Cohomological Hasse principle, finiteness of motivic cohomology, special values of zeta-functions

斎藤秀司 (東大数理)

Class number formula

K: number field, ζ_{K}(s): Dedekind zeta function of K

lim_{s→0} ζ_{K}(s) · s^{−ρ_{0}} = − \frac{|Cl(K)| · R_{K}}{|(O_{K}^{×})_{tors} |}

ρ_{0} := ord_{s=0} ζ_{K}(s) = rank (O_{K}^{×}) Cl(K) : ideal class group of K R_{K} : Dirichlet’s regulator View CNF as arithmetic index theorem?

index (analytic invariant) = characteristic class (e.g. Euler-Poincaré char.)

Motivic cohomology

X: scheme ⇝ H_{M}^{i}(X, ℤ(r)) (i, r ∈ ℤ_{≥0})

For X = Spec(O_{K}),

Cl(K) = H_{M}^{2}(X, ℤ(1)), O_{K}^{×} = H_{M}^{1}(X, ℤ(1))

Motivic cohomology

X: regular of finite type over field or

Dedekind ring CH^{r}(X, q) : Bloch’s higher Chow group Relation to Motivic cohomology :

CH^{r}(X, q) = H_{M}^{2r−q}(X, ℤ(r))

Relation to algebraic K-theory : E_{2}^{p,q} = CH^{−q/2}(X, −p − q) ⇒ K_{−p−q}(X)

Finiteness Conjecture

X: regular of finite type over 𝔽_{p} or ℤ ⇒ CH^{r}(X, q) is finitely generated for all q, r. Generalize finiteness results on

Cl(K) = CH^{1}(X) and O_{K}^{×} = CH^{1}(X, 1) for X = Spec(O_{K}) with number field K.

Review on homology in toplogy

X: topological space ⇒ H_{q}(X, ℤ) := H_{q}(z(X, •))

Singular complex z(X, •):

⋯ → z(X, q) ⟶^{∂} z(X, q − 1) ⟶^{∂} ⋯ → z(X, 0)

z(X, q) = ⨁_{Γ} ℤ[Γ] , Γ : Δ_{top}^{q} → X continuous map

Δ_{top}^{q} = {(x_{0}, x_{1}, …, x_{q}) ∈ ℝ^{q+1} | ∑_{0≤i≤q}x_{i} = 1, x_{i} ≥ 0}

∂: alternating sum of restrictions to faces

A quick review on CH^{r}(X, q)

z^{r}(X, q) = ⨁_{Γ⊂X×Δ^{q}}ℤ[Γ]

Γ ⊂ X × Δ^{q} closed integral of codimension r intersecting all faces Δ^{s} ⊂ Δ^{q} properly

Δ^{q} = Spec(ℤ[t_{0}, …, t_{q}]/(∑_{i=0}^{q} t_{i} − 1))

face : Δ^{s} = {t_{i_{1}} = ⋯ = t_{i_{q−s}} = 0} ⊂ Δ^{q}

Bloch’s cycle complex z^{r}(X, •)

⋯ → z^{r}(X, q) ⟶^{∂} z^{r}(X, q − 1) → ⋯

⋯ → z^{r}(X, 1) ⟶^{∂} z^{r}(X, 0),

∂ = alternating sum of pull-back maps to faces of codimension 1

Definition CH^{r}(X, q) := H_{q}(z^{r}(X, •))

Fact : CH^{r}(X, 0) = CH^{r}(X): Chow group Remark : z^{r}(X, q) of infinite rank

Known results on Finiteness Conjecture

X regular of finite type over 𝔽_{p} or ℤ

Theorem 1 (Minkowski, Dirichlet, Mordell-Weil)

CH^{1}(X, q) is finitely generated

CH^{1}(X, q) = \{ \begin{matrix}Pic(X) \\ Γ(X, O_{X}^{×}) \\ 0 \end{matrix} \begin{matrix}q = 0 \\ q = 1 \\ q > 1 \end{matrix}

Theorem 2 (Quillen (1972)) CH^{r}(X, q) is finitely generated if dim (X) = 1 Theorem 3 (Bloch, Kato-S.(1985)) CH_{0}(X) = CH^{d}(X) is finitely generated (d = dim (X)).

