_rat 0 iff X

ajPj =div(f)

for some functionf on C.

I In general : similar, ∼_rat is generated by

Xa_jY_j ∼_rat 0 if X

a_jY_j =div(f) for some functionf on W ⊂X of dimensiond −i+1(better : the normalization of W).

Subvarieties of smaller dimension

Look at all Y ⊂X irreducibles of dimensiond −i. Acycleis a formal linear combination of suchY’s:

Thegroup of cyclesof

codimensioni is Zⁱ(X) =⊕ZY

too huge!

Equivalence relations:

I ForX =C a curve andi =1 define : XajPj ∼_rat 0 iff X

ajPj =div(f)

for some functionf on C.

I In general : similar, ∼_rat is generated by

Xa_jY_j ∼_rat 0 if X

a_jY_j =div(f) for some functionf on W ⊂X of dimensiond −i+1(better : the normalization of W).

Subvarieties of smaller dimension

Look at all Y ⊂X irreducibles of dimensiond −i. Acycleis a formal linear combination of suchY’s:

Thegroup of cyclesof

codimensioni is Zⁱ(X) =⊕ZY too huge!

Equivalence relations:

I ForX =C a curve andi =1 define : XajPj ∼_rat 0 iff X

ajPj =div(f)

for some functionf on C.

I In general : similar, ∼_rat is generated by

Xa_jY_j ∼_rat 0 if X

a_jY_j =div(f) for some functionf on W ⊂X of dimensiond −i+1(better : the normalization of W).

Subvarieties of smaller dimension

Look at all Y ⊂X irreducibles of dimensiond −i. Acycleis a formal linear combination of suchY’s:

Thegroup of cyclesof

codimensioni is Zⁱ(X) =⊕ZY too huge!

Equivalence relations:

I ForX =C a curve andi =1 define : XajPj ∼_rat 0 iff

XajPj =div(f)

for some functionf on C.

I In general : similar, ∼_rat is generated by

Xa_jY_j ∼_rat 0 if X

a_jY_j =div(f) for some functionf on W ⊂X of dimensiond −i+1(better : the normalization of W).

Subvarieties of smaller dimension

Look at all Y ⊂X irreducibles of dimensiond −i. Acycleis a formal linear combination of suchY’s:

Thegroup of cyclesof

codimensioni is Zⁱ(X) =⊕ZY too huge!

Equivalence relations:

I ForX =C a curve andi =1 define : XajPj ∼_rat 0 iff X

ajPj =div(f) for some functionf on C.

I In general : similar, ∼_rat is generated by

Xa_jY_j ∼_rat 0 if X

a_jY_j =div(f) for some functionf on W ⊂X of dimensiond −i+1(better : the normalization of W).

Subvarieties of smaller dimension

Look at all Y ⊂X irreducibles of dimensiond −i. Acycleis a formal linear combination of suchY’s:

Thegroup of cyclesof

codimensioni is Zⁱ(X) =⊕ZY too huge!

Equivalence relations:

I ForX =C a curve andi =1 define : XajPj ∼_rat 0 iff X

ajPj =div(f)

for some functionf on C.

I In general : similar, ∼_rat is generated by

Xa_jY_j ∼_rat 0 if

Xa_jY_j =div(f)

for some functionf on W ⊂X of dimensiond −i+1(better : the normalization of W).

Subvarieties of smaller dimension

Look at all Y ⊂X irreducibles of dimensiond −i. Acycleis a formal linear combination of suchY’s:

Thegroup of cyclesof

codimensioni is Zⁱ(X) =⊕ZY too huge!

Equivalence relations:

I ForX =C a curve andi =1 define : XajPj ∼_rat 0 iff X

ajPj =div(f)

for some functionf on C.

I In general : similar, ∼_rat is generated by

Xa_jY_j ∼_rat 0 if X

a_jY_j =div(f) for some functionf on W ⊂X of dimensiond −i+1

(better : the normalization of W).

Subvarieties of smaller dimension

Look at all Y ⊂X irreducibles of dimensiond −i. Acycleis a formal linear combination of suchY’s:

Thegroup of cyclesof

codimensioni is Zⁱ(X) =⊕ZY too huge!

Equivalence relations:

I ForX =C a curve andi =1 define : XajPj ∼_rat 0 iff X

ajPj =div(f)

for some functionf on C.

I In general : similar, ∼_rat is generated by

Xa_jY_j ∼_rat 0 if X

a_jY_j =div(f) for some functionf on W ⊂X of dimensiond −i+1(better : the normalization of W).

Chow groups

Recall:

I Zⁱ(X) =⊕ZY

I ∼_rat is generated by PajYj ∼_rat 0 if PajYj =div(f).

Chow groups :

CHⁱ(X) =Zⁱ(X)/∼_rat; write[Y]∈CHⁱ(X)for the class ofY.

Examples:

I i =0 : CH⁰(X) =Z[X].

I i =1 : CH¹(X) =

divisors/divisors of functions= PicX.

I i =dimX,write

CH^d(X) =CH₀(X) zero-cycles.

