類体論の高次元化と高次化

斎藤秀司 (東大数理)

Higher higher class field theory Shuji SAITO (University of Tokyo)

Plan of talk

Part I: Class field theory for undergraduate Part II: Brief introduction to Higher CFT Part III: Enters Motivic (co)homology Part IV: Higher higher CFT and Hasse principle Part V: Application to special values Apology: No nonabelian CFT

Galois theory: \begin{matrix}Gal (ℚ(√−1)/ℚ) ≃ ℤ/2ℤ \\ Gal (ℚ(√2)/ℚ) ≃ ℤ/2ℤ \end{matrix}

⇓

Number theory: \begin{matrix}Gal (ℚ(√−1)/ℚ) ≃ (ℤ/4ℤ)^{×} \\ Gal (ℚ(√2)/ℚ) ≃ (ℤ/8ℤ)^{×}/\{±1\} \end{matrix}

⇓ Theorem (Gauss) p: odd prime

∃x, y ∈ ℤ, p = x^{2} + y^{2} ⇔ p ≡ 1 mod 4 ∃x, y ∈ ℤ, p = x^{2} − 2y^{2} ⇔ p ≡ ±1 mod 8

Remark ℤ[√−1] PID ⇒ ∃x, y ∈ ℤ, p = x^{2} + y^{2} ⇔ (p) = 𝔭_{1} · 𝔭_{2} ⊂ ℤ[√−1]

𝔭_{1}, 𝔭_{2}: prime ideals

Same for ℤ[√2] and p = x^{2} − 2y^{2}

General fact

[K : ℚ] < ∞, 𝒪_{K}: ring of integers in K ⇒ any ideal 𝔞 ⊂ 𝒪_{K} admits prime decomposition

𝔞 = 𝔭_{1}^{e_{1}} ⋯ 𝔭_{r}^{e_{r}}

Problem 1

[K : ℚ] < ∞, L/K finite abelian extension Describe Gal (L/K) in such way that prime decomposition law (PDL) well-understood (i.e. law of PD of 𝔭𝒪_{L} for prime 𝔭 ⊂ 𝒪_{K})

Example 1 (quadratic extension of K = ℚ) Proposition

L = ℚ(√a) (a ∈ ℤ square free) p: rational prime with (p, 2a) = 1 p𝒪_{L} = 𝔭_{1} · 𝔭_{2} in 𝒪_{L} ⇔ (\frac{a}{p}) = 1 p𝒪_{L} prime in 𝒪_{L} ⇔ (\frac{a}{p}) = −1 (\frac{a}{p}): Legendre symbol

Conclusion PDL for ℚ(√a)/ℚ follows from Quadratic reciprocity law

Example 2 (Cyclotomic extension) L = ℚ(ζ_{n}), 𝒪_{L} = ℤ[ζ_{n}] (n ∈ ℕ fixed)

\begin{matrix}(ℤ/nℤ)^{×} & ≃ & Gal (ℚ(ζ_{n})/ℚ) \\ a & → & σ_{a} \end{matrix} (σ_{a}(ζ_{n}) = ζ_{n}^{a})

Theorem p: rational prime with (p, n) = 1 f = order of σ_{p} = min{a ∈ ℕ| p^{a} ≡ 1 mod n} ⇒ p𝒪_{L} = 𝔭_{1} ⋯ 𝔭_{r} in ℤ[ζ_{n}] (r = |(ℤ/nℤ)^{×}|/f)

𝒪_{L}/𝔭_{i} = 𝔽_{p^{f}}

How about general abelian extension L/K?