Observation : These results obtained via

regulator maps or cycle maps

e.g. higher reciprocity map

ρ_{X} : CH^{d}(X) = CH_{0}(X) → π_{1}^{ab}(X)

π_{1}^{ab}(X) : abelian fundamental group of X Higher dimensional class field theory

Theorem (Kato-S., Colliot-Thélène-Sansuc-Soulé)

X: regular proper over 𝔽_{p} or ℤ ⇒ ρ_{X} : CH_{0}(X) → π_{1}^{ab}(X) ‘almost isomorphism’

X = Spec(O_{K}), [K : ℚ] < ∞ ⇒ CH_{0}(X) = Cl(K) (Hilbert class field theory) Finiteness result of CH_{0}(X) follows from “Finiteness result of π_{1}^{ab}(X)” (Katz-Lang)

Generalization étale cycle map ρ_{X}^{r,q} : CH^{r}(X, q; ℤ/nℤ) → H_{ét}^{2r−q}(X, ℤ/nℤ(r))

(Geisser-Levine, Levine, K.Sato)

CH^{r}(X, q; ℤ/nℤ) = H_{q}(z^{r}(X, •) ⊗ ℤ/nℤ)

Higher Chow group with finite coefficient 0 → CH^{r}(X, q)/n → CH^{r}(X, q; ℤ/nℤ) →

CH^{r}(X, q + 1)[n] → 0

étale cohomology

If n invertible on X,

H_{ét}^{t} (X, ℤ/nℤ(r)) = H_{ét}^{t} (X, μ_{n}^{⊗r})

μ_{n} : étale sheaf of m-th roots of unity If n not invert on X, need special care in char p

Theorem (M.Artin, Milne, Sato) X regular of finite type over 𝔽_{p} or ℤ

⇒ |H_{ét}^{t} (X, ℤ/nℤ(r))| < ∞

Theorem (Suslin-Voevodsky, Geisser-Levine)

X regular (+ technical condition)

⇒ ρ_{X}^{r,q} : CH^{r}(X, q; ℤ/nℤ) → H_{ét}^{2r−q}(X, ℤ/nℤ(r)) isomorphism if r ≤ q, injective if r = q + 1. ⇒ |CH^{r}(X, q; ℤ/nℤ)| < ∞ if r ≤ q + 1

if X of finite type over 𝔽_{p} or ℤ Proof: Use the Bloch-Kato conjecture proved by Rost and Voevodsky ρ_{X}^{r,q} for r ≥ d = dim (X) Theorem 1 (Jannsen-S.)

X: smooth projective over 𝔽 For r ≥ d = dim (X),

⇒ ρ_{X}^{r,q} : CH^{r}(X, q; ℤ/nℤ) ⟶^{≅} H_{ét}^{2r−q}(X, ℤ/nℤ(r))

if q ≤ 2 or (RES) _{q,𝔽} holds

Resolution of singularities

(RES) _{q,S}: (S base scheme) Z: regular of fintie type over S, W ⊂ Z: integral closed of dimension ≤ q, ⇒ ∃π : Z′ → Z: birational morphism s.t.

• Z′ regular and π^{−1}(Z − W) ≃ Z − W,

• W′ = π^{−1}(W)_{red} ⊂ Z′ is SNCD Theorem (Hironaka, Cossart-Jannsen-S.)

(RES) _{q,S} holds if q ≤ 2 and S is excellent.

Theorem 2 (Kerz-S.) X: smooth projective over 𝔽 For r ≥ d = dim (X),

⇒ ρ_{X}^{r,q} : CH^{r}(X, q; ℤ/nℤ) ⟶^{≅} H_{ét}^{2r−q}(X, ℤ/nℤ(r))

if n is prime to p = ch(𝔽)

Corollary

X: quasi-projective over 𝔽 For r ≥ d = dim (X), |CH^{r}(X, q; ℤ/nℤ)| < ∞ if p |̸n

Theorem 3 (Kerz-S.) X: regular projective over Spec(O_{k}), [k : ℚ_{p}] < ∞

quasi-semistable

(i.e. reduced closed fiber is SNCD on X) For r ≥ d = dim (X),

ρ_{X}^{r,q} : CH^{r}(X, q; ℤ/nℤ) ⟶^{≅} H_{ét}^{2r−q}(X, ℤ/nℤ(r))

if n is prime to p.

Gabber’s refined alteration

Theorem (Gabber) Z: smooth over perfect field F W ⊂ Z: closed, nowhere dense ℓ: prime≠ ch(F) ⇒ ∃π : Z′ → Z proper surjective s.t.