In general difficult to determine!

Chow groups

Recall:

I Zⁱ(X) =⊕ZY

I ∼_rat is generated by PajYj ∼_rat 0 if PajYj =div(f).

Chow groups :

CHⁱ(X) =Zⁱ(X)/∼_rat; write[Y]∈CHⁱ(X)for the class ofY.

Examples:

I i =0 : CH⁰(X) =Z[X].

I i =1 : CH¹(X) =

divisors/divisors of functions= PicX.

I i =dimX,write

CH^d(X) =CH₀(X) zero-cycles.

In general difficult to determine!

Chow groups

Recall:

I Zⁱ(X) =⊕ZY

I ∼_rat is generated by PajYj ∼_rat 0 if PajYj =div(f).

Chow groups :

CHⁱ(X) =Zⁱ(X)/∼_rat; write[Y]∈CHⁱ(X)for the class ofY.

Examples:

I i =0 :

CH⁰(X) =Z[X].

I i =1 : CH¹(X) =

divisors/divisors of functions= PicX.

I i =dimX,write

CH^d(X) =CH₀(X) zero-cycles.

In general difficult to determine!

Chow groups

Recall:

I Zⁱ(X) =⊕ZY

I ∼_rat is generated by PajYj ∼_rat 0 if PajYj =div(f).

Chow groups :

CHⁱ(X) =Zⁱ(X)/∼_rat; write[Y]∈CHⁱ(X)for the class ofY.

Examples:

I i =0 : CH⁰(X) =Z[X].

I i =1 : CH¹(X) =

divisors/divisors of functions= PicX.

I i =dimX,write

CH^d(X) =CH₀(X) zero-cycles.

In general difficult to determine!

Chow groups

Recall:

I Zⁱ(X) =⊕ZY

I ∼_rat is generated by PajYj ∼_rat 0 if PajYj =div(f).

Chow groups :

CHⁱ(X) =Zⁱ(X)/∼_rat; write[Y]∈CHⁱ(X)for the class ofY.

Examples:

I i =0 : CH⁰(X) =Z[X].

I i =1 :

CH¹(X) =

divisors/divisors of functions= PicX.

I i =dimX,write

CH^d(X) =CH₀(X) zero-cycles.

In general difficult to determine!

Chow groups

Recall:

I Zⁱ(X) =⊕ZY

I ∼_rat is generated by PajYj ∼_rat 0 if PajYj =div(f).

Chow groups :

CHⁱ(X) =Zⁱ(X)/∼_rat; write[Y]∈CHⁱ(X)for the class ofY.

Examples:

I i =0 : CH⁰(X) =Z[X].

I i =1 : CH¹(X) =

divisors/divisors of functions= PicX.

I i =dimX,write

CH^d(X) =CH₀(X) zero-cycles.

In general difficult to determine!

Chow groups

Recall:

I Zⁱ(X) =⊕ZY

I ∼_rat is generated by PajYj ∼_rat 0 if PajYj =div(f).

Chow groups :

CHⁱ(X) =Zⁱ(X)/∼_rat; write[Y]∈CHⁱ(X)for the class ofY.

Examples:

I i =0 : CH⁰(X) =Z[X].

I i =1 : CH¹(X) =

divisors/divisors of functions= PicX.

I i =dimX,write

CH^d(X) =CH₀(X) zero-cycles.

In general difficult to determine!

Chow groups

Recall:

I Zⁱ(X) =⊕ZY

I ∼_rat is generated by PajYj ∼_rat 0 if PajYj =div(f).

Chow groups :

CHⁱ(X) =Zⁱ(X)/∼_rat; write[Y]∈CHⁱ(X)for the class ofY.

Examples:

I i =0 : CH⁰(X) =Z[X].

I i =1 : CH¹(X) =

divisors/divisors of functions= PicX.

I i =dimX,write

CH^d(X) =CH₀(X) zero-cycles.

In general difficult to determine!

Cohomology

Notation : X¯ is the base change ofX to an algebraic closure ¯FofF. Étale cohomology groups : H_et´ⁱ (X,Z/`),H<sub>et</sub><sup>i</sup><sub>´</sub>(X, µ<sup>⊗</sup><sub>n</sub><sup>j</sup>),H<sub>et</sub><sup>i</sup><sub>´</sub> ( ¯X, µ<sup>⊗</sup><sub>n</sub><sup>j</sup>) H<sub>et</sub><sup>i</sup><sub>´</sub> (X,Z`(j)) =lim←−rH_et´ⁱ (X, µ^⊗_`r^j),

H_etⁱ_´ (X,Q`(j)) =H<sub>et</sub><sup>i</sup><sub>´</sub>(X,Z`(j))⊗Q` (n,`are prime to the characteristic ofF).Properties :

1. H_etⁱ_´ (X, µ^⊗_n^j) are finite, H_etⁱ_´ (X,Z`(j))areZ`-modules of finite type (resp. with X¯);H_etⁱ_´( ¯X,Z`) have no torsion for almost all

` (Gabber, difficult);

2. Hochschild-Serre spectral sequence relates X andX¯:

0→H¹(G,H_´etⁱ⁻¹( ¯X,Z`(j))→H<sub>et</sub><sup>i</sup><sub>´</sub>(X,Z`(j))→H_etⁱ_´( ¯X,Z`(j))^G →0

3. there is a cycle class mapCHⁱ(X)⊗Z` →H<sub>et´</sub><sup>2i</sup>(X,Z`(i)).

