Conjecture of Colliot-Thélèneon zero-cycles over local fields
Shuji Saito
December 7, 2007, Luminy

1

2

k : a field,
X/k : projective smooth variety,CH0(X) =!

Mx∈XZ/∼rat. equiv.

closed point

Chow group of zero-cycles on X modulo rational equivalence.

deg!

A0(X) = Ker CH0(X) −→ Z

Chow group of zero-cycles of degree zero

albX : A0(X) → AlbX/k (k)Albanese map

3Structure of AlbX/k (k) :A/k : abelian variety,◦ k = C ⇒ A(k) = Cd/Γ (Γ lattice)◦ [k : Q] < ∞ ⇒ A(k) = Zr L (finite)(Mordell-Weil)◦ [k : Qp] < ∞ ⇒ A(k) = Zrp L (finite)(Mattuck)4

Known facts on Ker(albX ) :dim(X) = 1 ⇒ Ker(albX ) = 0 (Abel)
H 0(X, Ω2X ) 6= 0 ⇒ dim Ker(albX ) = ∞(Mumford)
Bloch-Beilinson Conjecture :[k : Q] < ∞ ⇒ Ker(albX ) finite ??

Structure of Ker(albX ) depends onarithmetic nature of k

How about the case [k : Qp] < ∞?

5[k : Qp] < ∞Ok : ring of integers in k
F = Ok /mk : residue fieldp = ch(F ): residue characteristic
Question of Colliot-Thélène :(CT 1) Ker(albX ) = D(X)M(finite)??

D(X) : maximal divisible subgroup of A0(X).

(CT 2)

D(X)tor = 0 ??

(CT 1) + (CT 2) ⇒ A0(X)tor finite

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Affirmative results :◦ surfaces (Saito-Sujatha)◦ Varieties fibered over curves in quadrics(Colliot-Thélène/Skorobogatov, Parimala/Suresh)◦ Varieties fibered over curves in Severi-Brauervarieties (Salberger)◦ Products of curves (Raskind-Spiess)

Theorem (Colliot-Thélène)
X : smooth compactification of connected
linear algebraic group over k. ThenA0(X) = (finite)M

p-primary torsionof finite exponent

Theorem (Colliot-Thélène/Raskind, Salberger)
H 0(X, OX ) = 0 ⇒ CH2(X)tor finite

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Theorem (Saito-Sato) Assume(∗) ∃X /Ok , regular projective model of X
s.t. Y := (X ×Ok Ok /mk )red is a SNCD on X .

⇒A0(X) = D0(X) (f inite)M

D0(X): maximal p0-divisible subgroup of A0(X)(p0-divisible ⇔ divisible by ∀n prime to p)
Remark: The conlucsion of the theorem would
be implied by (CT 1) and Mattuck’s theorem:
MAlbX (k) = Zrp (finite)8

Corollary 1 ((∗) not used)
A0(X) is `-divisible for almost all primes `.
Corollary 2
Assume (∗) and X rationally connected.
⇒A0(X) = (finite)

M









p-primary torsionof finite exponent

Note the exact sequence:
CH1(Y ) → CH1(X ) → CH0(X) → 0|CH1(X )/n| < ∞ ⇒ |CH0(X)/n| < ∞9R : henselian discrete valuation ring s.t.
F = R/mR : finite or separably closed
X : regular projective flat over R s.t.
Y := (X ×R R/mR)red : SNCD on X .
d = dim(X /R) : relative dimension
Theorem (Saito-Sato)'2d(X , Z/nZ(d))
CH1(X )/n −→Hét(∀n prime to p)

Corollary Assume R/mR finite.
CH1(X ) = D(X ) ⊕ (finite)
D(X ) : maximal p0-divisible subgroup of CH1(X )10

Key ingredients in the proof:◦ Formalism of cycle class map in étale homology for Sch/R◦ Absolute purity (Gabber)◦ Affine Lefschetz theorem (Gabber)◦ Theorem of Deligne (Weil conjecture)◦ Bertini’s theorem for Sch/R (Jannsen-Saito)◦ Classical results on Brauer groups of arithmetic surfaces (Grothendieck, Artin)◦ Theorem of Merkur’ev-Suslin