• Z′ smooth over F

• W′ = π^{−1}(W)_{red} ⊂ Z′ is SNCD

• π is generically finite of degree prime to ℓ

高次元類体論の高次化

X: smooth projective over 𝔽 of d = dim (X) For n prime to p = ch𝔽),

ρ_{X}^{d,q} : CH^{d}(X, q; ℤ/nℤ) ⟶^{≅} H^{2d−q}(X_{ét} , ℤ/nℤ(d))

Case q = 0:

CH^{d}(X, 0; ℤ/nℤ) ⟶^{≃} H^{2d}(X_{ét} , ℤ/nℤ(d))
∥ ↓^{≃}
CH_{0}(X)/n  π_{1}^{ab}(X)/n

⇒ higher dimensional unramified CFT

Tame higher dimensional CFT

Theorem (Schmidt-Spiess) For U smooth over ch(𝔽) and n prime to ch(𝔽),

H_{0}^{S}(U, ℤ/nℤ) ⟶^{≅} π_{1}^{ab}(U)/n

H_{i}^{S}(U, ℤ) (i ≥ 0): Suslin homology of U U smooth projective over 𝔽 ⇒

H_{i}^{S}(U, ℤ) ≃ CH^{d}(X, d − i) (d = dim (U)) dim (U) = 1 ⇒ H_{0}^{S}(U, ℤ/nℤ) =

Coker(𝔽(U)^{×} → ⨁_{x∈U}ℤ ⊕ ⨁_{v∈U_{∞}}𝔽(U)_{v}^{×})/n

U_{∞} = {points at infinity of compactification of U }

dim (U) > 1 ⇒ H_{0}^{S}(U, ℤ/nℤ) =

Coker( ⨁_{C⊂U}𝔽(C)^{×} → ⨁_{x∈U}ℤ ⊕ ⨁_{C⊂U}( ⨁_{v∈C_{∞}}𝔽(C)_{v}^{×}))/n

(C runs over all curves on U)

Higher higher CFT Theorem

For U smooth over ch(𝔽) and n prime to ch(𝔽),

H_{i}^{S}(U, ℤ/nℤ) ⟶^{≅} H_{c}^{2d−i}(U_{ét} , ℤ/nℤ(d))

Note

π_{1}^{ab}(U)/n ≃ H_{c}^{2d}(U_{ét} , ℤ/nℤ(d))

(étale cohomology with compact support) Case i = 0 ⇒ Theorem of Schmidt-Spiess

Zeta functions

X: schemes of finite type over 𝔽_{p} or ℤ

ζ(X, s) = ∏_{x∈X_{(0)}}\frac{1}{1 − N(x)^{−s}} (s ∈ ℂ)

X_{(0)}: set of closed points of X N(x) = |κ(x)| (κ(x): residue field of x) Conjecturally, ζ(X, s) mormorphic on ℂ True if X is of finite type over 𝔽_{p}

Class number formula

X = Spec(O_{K}) with [K : ℚ] < ∞

⇒ lim_{s→0} ζ(X, s) · s^{−ρ_{0}} = −\frac{|CH^{1}(X)_{tors} |}{|CH^{1}(X, 1)_{tors} |} · R_{K}

ρ_{0} := ord_{s=0} ζ_{K}(s) = rank (CH^{1}(X, 1)) CH^{1}(X) = Cl(K) : ideal class group of K CH^{1}(X, 1) = O_{K}^{×} : unit group of O_{K} R_{K} : Dirichlet’s regulator

Special values via motivic cohomology

X smooth projective over 𝔽_{q}. For r ∈ ℤ,

ζ(X, r)^{∗} := lim_{s→r} ζ(X, s) · (1 − q^{r−s})^{ρ_{r}}

ρ_{r} := −ord_{s=r}ζ(X, s) ρ_{r} = multiplicity of (1 − q^{r}t) in

Fact: ζ(X, r)^{∗} ∈ ℚ^{×}

Theorem

X: smooth projective over 𝔽 with p = ch(𝔽) ℓ prime≠ p, d = dim (X) ⇒ |CH^{d}(X, i){ℓ}| < ∞ for all i,