Hint from (ℤ/nℤ)^{×} ≃ Gal (ℚ(ζ_{n})/ℚ)

P: set of the rational primes Σ: subset of the primes p|n ⇒ exact sequence

ℚ(n) ⟶^{δ} ⨁_{p∈P −Σ}ℤ ⟶^{ρ} Gal (ℚ(ζ_{n})/ℚ) → 0

ℚ(n) := \{a ∈ ℚ^{×}| \begin{matrix}a = 1 + \frac{nc}{b} > 0 \\ b, c ∈ ℤ, (b, n) = 1 \end{matrix}\}

(ℤ/nℤ)^{×} ∥

ℚ(n) ⟶^{δ}  ⨁_{p∈P −Σ}ℤ  ⟶^{ρ} Gal (ℚ(ζ_{n})/ℚ)

1 at p  →  σ_{p}
a → (v_{p}(a))_{p∈P −Σ}

Point Existence of distinguished element

σ_{p} ∈ Gal (ℚ(ζ_{n})/ℚ) for ∀p |̸n L/K: finite abelian P_{K}: set of the primes in 𝒪_{K}

Proposition 𝔭 ∈ P_{K} unram in L/K, i.e.

𝔭𝒪_{L} = 𝔓_{1} ⋯ 𝔓_{r} ⊂ 𝒪_{L}, 𝔓_{i} ≠ 𝔓_{j} (i ≠ j) ⇒ ∃_{!} σ_{𝔭} ∈ Gal (L/K) s.t.

σ_{𝔭}(x) ≡ x^{q} mod 𝔓_{i} (∀x ∈ 𝒪_{L}, ∀i)

q = |𝒪_{K}/𝔭|

Definition σ_{𝔭} ∈ Gal (L/K): Frobenius at 𝔭 Σ: set of the primes ramified in L/K (|Σ| < ∞)

\begin{aligned}ρ_{L/K} : & ⨁_{𝔭∈P_{K}−Σ}ℤ & → & Gal (L/K) \\ & 1 at 𝔭 & → & σ_{𝔭} \end{aligned}

Fact (1) ρ_{L/K} surjective (Tchevotarev)

(2) 𝔭 ∈ P_{K} − Σ, f_{𝔭} = order of σ_{𝔭} in Gal (L/K) ⇒ 𝔭𝒪_{L} = 𝔓_{1} ⋯ 𝔓_{r} ⊂ 𝒪_{L} (r = [L : K]/f_{𝔭})

[𝒪_{L}/𝔓_{i} : 𝒪_{K}/𝔭] = f_{𝔭} Problem 1 is reduced to

Problem 2 Determine Ker(ρ_{L/K}) (⇔ Determine all relations among σ_{𝔭}’s)

Fix K and finite subset Σ ⊂ P_{K} and n ∈ ℕ

L_{Σ,n} := ∪_{L/K}L

L/K abelian unram outside Σ and [L : K]|n

Fact [L_{Σ,n} : K] < ∞

Consider

ρ_{L_{Σ,n}/K} : ⨁_{𝔭∈P_{K}−Σ}ℤ → Gal (L_{Σ,n}/K)

Theorem (Artin-Takagi) ρ_{L_{Σ,n}/K} induces

Coker (K_{Σ,n} ⟶^{δ} ⨁_{𝔭∈P_{K}−Σ}ℤ) ≃ Gal (L_{Σ,n}/K)

K_{Σ,n} = {a ∈ K^{×}| a ∈ (K_{𝔭}^{×})^{n} for ∀𝔭 ∈ Σ ∪ P_{K,∞}} P_{K,∞}: set of infinite places of K K_{𝔭}: completion of K at 𝔭 δ(a) = (v_{𝔭}(a))_{p∈P_{K}−Σ} (a ∈ K_{Σ,n})

Unramified CFT Corollary (Hilbert-Furtwengler)

Cl(K) ≃ Gal (H_{K}/K)

Cl(K) = Coker (K^{×} ⟶^{δ} ⨁_{𝔭∈P_{K}}ℤ)

(ideal class group of 𝒪_{K})

H_{K}: max unramified abelian ext of K s.t. H_{K} ⊗_{K} ℝ ≃ ℝ × ⋯ × ℝ for ∀τ : K ↪ ℝ