Cohomology

Notation : X¯ is the base change ofX to an algebraic closure ¯FofF.

Étale cohomology groups : H_et´ⁱ (X,Z/`),H<sub>et</sub><sup>i</sup><sub>´</sub>(X, µ<sup>⊗</sup><sub>n</sub><sup>j</sup>),H<sub>et</sub><sup>i</sup><sub>´</sub> ( ¯X, µ<sup>⊗</sup><sub>n</sub><sup>j</sup>) H<sub>et</sub><sup>i</sup><sub>´</sub> (X,Z`(j)) =lim←−rH_et´ⁱ (X, µ^⊗_`r^j),

H_etⁱ_´ (X,Q`(j)) =H<sub>et</sub><sup>i</sup><sub>´</sub>(X,Z`(j))⊗Q` (n,`are prime to the characteristic ofF).Properties :

1. H_etⁱ_´ (X, µ^⊗_n^j) are finite, H_etⁱ_´ (X,Z`(j))areZ`-modules of finite type (resp. with X¯);H_etⁱ_´( ¯X,Z`) have no torsion for almost all

` (Gabber, difficult);

2. Hochschild-Serre spectral sequence relates X andX¯:

0→H¹(G,H_´etⁱ⁻¹( ¯X,Z`(j))→H<sub>et</sub><sup>i</sup><sub>´</sub>(X,Z`(j))→H_etⁱ_´( ¯X,Z`(j))^G →0

3. there is a cycle class mapCHⁱ(X)⊗Z` →H<sub>et´</sub><sup>2i</sup>(X,Z`(i)).

Cohomology

Notation : X¯ is the base change ofX to an algebraic closure ¯FofF. Étale cohomology groups : H_et´ⁱ (X,Z/`),H<sub>et</sub><sup>i</sup><sub>´</sub>(X, µ<sup>⊗</sup><sub>n</sub><sup>j</sup>),H<sub>et</sub><sup>i</sup><sub>´</sub> ( ¯X, µ<sup>⊗</sup><sub>n</sub><sup>j</sup>) H<sub>et</sub><sup>i</sup><sub>´</sub> (X,Z`(j)) =lim←−rH_et´ⁱ (X, µ^⊗_`r^j),

H_etⁱ_´ (X,Q`(j)) =H<sub>et</sub><sup>i</sup><sub>´</sub>(X,Z`(j))⊗Q` (n,`are prime to the characteristic ofF).

Properties :

1. H_etⁱ_´ (X, µ^⊗_n^j) are finite, H_etⁱ_´ (X,Z`(j))areZ`-modules of finite type (resp. with X¯);H_etⁱ_´( ¯X,Z`) have no torsion for almost all

` (Gabber, difficult);

2. Hochschild-Serre spectral sequence relates X andX¯:

0→H¹(G,H_´etⁱ⁻¹( ¯X,Z`(j))→H<sub>et</sub><sup>i</sup><sub>´</sub>(X,Z`(j))→H_etⁱ_´( ¯X,Z`(j))^G →0

3. there is a cycle class mapCHⁱ(X)⊗Z` →H<sub>et´</sub><sup>2i</sup>(X,Z`(i)).

Cohomology

Notation : X¯ is the base change ofX to an algebraic closure ¯FofF. Étale cohomology groups : H_et´ⁱ (X,Z/`),H<sub>et</sub><sup>i</sup><sub>´</sub>(X, µ<sup>⊗</sup><sub>n</sub><sup>j</sup>),H<sub>et</sub><sup>i</sup><sub>´</sub> ( ¯X, µ<sup>⊗</sup><sub>n</sub><sup>j</sup>) H<sub>et</sub><sup>i</sup><sub>´</sub> (X,Z`(j)) =lim←−rH_et´ⁱ (X, µ^⊗_`r^j),

H_etⁱ_´ (X,Q`(j)) =H<sub>et</sub><sup>i</sup><sub>´</sub>(X,Z`(j))⊗Q` (n,`are prime to the characteristic ofF).Properties :

1. H_etⁱ_´(X, µ^⊗_n^j) are finite, H_etⁱ_´ (X,Z`(j))areZ`-modules of finite type (resp. with X¯);H_etⁱ_´( ¯X,Z`) have no torsion for almost all

`(Gabber, difficult);

2. Hochschild-Serre spectral sequence relates X andX¯:

0→H¹(G,H_´etⁱ⁻¹( ¯X,Z`(j))→H<sub>et</sub><sup>i</sup><sub>´</sub>(X,Z`(j))→H_etⁱ_´( ¯X,Z`(j))^G →0

3. there is a cycle class mapCHⁱ(X)⊗Z` →H<sub>et´</sub><sup>2i</sup>(X,Z`(i)).