ℓ-part of ζ(X, 0)^{∗} = ∏_{0≤i≤2d}|CH^{d}(X, i){ℓ}|^{(−1)^{i}}

It holds even for ℓ = p if d = dim (X) ≤ 4 How to show For r ≥ d = dim (X),

ρ_{X}^{r,q} : CH^{r}(X, q; ℤ/nℤ) ⟶^{≅} H_{ét}^{2r−q}(X, ℤ/nℤ(r))

Case r > d = dim (X) is reduced to case r = d by projective bundle formula (replace X by ℙ_{X}^{r−d} = X × ℙ_{ℤ}^{r−d}) Reduced to show case r = d = dim (X)

ρ_{X}^{d,q} : CH^{d}(X, q; ℤ/nℤ) ⟶^{≅} H_{ét}^{2d−q}(X, ℤ/nℤ(r))

Proof in case d := dim (X) = 1 X: projective smooth curve over 𝔽 or X = Spec(O_{k}) for number field k

ρ_{X}^{1,0} : CH^{1}(X)/n → H_{ét}^{2} (X, ℤ/nℤ(1)) Lemma ρ_{X}^{1,0} isomorphism ⇔

Br(k)[n] ↪ ⨁_{x∈X_{(0)}}Br(k_{x})[n]

X_{(0)}: set of the closed points of X k_{x}: completion of k = k(X) at x ∈ X_{(0)}

Hasse principle for Br(k) ⇒ ρ_{X}^{1,0} isom.

Hasse principle for Br(k) k: number field or function field of curve over finite field P (k): set of all places of k k_{v}: completion of k at v ∈ P (k) A: central simple algebra over k

A ≃ M_{n}(k) ⇔ A ⊗_{k} k_{v} ≃ M_{n}(k_{v}) for ∀v ∈ P (k)

A = (\frac{a,b}{k}) (a, b ∈ k) ⇒ Global-local principle for X^{2} − aY^{2} − bZ^{2} + abW^{2} = 0 over k

Cohomological Hasse principle

X of finite type over 𝔽 or ℤ or ℤ_{p} Kato complex KC_{•}(X, ℤ/nℤ) =

⋯ → ⨁_{x∈X_{(a)}}H^{a+1}(x, ℤ/nℤ(a)) → ⨁_{x∈X_{(a−1)}}H^{a}(x, ℤ/nℤ(a−1)) → ⋯

⋯ → ⨁_{x∈X_{(1)}}H^{2}(x, ℤ/nℤ(1)) → ⨁_{x∈X_{(0)}}H^{1}(x, ℤ/nℤ)

X_{(a)} = \{x ∈ X | dim \overline{\{x\}} = a\} X: projective smooth curve over 𝔽 or X = Spec(O_{k}) for number field k

η ∈ X_{(1)}: generic point of X

⇒ KC_{•}(X, ℤ/nℤ) =

H^{2}(η, ℤ/nℤ(1)) ⟶ ⨁_{x∈X_{(0)}}H^{1}(x, ℤ/nℤ)
∥  _{residue} ↑^{≃}
Br(k)[n]  ⟶ ⨁_{x∈X_{(0)}}Br(k_{x})[n]

H_{1}(KC_{•}(X, ℤ/nℤ)) = 0 ⇔ H.P. for Br(k)

Kato homology

KH_{a}(X, ℤ/nℤ) = H_{a}(KC_{•}(X, ℤ/nℤ)) (a ≥ 0)

Conjecture (Kato’s Hasse principle) X: smooth proper over 𝔽, or X: regular proper flat over Spec(O_{k}), where k: number field or 𝔭-adic field

⇒ K̃H_{a}(X, ℤ/nℤ) = 0 for a > 0.

K̃H_{a}(X) = modification of KH_{a}(X) K̃H_{a}(X) = KH_{a}(X) unless ∃Spec(ℝ) → X & 2|n

Kato’s HS and motivic cohomology Lemma We have exact sequence: KH_{q+2}(X, ℤ/nℤ) →

CH^{d}(X, q; ℤ/nℤ) ⟶^{ρ_{X}^{d,q}} H_{ét}^{2d−q}(X, ℤ/nℤ(d))

→ KH_{q+1}(X, ℤ/nℤ)

(d = dim (X))

Known cases of Kato conjecture

• dim (X) = 1: Classical (H.P. for Brauer group)

• dim (X) = 2: Kato (1986)

Theorem (Colliot-Thélène (1993), Suwa (1995))

X: smooth projective over Spec(𝔽), ⇒ KH_{a}(X, ℚ/ℤ) = 0 for 0 < a ≤ 3

KH_{a}(X, ℚ/ℤ) = lim_{⟶_{n}} KH_{a}(X, ℤ/nℤ)

Theorem 3 (Jannsen-S. (2006)) X: smooth projective over 𝔽

⇒ KH_{a}(X, ℤ/nℤ) = 0 for a > 0

if a ≤ 4 or (RES) _{a−2,𝔽} holds.