Geometrization of CFT via schemes X = Spec (𝒪_{K}) with [K : ℚ] < ∞ X_{(0)}: set of the closed points of X ⇒ X_{(0)} = P_{K} := set of the primes in 𝒪_{K} U ⊂ X: (non-empty) open subscheme π_{1}^{ab}(U): abelian fundamental group of U (classifies abelian finite étale coverings V → U )

π_{1}^{ab}(U)/n = Gal (L_{Σ,n}/K), Σ = X − U

[figure: (2); (p); Spec (ℤ[i]); Spec (ℤ); V = Spec (ℤ[i]) − {(1 + i)}; U = Spec (ℤ) − {(2)}]

Geometric construction of Frobenius For x ∈ U_{(0)}, x → U induces

ρ_{x} : π_{1}^{ab}(x) → π_{1}^{ab}(U)

π_{1}^{ab}(x) = Galois group of κ(x), residue field of x σ_{x} ∈ π_{1}^{ab}(U): image of q-th power map∈ π_{1}^{ab}(x)

(q = |κ(x)|)

Same construction works in higher dimension! U may be scheme of fintie type over ℤ or 𝔽_{p}

Generalization to higher dimension U: regular scheme of finite type over ℤ or 𝔽_{p} U_{(0)}: set of the closed points of U Z_{0}(U) = ⨁_{x∈U_{(0)}}ℤ : group of zero-cycles on U

\begin{aligned}ρ_{U} : & Z_{0}(U) & → & π_{1}^{ab}(U) \\ & 1 at x & → & σ_{x} \end{aligned}

Fact (Lang) Image(ρ_{U}) dense in π_{1}^{ab}(U) Problem Determine Ker(ρ_{U})

Higher CFT à la Kato-Saito U: regular scheme of finite type over ℤ or 𝔽_{p} Choose X integral proper over ℤ or 𝔽_{p} with open immersion U ↪ X. K-S constructs:

• For ideal I ⊂ 𝒪_{X},

higher idele class group of modulus I

C_{I}(X) = H^{d}(X_{Nis}, K_{d}^{M}(𝒪_{X}, I)),

(It has idele presentation via higher local fields)

• surjective map

Z_{0}(U) = ⨁_{x∈U_{(0)}}ℤ ↠ C_{I}(X)

if Supp(𝒪_{X}/I) ⊂ Σ := X − U,

• Reciprocity map

ρ_{U} : lim_{⟵_{Supp(𝒪X/I)⊂Σ}} C_{I}(X) → π_{1}^{ab}(U)

Theorem (Kato-S) ρ_{U} is (almost) isomorphism

Caution Difficult to describe Ker(Z_{0}(U) → C_{I}(X)),

so is Ker(Z_{0}(U) ⟶^{ρ_{U}} π_{1}^{ab}(U))

except the case U = X (case of unramified class field theory)

Theorem (Higher dimensional unram CFT) X regular projective over ℤ or 𝔽_{p} (Assume X(ℝ) ≠ ∅ for simplicity) ⇒ ρ_{X} : CH _{0}(X) → π_{1}^{ab}(X) (almost) isomorphism CH _{0}(X): Chow group of zero-cycles on X

modulo rational equivalence

X = Spec (𝒪_{K}) ([K : ℚ] < ∞) ⇒ CH _{0}(X) = Cl(K) ideal class group of K

CH _{0}(X) = Coker ( ⨁_{y∈X_{(1)}}κ(y)^{×} ⟶^{δ} ⨁_{x∈X_{(0)}}ℤ)

X_{(1)}: set of the generic points of curves C ⊂ X i : C ↪ X curve on X with generic point y κ(y): residue field of y = function field of C ⇒ [κ(y) : ℚ] < ∞ or trdeg_{𝔽_{p}}(κ(y)) = 1