Cohomology

Notation : X¯ is the base change ofX to an algebraic closure ¯FofF. Étale cohomology groups : H_et´ⁱ (X,Z/`),H<sub>et</sub><sup>i</sup><sub>´</sub>(X, µ<sup>⊗</sup><sub>n</sub><sup>j</sup>),H<sub>et</sub><sup>i</sup><sub>´</sub> ( ¯X, µ<sup>⊗</sup><sub>n</sub><sup>j</sup>) H<sub>et</sub><sup>i</sup><sub>´</sub> (X,Z`(j)) =lim←−rH_et´ⁱ (X, µ^⊗_`r^j),

H_etⁱ_´ (X,Q`(j)) =H<sub>et</sub><sup>i</sup><sub>´</sub>(X,Z`(j))⊗Q` (n,`are prime to the characteristic ofF).Properties :

1. H_etⁱ_´(X, µ^⊗_n^j) are finite, H_etⁱ_´ (X,Z`(j))areZ`-modules of finite type (resp. with X¯);H_etⁱ_´( ¯X,Z`) have no torsion for almost all

`(Gabber, difficult);

2. Hochschild-Serre spectral sequence relates X andX¯:

0→H¹(G,H_´etⁱ⁻¹( ¯X,Z`(j))→H<sub>et</sub><sup>i</sup><sub>´</sub>(X,Z`(j))→H_´etⁱ ( ¯X,Z`(j))^G →0

3. there is a cycle class mapCHⁱ(X)⊗Z` →H<sub>et´</sub><sup>2i</sup>(X,Z`(i)).

Cohomology

Notation : X¯ is the base change ofX to an algebraic closure ¯FofF. Étale cohomology groups : H_et´ⁱ (X,Z/`),H<sub>et</sub><sup>i</sup><sub>´</sub>(X, µ<sup>⊗</sup><sub>n</sub><sup>j</sup>),H<sub>et</sub><sup>i</sup><sub>´</sub> ( ¯X, µ<sup>⊗</sup><sub>n</sub><sup>j</sup>) H<sub>et</sub><sup>i</sup><sub>´</sub> (X,Z`(j)) =lim←−rH_et´ⁱ (X, µ^⊗_`r^j),

H_etⁱ_´ (X,Q`(j)) =H<sub>et</sub><sup>i</sup><sub>´</sub>(X,Z`(j))⊗Q` (n,`are prime to the characteristic ofF).Properties :

1. H_etⁱ_´(X, µ^⊗_n^j) are finite, H_etⁱ_´ (X,Z`(j))areZ`-modules of finite type (resp. with X¯);H_etⁱ_´( ¯X,Z`) have no torsion for almost all

`(Gabber, difficult);

2. Hochschild-Serre spectral sequence relates X andX¯:

0→H¹(G,H_´etⁱ⁻¹( ¯X,Z`(j))→H<sub>et</sub><sup>i</sup><sub>´</sub>(X,Z`(j))→H_´etⁱ ( ¯X,Z`(j))^G →0

3. there is a cycle class mapCHⁱ(X)⊗Z` →H<sub>et´</sub><sup>2i</sup>(X,Z`(i)).

Computing cohomology

I H_et^2d_´ ( ¯X, µ^⊗_n^d)→^' Z/n;H_etⁱ_´ ( ¯X, µ^⊗n^j) =0,i >2n;H_etⁱ_´ ( ¯X, µ^⊗n^j) andH_et^2d−i_´ ( ¯X, µ^⊗n^(d−j)) are dual (resp. withQ`-coefficients).

I H_etⁱ_´ (Pⁿ¯F, µ^⊗r ^j) = (

µ^⊗j−

i

r 2 i even,i ≤2n 0 i otherwise.

I X ⊂Pⁿ is a hypersurface. Same formulas as above forX¯, but for i =d :

H_et^d_´( ¯X, µ^⊗_r ^j) =H_et^d_´ (Pⁿ¯F, µ^⊗_r ^j)⊕H_et´^d( ¯X, µ^⊗_r ^j)⁰, H_et^d_´ ( ¯X, µ^⊗_r ^j)⁰ is of HUGE rank ^{(deg X−1)d+2}_{deg X}^{+(−1)d(deg X−1)}.

Computing cohomology

I H_et^2d_´ ( ¯X, µ^⊗_n^d)→^' Z/n;H_etⁱ_´ ( ¯X, µ^⊗n^j) =0,i >2n;H_etⁱ_´ ( ¯X, µ^⊗n^j) andH_et^2d−i_´ ( ¯X, µ^⊗n^(d−j)) are dual (resp. withQ`-coefficients).

I H_etⁱ_´(Pⁿ¯F, µ^⊗r ^j) = (

µ^⊗j−

i

r 2 i even,i ≤2n 0 i otherwise.