11Theorem For any prime ` 6= p,'2d−1(X , Q /Z (d))
CH1(X ){`} −→Hét` `(Analogue of Roitman’s theorem and geometric class field theory)
Corollary Assume R/mR finite.
CH1(X ) = D(X ) ⊕ (finite)
D(X ) uniquely p0-divisible⇔ n : D(X ) ' D(X ) for any n prime to p
In particular
CH1(X ){p0} =M`6=pCH1(X ){`} is finite12

Key ingredient of the proof:Theorem (Hironaka, Cossart-Jannsen-Saito)(RES)2 holds.
t ≥ 1 : an integer(RES)t : For given
Z : excellent regular scheme,
B : SNCD on Z,
W ⊂ Z: reduced closed of dimension≤ t,∃ : Z ← Z1 ← · · · ← Zm : sequence ofblowups s.t.
the centers of Zi+1 → Zi are regular andnormal crossing with the complete transforms
Bi of B in Zi,
the strict transform of W in Zm is regularand normal crossing with Bm

13CH1(Y ) → CH1(X ) → CH0(X) → 0
Infinite torsion in CH0(X):Fix:[k : Qp] < ∞,d ≥ 0,
X = {F (z) = 0} ⊂ P3k = Proj(k[z])F (z) = aI z I ∈ k[z] = k[z0, z1, z2, z3]
XI(homogeneous polynomial of degree d)iiii(z I = z00 z11 z22 z33 ,I = (i0, i1, i2, i3))

14Theorem (Asakura-Saito) Assume◦ X is generic, i.e.
{aI }I algebraically independent over Q◦ ∃X /Ok : projective smooth model of X◦d≥5'Mr−1⇒ CH0(X){`} −→ (Q`/Z`)(finite)r = dim(CH 1(Y ) ⊗ Q)
Y = X ×Ok Ok /mk15

Bloch’s exact sequence :3 (X, Q /Z (2))
0 → CH2(X, 1)⊗Q`/Z` → N Hét` `→ CH0(X){`} → 0⇒ (∗) CH0(X){`} ' (Q`/Z`)sM(finite)Localization sequence:∂CH2(X, 1) −→CH1(Y )→ CH1(X ) → CH0(X) → 0⇒CH0(X){`} → Coker(∂) ⊗ Q`/Z`ι→ CH1(X )⊗Q`/Z` −→CH0(X)⊗Q`/Z` → 0
Theorem of Saito and Sato ⇒ Ker(ι) finite⇒ s ≥ dim(Coker(∂) ⊗ Q) in (∗)

16Commutative diagram(Langer-Saito in case ` = p)
ρX2
CH (X, 1) −−→yy∂

CH1(Y )

Hg1(Gk , V )α−→H 1(Gk , V )/Hf1 (Gk , V )Gk = Gal(k/k),
2 (X , Q (2)),
V = Hét`k

Hf1 ⊂ Hg1 ⊂ H 1(Gk , V ) : subspaces introduced by Bloch-KatoρX : CH2(X, 1) → H 1(Gk , V )`-adic regulator map

Key fact: α is injective

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Put:
2 (X , Q (2))/[H ∩ X] ⊗ Q (1)Vprim = Hét``kH ⊂ P3k hyperplane
2 (X , Q (1))[H ∩ X] ∈ Hét`k

ρX,prim : CH2(X, 1) ⊗ Q` → H 1(Gk , V )→ H 1(Gk , Vprim)
Theorem
X ⊂ P3k : generic of degree d ≥ 5⇒ ρX,prim = 0

18(d+33 )PQ⊃ M : moduli of smooth surfaces ofdegree d in P3/Q
XM → M : universal family.
X generic ⇒ ∃t : Spec(k) → M s.t.
◦ t : dominant, i.e. Q(M ) ,→ k,◦ X ' XM ×M Spec(k) (base change via t)2(X , 1)
CH 2(X, 1) = lim
CHS−→(S,tS )tS