Theorem 4 (Jannsen-S. (2003)) X: regular projective flat log-smooth over Spec(O_{k}), k: number field or 𝔭-adic field

⇒ KH_{a}(X, ℚ/ℤ) = 0 for 0 < a ≤ 3

Theorem 5 (Kerz-S. (2009)) X: smooth projective over 𝔽

⇒ KH_{a}(X, ℤ/nℤ) = 0 for a > 0

if n is prime to p = ch(𝔽)

Theorem 6 (Kerz-S. (2009)) X: regular projective quasi-semistable over Spec(O_{k}) for 𝔭-adic field k

⇒ KH_{a}(X, ℤ/nℤ) = 0 for a ≥ 0

if n is prime to p

Variant of Kato complex

n: integer invertible on X, KC_{•}^{(j)}(X, ℤ/nℤ) =

⋯ → ⨁_{x∈X_{(a)}}H^{a+j+1}(x, ℤ/nℤ(a + j)) →

⨁_{x∈X_{(a−1)}}H^{a+j}(x, ℤ/nℤ(a + j − 1)) → ⋯

⋯ → ⨁_{x∈X_{(1)}}H^{j+2}(x, ℤ/nℤ(j + 1)) → ⨁_{x∈X_{(0)}}H^{j+1}(x, ℤ/nℤ(j))

Original Kato complex = KC_{•}^{(0)}(X, ℤ/nℤ)

Theorem 7 (S.-Sato) R: henselian DVR s.t. F = R/𝔪_{R} finite or separably closed, X: regular projective flat quasi-semistable over S = Spec(R) ℓ: prime≠ ch(F) ⇒ H_{a}(KC_{•}^{(−1)}(X, ℚ_{ℓ}/ℤ_{ℓ})) = 0

for 0 ≤ a ≤ 4, a ≠ 1

Application to 1-cycles Corollary 1

X/R: as above with d = dim (X) − 1 ≥ 1 (1) For n prime to p = ch(F),

CH_{1}(X)/n ≃ H_{ét}^{2d}(X, ℤ/nℤ(d)), F is finite ⇒ |CH_{1}(X)/n| < ∞

(2) For prime ℓ ≠ p = ch(F),

CH_{1}(X){ℓ} ≃ H_{ét}^{2d−1}(X, ℚ_{ℓ}/ℤ_{ℓ}(d)). F is finite ⇒ |CH_{1}(X){ℓ}| < ∞ X/R: as above, F = R/𝔪_{R} finite V = X_{η}: generic fiber (⇒ V smooth over local field Frac(R)) Y = X_{s}: closed fiber ⇒ exact sequence

CH_{1}(Y ) → CH_{1}(X) → CH_{0}(V ) → 0

We have shown |CH_{1}(X){ℓ}| < ∞ Question |CH_{0}(V ){ℓ}| < ∞?

General Question |CH^{r}(V )_{tor}| < ∞ for V smooth over 𝔭-adic field? Several affirmative results for special cases

Remark Finiteness conjecture implies |CH^{r}(V )_{tor}| < ∞ for V smooth over number field

Finiteness conjecture

V of finite type over ℚ ⇒ CH^{r}(V ) is finitely gen.

Anomaly over local field

Theorem (Asakura-S.) n ≥ 5: integer V ⊂ ℙ_{ℚ_{p}}^{3} = Proj(ℚ_{p}[z_{0}, z_{1}, z_{2}, z_{3}]) defined by

z_{0}^{n} − z_{1}^{n} + z_{2}^{n} − z_{3}^{n} + p ∑_{|I|=n}a_{I} · z^{I}

where a_{I} ∈ ℤ_{p} are alg. indep. over ℚ

⇒ CH_{0}(V ) ⊃ ℚ_{ℓ}/ℤ_{ℓ} for ∀ℓ ≠ p.

Proof Use transcendental method (Hodge theory)