I X ⊂Pⁿ is a hypersurface. Same formulas as above forX¯, but for i =d :

H_et^d_´( ¯X, µ^⊗_r ^j) =H_et^d_´ (Pⁿ¯F, µ^⊗_r ^j)⊕H_et´^d( ¯X, µ^⊗_r ^j)⁰, H_et^d_´ ( ¯X, µ^⊗_r ^j)⁰ is of HUGE rank ^{(deg X−1)d+2}_{deg X}^{+(−1)d(deg X−1)}.

Computing cohomology

I H_et^2d_´ ( ¯X, µ^⊗_n^d)→^' Z/n;H_etⁱ_´ ( ¯X, µ^⊗n^j) =0,i >2n;H_etⁱ_´ ( ¯X, µ^⊗n^j) andH_et^2d−i_´ ( ¯X, µ^⊗n^(d−j)) are dual (resp. withQ`-coefficients).

I H_etⁱ_´(Pⁿ¯F, µ^⊗r ^j) = (

µ^⊗j−

i

r 2 i even,i ≤2n 0 i otherwise.

I X ⊂Pⁿ is a hypersurface. Same formulas as above forX¯, but for i =d :

H_et^d_´( ¯X, µ^⊗_r ^j) =H_et^d_´ (Pⁿ¯F, µ^⊗_r ^j)⊕H_et´^d( ¯X, µ^⊗_r ^j)⁰, H_et^d_´( ¯X, µ^⊗_r ^j)⁰ is of HUGE rank ^{(deg X−1)d+2}_{deg X}^{+(−1)d(deg X−1)}.

Computing cohomology

In general :

Theorem (D.Madore and F. Orgogozo) There exists an algorithm which allows to compute the groups H_etⁱ_´( ¯X,Z/`) (so that the étale cohomology groups are computable in the sense of Church-Turing.)

Cycle class maps

Recall: we havecⁱ :CHⁱ(X)⊗Z` →H<sub>et´</sub><sup>2i</sup>(X,Z`(i)).

Other versions :

I tensoring withQ` : cⁱ

Q` :CH<sup>i</sup>(X)⊗Q`→H_et²ⁱ_´(X,Q`(i));

I geometric version : G =Gal(F/F) = ˆZ the absolute Galois group, generated by Frobenius

¯ c_Qⁱ

` :CH<sup>i</sup>(X)⊗Q`→H_et²ⁱ_´ ( ¯X,Q`(i))^G

I another geometric version :

cl_Qⁱ<sub>`</sub> :CH<sup>i</sup>( ¯X)⊗Q` →[

H_et´²ⁱ( ¯X,Q`(i))^H where the union is over all open subgroups H⊂G.

Cycle class maps

Recall: we havecⁱ :CHⁱ(X)⊗Z` →H<sub>et´</sub><sup>2i</sup>(X,Z`(i)).

Other versions :

I tensoring withQ` : cⁱ

Q` :CH<sup>i</sup>(X)⊗Q` →H_et´²ⁱ(X,Q`(i));

I geometric version : G =Gal(F/F) = ˆZ the absolute Galois group, generated by Frobenius

¯ c_Qⁱ

` :CH<sup>i</sup>(X)⊗Q`→H_et²ⁱ_´ ( ¯X,Q`(i))^G

I another geometric version :

cl_Qⁱ<sub>`</sub> :CH<sup>i</sup>( ¯X)⊗Q` →[

H_et´²ⁱ( ¯X,Q`(i))^H where the union is over all open subgroups H⊂G.

Cycle class maps

Recall: we havecⁱ :CHⁱ(X)⊗Z` →H<sub>et´</sub><sup>2i</sup>(X,Z`(i)).

Other versions :

I tensoring withQ` : cⁱ

Q` :CH<sup>i</sup>(X)⊗Q` →H_et´²ⁱ(X,Q`(i));

I geometric version : G =Gal(F/F) = ˆZ the absolute Galois group, generated by Frobenius

¯ c_Qⁱ

` :CH<sup>i</sup>(X)⊗Q`→H_et²ⁱ_´ ( ¯X,Q`(i))^G

I another geometric version :

cl_Qⁱ<sub>`</sub> :CH<sup>i</sup>( ¯X)⊗Q` →[

H_et´²ⁱ( ¯X,Q`(i))^H where the union is over all open subgroups H⊂G.

Cycle class maps

Recall: we havecⁱ :CHⁱ(X)⊗Z` →H<sub>et´</sub><sup>2i</sup>(X,Z`(i)).

Other versions :

I tensoring withQ` : cⁱ

Q` :CH<sup>i</sup>(X)⊗Q` →H_et´²ⁱ(X,Q`(i));

I geometric version : G =Gal(F/F) = ˆZ the absolute Galois group, generated by Frobenius

¯ c_Qⁱ

` :CH<sup>i</sup>(X)⊗Q`→H_et²ⁱ_´ ( ¯X,Q`(i))^G

I another geometric version :

cl_Qⁱ<sub>`</sub>:CH<sup>i</sup>( ¯X)⊗Q` →[

H_et´²ⁱ( ¯X,Q`(i))^H where the union is over all open subgroups H⊂G.