δ : κ(y)^{×} → Z_{0}(X) = ⨁_{x∈X_{(0)}} ℤ is the composite

κ(y)^{×} ⟶^{δ} Z_{0}(C̃) ⟶^{π_{∗}} Z_{0}(C) ⟶^{i_{∗}} Z_{0}(X)

π : C̃ → C normalization of C Higher CFT à la German

Theorem (Kerz-Schmidt-Wiesend, Schmidt-Spiess)

U regular scheme of finite type over ℤ or 𝔽_{p}, n ∈ ℕ prime to ch (K) (K function field of U) ρ_{U} : Z_{0}(U) → π_{1}^{ab}(U) induces

Coker ( ⨁_{y∈U_{(1)}}κ(y)_{Σ,n} ⟶^{δ} ⨁_{x∈U_{(0)}}ℤ/nℤ) ≃ π_{1}^{ab}(U)/n

κ(y)_{Σ,n} ⊂ κ(y)^{×} (defined in suitable manner) Theorem for dim (U) = 1 is classical CFT

Motivic interpretation of CFT

Theorem (Schmidt, Schmidt-Spiess) Under tame condition,

Coker ( ⨁_{y∈U_{(1)}}κ(y)_{Σ,n} ⟶^{δ} ⨁_{x∈U_{(0)}}ℤ/nℤ) ≃ H_{0}^{S}(U, ℤ/nℤ)

H_{i}^{S}(U, ℤ/nℤ) (i ∈ ℕ): Suslin homology of U

= motivic homology of U

defined by using algebraic cycles Higher CFT à la German ⇒

H_{0}^{S}(U, ℤ/nℤ) ≃ π_{1}^{ab}(U)/n

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))

Regulator maps

H_{M}^{i}(X, ℤ(r)) → H_{B}^{i}(X, ℤ(r)) (Betti cohomology) H_{D}^{i}(X, ℤ(r)) (Deligne cohomology) H_{ét}^{i} (X, ℤ_{ℓ}(r)) (étale cohomology) H_{crys}^{i}(X/W (k)) (crystalline cohomology) · · · · · · · · · ⇒ H_{M}^{i}(X, ℤ(r)) universal cohomology theory

Relation to algebraic K-theory

E_{2}^{p,q} = H_{M}^{p}(X, ℤ(−\frac{q}{2})) ⇒ K_{−p−q}(X)

Construction of motivic (co)homology

(1) Voevodsky, Hanamura, Levine construct DM(k): category of mixed motives over k equipped with functor

M : Sm/k → DM(k)

Sm/k: category of smooth schemes over k Define for X ∈ Sm/k and i, r ∈ ℤ_{≥0}

H_{M}^{i}(X, ℤ(r)) = Hom_{DM(k)}(M(X), ℤ(r)[i])

H_{i}^{M}(X, ℤ(r)) = Hom_{DM(k)}(ℤ(r)[i], M(X))

(2) X of finite type over field or Dedekind ring CH ^{r}(X, q) : Bloch’s higher Chow group H_{i}^{S}(X, ℤ) : Suslin homology

Theorem (1) X smooth over field H_{i}^{M}(X, ℤ(0)) = H_{i}^{S}(X, ℤ)

(2) X smooth over perfect field H_{M}^{i}(X, ℤ(r)) = CH ^{r}(X, 2r − i),

Review on homology in toplogy

X: topological space ⇝

H_{q}(X, ℤ) := H_{q}(s(X, •))

s(X, •) = Singular complex:

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

s(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

Quick review on CH ^{r}(X, q) and H_{q}^{S}(X, bZ)

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}

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

c_{0}(X, q) = ⨁_{Ξ⊂X×Δ^{q}}ℤ[Γ]