Spec(k) −→ St & ↓ smoothM
XS := XM ×M S

19Commutative diagram
ρét
X ,primS
1 (S, V ét−−→ Hét
CH2(XS , 1) −−−S,prim)


y




y

CH2(X, 1)

ρX,prim−−−−→ H 1(Gk , Vprim)2(X , 1) → H 1 (S, V ét
ρét
:CHS
XS ,primétS,prim)

fS : XS := XM ×M S → Sét
VS,prim= R2(fS )∗Q`(2))prim(smooth Q`-sheaf on Sét)
Reduced to showing ρét
XS ,prim = 0

20First key step : Kill Gal(Q/Q).
Put SQ = S ×Q Q.
Theoremétét
H 0(S, VS,prim) = (VS,prim)π1(SQ) = 0.
Proof: monodromy argument, or Lefschetz
principle + infinitesimal method in Hodge theory
Corollary
1 (S, V ét
1 (S , V ét
Hét),→HS,primét Q S,prim)

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Commutative diagram
ρét
X ,prim

S−−→CH2(XS , 1) −−−

1 (S , V étHétQ S,prim)yy'ρan
X ,primS
1 (S(C), V an−−→ Han
CH2(XSC , 1) −−−S,prim) ⊗ Q`

fSC : XSC = X ×S SC → SC = S ×Q Can
VS,prim= R2(fSC )an∗ Q(2) prim!

(local system on S(C))
2(X , 1) → H 1 (S(C), V an
ρan:CHSCXS ,primanS,prim)

analytic regulator map
Reduced to showing ρan
XS ,prim = 0.

22Theory of mixed Hodge modules (M. Saito) :1 (S(C), V an
◦ H := Han
S,prim) ∈ M HS2H)⊂F◦ Image(ρanC
XS ,prim(Hodge filtration on HC = H ⊗ C)

Reduced to showing:
Lemma(1) F 2HC ,→ cohomolgy of de Rham complex:∇2 (X /S)F 2HdR⊗O−→
primSSC∇2 (X /S)
1
F 1HdR
⊗
Ω−→
prim
S
SC/C

2 (X /S)
2HdR
⊗
Ω
prim
S
SC/C.

(2) The complex is exact.

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Ingredients of proof :(1)◦ Degeneration of Hodge spectral sequencewith coefficients in local systems◦ Deligne’s GAGA for vector bundles with integrable connections with regular singularities(2)
Infinitesimal method in Hodge theory(d = deg(X) ≥ 5 is used)
Open Problem :How about the case d = 4 (K3 surfaces) ?

24Implication to Bloch-Kato conjecture
X/k : projective smooth over k[k : Q] < ∞ or [k : Qp] < ∞.
r,q

ρX : CHr (X, q) ⊗ Q` → H 1(Gk , V )`-adic regulator map
2r−q−1

V = Hét(Xk , Q`(r))

p-adic Hodge theory ⇒r,qImage(ρX ) ⊂ Hg1(Gk , V )(Faltings, Hyodo-Kato, Tsuji, Yamashita)
Conjecture of Bloch-Kato(part of Tamagawa number conjecture):r,q[k : Q] < ∞ ⇒ Image(ρX ) = Hg1(Gk , V )25

Recall:H 1(Gk , V ) = Ext1R`(Gk )(Q`, V )R`(Gk ) : category of finite dimensional Q`spaces with continuous Gk -actions[0 → V → E → Q` → 0] ∈ Hg1(Gk , V )⇔ E : de Rham representation of Gk[0 → V → E → Q` → 0] ∈ Hf1 (Gk , V )⇔ E : crystalline representation of Gk
Heuristically BK conjecture implies :E : de Rham ⇒ realization of mixed motive
Corollary
Variant of BK conjecture over p-adic field is
FALSE.