The cycle class maps allow to relate the Chow groups (defined in a more geometric way, but difficult to compute) and étale

cohomology groups (which are probably easier to understand.)

Conjecture (J. Tate) The cycle class map

¯ cⁱ

Q` :CH<sup>i</sup>(X)⊗Q`→H_et²ⁱ_´( ¯X,Q`(i))<sup>G</sup> is surjective (for any` andi).

Still widely open, even fori =1 (for divisors).

Integral versions: understand if we have the surjectivity with Z`-coefficients (counterexamples exist).

Remark: using Weil conjectures, one can show that the map H_et²ⁱ_´ (X,Q`(i))→H<sub>et</sub><sup>2i</sup><sub>´</sub> ( ¯X,Q`(i))^G is an isomorphism (in fact the kernelH¹(G,H_et²ⁱ_´⁻¹( ¯X,Z`(i))of the map with Z`-coefficients is finite). So that we can identifycⁱ

Q` andc¯ⁱ

Q`.

The cycle class maps allow to relate the Chow groups (defined in a more geometric way, but difficult to compute) and étale

cohomology groups (which are probably easier to understand.) Conjecture (J. Tate) The cycle class map

¯ cⁱ

Q` :CH<sup>i</sup>(X)⊗Q` →H_et´²ⁱ( ¯X,Q`(i))<sup>G</sup> is surjective (for any` andi).

Still widely open, even fori =1 (for divisors).

Integral versions: understand if we have the surjectivity with Z`-coefficients (counterexamples exist).

Remark: using Weil conjectures, one can show that the map H_et²ⁱ_´ (X,Q`(i))→H<sub>et</sub><sup>2i</sup><sub>´</sub> ( ¯X,Q`(i))^G is an isomorphism (in fact the kernelH¹(G,H_et²ⁱ_´⁻¹( ¯X,Z`(i))of the map with Z`-coefficients is finite). So that we can identifycⁱ

Q` andc¯ⁱ

Q`.

The cycle class maps allow to relate the Chow groups (defined in a more geometric way, but difficult to compute) and étale

cohomology groups (which are probably easier to understand.) Conjecture (J. Tate) The cycle class map

¯ cⁱ

Q` :CH<sup>i</sup>(X)⊗Q` →H_et´²ⁱ( ¯X,Q`(i))<sup>G</sup> is surjective (for any` andi).

Still widely open, even fori =1 (for divisors).

Integral versions: understand if we have the surjectivity with Z`-coefficients (counterexamples exist).

Remark: using Weil conjectures, one can show that the map H_et²ⁱ_´ (X,Q`(i))→H<sub>et</sub><sup>2i</sup><sub>´</sub> ( ¯X,Q`(i))^G is an isomorphism (in fact the kernelH¹(G,H_et²ⁱ_´⁻¹( ¯X,Z`(i))of the map with Z`-coefficients is finite). So that we can identifycⁱ

Q` andc¯ⁱ

Q`.

The cycle class maps allow to relate the Chow groups (defined in a more geometric way, but difficult to compute) and étale

cohomology groups (which are probably easier to understand.) Conjecture (J. Tate) The cycle class map

¯ cⁱ

Q` :CH<sup>i</sup>(X)⊗Q` →H_et´²ⁱ( ¯X,Q`(i))<sup>G</sup> is surjective (for any` andi).

Still widely open, even fori =1 (for divisors).

Integral versions: understand if we have the surjectivity with Z`-coefficients (counterexamples exist).

Remark: using Weil conjectures, one can show that the map H_et²ⁱ_´ (X,Q`(i))→H<sub>et</sub><sup>2i</sup><sub>´</sub>( ¯X,Q`(i))^G is an isomorphism (in fact the kernelH¹(G,H_et²ⁱ_´⁻¹( ¯X,Z`(i))of the map with Z`-coefficients is finite). So that we can identifycⁱ andc¯ⁱ .

More conjectures

I (follows from Bass conjecture) the Chow groupsCHⁱ(X) are of finite type;

I the kernel of Zⁱ(X)→H_et²ⁱ_´(X,Z`(i))consists of classes numerically equivalent to zero, i.e. having zero intersection with any cycle of complimentary dimension (Tate); with Q`-coefficients rational and numerical equivalence coincide (Beilinson conjecture), so thatcⁱ

Q` is also injective (conjecturally).

More conjectures

I (follows from Bass conjecture) the Chow groupsCHⁱ(X) are of finite type;

I the kernel of Zⁱ(X)→H_et²ⁱ_´(X,Z`(i))consists of classes numerically equivalent to zero, i.e. having zero intersection with any cycle of complimentary dimension (Tate); with Q`-coefficients rational and numerical equivalence coincide (Beilinson conjecture), so thatcⁱ

Q` is also injective (conjecturally).

More conjectures

I (follows from Bass conjecture) the Chow groupsCHⁱ(X) are of finite type;

I the kernel of Zⁱ(X)→H_et´²ⁱ(X,Z`(i))consists of classes numerically equivalent to zero, i.e. having zero intersection with any cycle of complimentary dimension (Tate); with Q`-coefficients rational and numerical equivalence coincide (Beilinson conjecture), so thatcⁱ

Q` is also injective (conjecturally).