Ξ ⊂ X × Δ^{q} closed integral finite surjective over Δ^{q}

Cycle complex

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

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

Definition CH ^{r}(X, q) := H_{q}(z^{r}(X, •)) H_{q}^{S}(X, ℤ) := H_{q}(c_{0}(X, •)) H_{q}^{S}(X, ℤ/nℤ) := H_{q}(c_{0}(X, •) ⊗ ℤ/nℤ)

Higher higher CFT

Theorem (Kerz-S.) U smooth scheme over 𝔽_{p}, n prime to p

H_{i}^{S}(U, ℤ/nℤ) ≃ H_{c}^{2d−i}(U_{ét} , μ_{n}^{⊗d})

H_{c}^{j}(U_{ét} , μ_{n}^{⊗d}): étale cohomology with compact support

Remark H_{c}^{2d}(U_{ét} , μ_{n}^{⊗d}) ≃ π_{1}^{ab}(U)/n ⇒ Case i = 0 is Higher CFTà la German (geometric case)

H_{0}^{S}(U, ℤ/nℤ) ≃ π_{1}^{ab}(U)/n Key to proof of Theorem:

Cohomological Hasse principle

Hasse principle (Hasse-Minkowski)

a_{1}X_{1}^{2} + ⋯ a_{n}X_{n}^{2} = 0 (a_{1}, …, a_{n} ∈ ℚ) ∃solution in ℚ ⇔ ∃solution in ℝ and ℚ_{p} for ∀p Remark: n = 3 is the most essential case.

Cohomological interpretation k: field

X^{2} − aY^{2} − bZ^{2} = 0 (a, b ∈ k^{×})

∃solution in k ⇔ {a, b} = 0 ∈ H^{2}(k, ℤ/2ℤ) H^{∗}(k, ℤ/2ℤ): Galois cohomology of k {a, b} = h(a) ∪ h(b) h : k^{×}/2 ≃ H^{1}(k, ℤ/2ℤ) (Kummer theory) ∪ : H^{1}(k, ℤ/2ℤ) × H^{1}(k, ℤ/2ℤ) → H^{2}(k, ℤ/2ℤ)

Hasse principle ⇔ injectivity of

H^{2}(ℚ, ℤ/2ℤ) → ⨁_{p prime} H^{2}(ℚ_{p}, ℤ/2ℤ) ⊕ H^{2}(ℝ, ℤ/2ℤ)

residue isomorphism ∂_{p} : H^{2}(ℚ_{p}, ℤ/2ℤ) ≃ H^{1}(𝔽_{p}, ℤ/2ℤ)

Hasse principle ⇔ injectivity of

H^{2}(ℚ, ℤ/2ℤ) ⟶^{∂} ⨁_{p prime} H^{1}(𝔽_{p}, ℤ/2ℤ) ⊕ H^{2}(ℝ, ℤ/2ℤ)

Generalization X = Spec (𝒪_{K}) ([K : ℚ] < ∞), or X = smooth proper curve over 𝔽_{q}, K = 𝔽_{q}(X) (Assume ∄Spec (ℝ) → X for simplicity)

H^{2}(K, ℤ/nℤ(1)) ⟶^{∂} ⨁_{x∈X_{(0)}}H^{1}(x, ℤ/nℤ)

ℤ/nℤ(1) = μ_{n} (if ch (K) |̸n)

Remark H^{2}(K, ℤ/nℤ(1)) ≃ Br(K)[n]

Hasse principle for Br(K) ⇔ ∂ injective

HP for Br(K) (Brauer-Hasse-Noether) K: number field, or function field of curve over finite field \overline{P}_{K}: set of the places of K K_{v}: completion of K at v ∈ \overline{P}_{K} A: central simple algebra over K

A ≃ M_{n}(K) ⇔ A ⊗_{K} K_{v} ≃ M_{n}(K_{v}) ∀v ∈ \overline{P}_{K}

A = (\frac{a,b}{K}) (a, b ∈ K) ⇒ Hasse principle for X^{2} − aY^{2} − bZ^{2} = 0 over K