Zeta functions

IfF=F_q is a finite field withq elements, define

Z(X,T) =exp(X

n≥1

|X(F_qⁿ)|Tⁿ n )

ζ(X,s) =Z(X,q^−s),

From Weil conjectures (proved by Deligne), the poles ofζ are on the linesRes =0,1. . .d.

Tate conjecture, the strong form ord_s=iζ(X,s) =−dim(Zⁱ(X)/∼_num)⊗Q.

Zeta functions

IfF=F_q is a finite field withq elements, define

Z(X,T) =exp(X

n≥1

|X(F_qⁿ)|Tⁿ n )

ζ(X,s) =Z(X,q^−s),

From Weil conjectures (proved by Deligne), the poles ofζ are on the linesRes =0,1. . .d.

Tate conjecture, the strong form ord_s=iζ(X,s) =−dim(Zⁱ(X)/∼_num)⊗Q.

The case of divisors

I One has an exact sequence

0→PicX ⊗Z` →H<sub>et</sub><sup>2</sup><sub>´</sub> (X,Z`(1))→Hom(Q`/Z`,BrX)→0 where the last group has NO torsion : it follows that

PicX ⊗Z`→H<sub>et</sub><sup>2</sup><sub>´</sub>(X,Z`(1))is surjective⇔

PicX ⊗Q`→H<sub>et</sub><sup>2</sup><sub>´</sub>(X,Q`(1))is surjective⇔ BrX is finite.

Zero-cycles

Theorem

(J.-L. Colliot-Thélène, J.-J. Sansuc, C.Soulé) The cycle class induces an isomorphism

CH^d(X)⊗Z`

→∼ H^2d(X,Z`(d)).

Torsion

I (J.-L. Colliot-Thélène, J.-J. Sansuc, C.Soulé) the torsion subgroup CH²(X)tors is finite and the map

CH²(X)tors →H⁴(X,Z`(2))is injective.

I could one have that the kernel of the map CHⁱ(X){`} →H<sup>2i</sup>(X,Z`(i))is nonzero?

Known cases of Tate’s conjecture

I Divisors (i =1) on abelian varieties, precise version:

Hom(A,B)⊗Z`→Hom<sub>Q`</sub>(T_{`</sub>(A),T<sub>`}(B)) (whereT<sub>`</sub>(A) =lim←−r A[`^r].)

I Note : over a finite field, there exist abelian varieties with

’exotic’ Tate classes : not coming (by cup-product) from H¹.

I K3 surfaces in caracteristic different from 2 (F. Charles, D. Maulik, K. Madapusi Pera), examples : X ⊂P³ a quartic;X a double cover w² =f6(x,y,z) withf6 of degree 6.

I some other specific varieties.

Known cases of Tate’s conjecture

I Divisors (i =1) on abelian varieties, precise version:

Hom(A,B)⊗Z`→Hom<sub>Q`</sub>(T_{`</sub>(A),T<sub>`}(B)) (whereT<sub>`</sub>(A) =lim←−r A[`^r].)

I Note : over a finite field, there exist abelian varieties with

’exotic’ Tate classes : not coming (by cup-product) from H¹.

I K3 surfaces in caracteristic different from 2 (F. Charles, D. Maulik, K. Madapusi Pera), examples : X ⊂P³ a quartic;X a double cover w² =f6(x,y,z) withf6 of degree 6.

I some other specific varieties.

Known cases of Tate’s conjecture

I Divisors (i =1) on abelian varieties, precise version:

Hom(A,B)⊗Z`→Hom<sub>Q`</sub>(T_{`</sub>(A),T<sub>`}(B)) (whereT<sub>`</sub>(A) =lim←−r A[`^r].)

I Note : over a finite field, there exist abelian varieties with

’exotic’ Tate classes : not coming (by cup-product) from H¹.

I K3 surfaces in caracteristic different from 2 (F. Charles, D.

Maulik, K. Madapusi Pera), examples : X ⊂P³ a quartic;X a double cover w² =f6(x,y,z)with f6 of degree 6.

I some other specific varieties.

Products

I Divisors (i =1) forX rationally dominated by products of abelian varieties and curves (in fact, Tate conjecture holds for i =1 onX ×Y iff it holds for i =1 for X and for Y).

I Example : a₀x₀ⁿ+a₁x₁ⁿ+a₂x₂ⁿ+a₃x₃ⁿ=0 is dominated by the product of a0x₀ⁿ+a1x₁ⁿ=yⁿ anda2x₂ⁿ+a3x₃ⁿ=zⁿ.

I Remark 1: for curves Pic(X×Y) =Pic(X)⊕PicY⊕Hom(JX,JY)

(similar formula in higher dimension).

I Remark 2, reductions : if E,E⁰ are two elliptic curves over a number fieldk, then there are infinitely many places where the reductions ofE andE⁰ are geometrically isogeneous (F. Charles). In particular, for a given elliptic curve E overk either E is supersingular at infinitely many places, or has complex multiplication at inifinitely many places.

Products

I Divisors (i =1) forX rationally dominated by products of abelian varieties and curves (in fact, Tate conjecture holds for i =1 onX ×Y iff it holds for i =1 for X and for Y).

I Example : a0x₀ⁿ+a1x₁ⁿ+a2x₂ⁿ+a3x₃ⁿ=0 is dominated by the product of a0x₀ⁿ+a1x₁ⁿ=yⁿ anda2x₂ⁿ+a3x₃ⁿ=zⁿ.

I Remark 1: for curves Pic(X×Y) =Pic(X)⊕PicY⊕Hom(JX,JY)

(similar formula in higher dimension).

I Remark 2, reductions : if E,E⁰ are two elliptic curves over a number fieldk, then there are infinitely many places where the reductions ofE andE⁰ are geometrically isogeneous (F. Charles). In particular, for a given elliptic curve E overk either E is supersingular at infinitely many places, or has complex multiplication at inifinitely many places.

Products

I Divisors (i =1) forX rationally dominated by products of abelian varieties and curves (in fact, Tate conjecture holds for i =1 onX ×Y iff it holds for i =1 for X and for Y).

I Example : a0x₀ⁿ+a1x₁ⁿ+a2x₂ⁿ+a3x₃ⁿ=0 is dominated by the product of a0x₀ⁿ+a1x₁ⁿ=yⁿ anda2x₂ⁿ+a3x₃ⁿ=zⁿ.

I Remark 1: for curves Pic(X×Y) =Pic(X)⊕PicY⊕Hom(JX,JY)

(similar formula in higher dimension).

I Remark 2, reductions : if E,E⁰ are two elliptic curves over a number fieldk, then there are infinitely many places where the reductions ofE andE⁰ are geometrically isogeneous (F. Charles). In particular, for a given elliptic curve E overk either E is supersingular at infinitely many places, or has complex multiplication at inifinitely many places.

Products

I Divisors (i =1) forX rationally dominated by products of abelian varieties and curves (in fact, Tate conjecture holds for i =1 onX ×Y iff it holds for i =1 for X and for Y).

I Example : a0x₀ⁿ+a1x₁ⁿ+a2x₂ⁿ+a3x₃ⁿ=0 is dominated by the product of a0x₀ⁿ+a1x₁ⁿ=yⁿ anda2x₂ⁿ+a3x₃ⁿ=zⁿ.

I Remark 1: for curves Pic(X×Y) =Pic(X)⊕PicY⊕Hom(JX,JY)

(similar formula in higher dimension).

I Remark 2, reductions : if E,E⁰ are two elliptic curves over a number fieldk, then there are infinitely many places where the reductions ofE andE⁰ are geometrically isogeneous (F.

Charles).

In particular, for a given elliptic curve E overk either E is supersingular at infinitely many places, or has complex multiplication at inifinitely many places.

Products

I Divisors (i =1) forX rationally dominated by products of abelian varieties and curves (in fact, Tate conjecture holds for i =1 onX ×Y iff it holds for i =1 for X and for Y).

I Example : a0x₀ⁿ+a1x₁ⁿ+a2x₂ⁿ+a3x₃ⁿ=0 is dominated by the product of a0x₀ⁿ+a1x₁ⁿ=yⁿ anda2x₂ⁿ+a3x₃ⁿ=zⁿ.

I Remark 1: for curves Pic(X×Y) =Pic(X)⊕PicY⊕Hom(JX,JY)

(similar formula in higher dimension).

I Remark 2, reductions : if E,E⁰ are two elliptic curves over a number fieldk, then there are infinitely many places where the reductions ofE andE⁰ are geometrically isogeneous (F.

Charles). In particular, for a given elliptic curve E overk either E is supersingular at infinitely many places, or has complex multiplication at inifinitely many places.

Integral versions

Goal : understand the surjectivity of

I cⁱ :CHⁱ(X)⊗Z`→H<sub>et</sub><sup>2i</sup><sub>´</sub>(X,Z`(i)).

I c¯ⁱ :CHⁱ(X)⊗Z`→H<sub>et</sub><sup>2i</sup><sub>´</sub>( ¯X,Z`(i))^G

I clⁱ :CHⁱ( ¯X)⊗Z`→S

H_et²ⁱ_´ ( ¯X,Z`(i))^H,where the union is over all open subgroups H⊂G.

None of these maps need be surjective!

Integral versions

Goal : understand the surjectivity of

I cⁱ :CHⁱ(X)⊗Z`→H<sub>et</sub><sup>2i</sup><sub>´</sub>(X,Z`(i)).

I c¯ⁱ :CHⁱ(X)⊗Z`→H<sub>et</sub><sup>2i</sup><sub>´</sub>( ¯X,Z`(i))^G

I clⁱ :CHⁱ( ¯X)⊗Z`→S

H_et²ⁱ_´ ( ¯X,Z`(i))^H,where the union is over all open subgroups H⊂G.

None of these maps need be surjective!