Cohomological HP in higher dimension X of finite type over 𝔽 or ℤ

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 ∈ X_{(a)} ⇔ trdeg_{𝔽_{p}}(κ(x)) = a or trdeg_{ℚ}(κ(x)) = a − 1 ℤ/nℤ(a) = μ_{n}^{⊗a} if ch (κ(x)) |̸n

dim (X)=1 X = Spec (𝒪_{K}) for [K : ℚ] < ∞, or X = smooth proper curve over 𝔽_{q}, K = 𝔽_{q}(X) KC_{•}(X, ℤ/nℤ) =

⨁_{η∈X_{(1)}}H^{2}(η, ℤ/nℤ(1)) ⟶ ⨁_{x∈X_{(0)}}H^{1}(x, ℤ/nℤ)
∥  _{residue} ↑^{≃}
H^{2}(K, ℤ/nℤ(1)) ⟶ ⨁_{x∈X_{(0)}}H^{2}(K_{x}, ℤ/nℤ(1))

H_{1}(KC_{•}(X, ℤ/nℤ)) = 0 ⇔ HP 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}), [k : ℚ] < ∞ Assume X(ℝ) = ∅ for simplicity

⇒ KH_{a}(X, ℤ/nℤ) = 0 for a > 0.

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

H_{M}^{i}(X, ℤ/nℤ(d)) → H_{ét}^{i} (X, ℤ/nℤ(d))

→ KH_{2d−i+1}(X, ℤ/nℤ)
d = dim (X)

The proof of Lemma uses Bloch-Kato conjecture proved by Rost-Voevodsky

Remark H_{ét}^{i} (X, ℤ/nℤ(d)) known to be finite Kato’s HP ⇒ Finiteness of motivic cohomolgy

Known cases of Kato conjecture

• dim (X) = 1: Classical (HP for Brauer group)

• dim (X) = 2: Kato (1986)

Theorem KH_{a}(X, ℚ/ℤ) = 0 for 0 < a ≤ 3 under the following assumptions:

• X smooth projective over Spec (𝔽_{p}) (Colliot-Thélène (1993), Suwa (1995))

• X regular projective semistable over Spec (𝒪_{k}) (Jannsen-S. (2003))

Theorem (Jannsen-S. (2006)) X: smooth projective over 𝔽_{p}

⇒ KH_{a}(X, ℤ/nℤ) = 0 for a > 0

if a ≤ 4 or (RES) _{a−2,𝔽_{p}} holds.

Key ingredients in the proof:

• Weil conjecture (Deligne’s theorem)

• Resolution of singularities

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.

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

Theorem (Kerz-S. (2009)) X: smooth proper over 𝔽_{p}

⇒ KH_{a}(X, ℤ/nℤ) = 0 for a > 0

if n is prime to p

Remark Theorem implies Higher higher CFT

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

by using Voevodsky’s theory of mixed motives

Application to special values

X smooth proper over 𝔽_{q}

ζ(X, s) = exp ( ∑_{n=1}^{∞} |X(𝔽_{q^{n}})| \frac{q^{−ns}}{n}) (s ∈ ℂ)

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)

⇒ ζ(X, r)^{∗} ∈ ℚ^{×} (Grothendieck, Deligne)

Theorem

X: smooth projective over 𝔽_{q} ℓ prime Assume ℓ ≠ ch (𝔽_{q}) or d = dim (X) ≤ 4 ⇒ |H_{M}^{i}(X, ℤ(d)){ℓ}| < ∞ for all i

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

Class number formula

X = Spec (𝒪_{K}) with [K : ℚ] < ∞

⇒ lim_{s→0} ζ(X, s) · s^{−ρ_{0}} = − \frac{|H_{M}^{2}(X, ℤ(1))_{tors} |}{|H_{M}^{1}(X, ℤ(1))_{tors} |} · R_{K}

ρ_{0} := ord_{s=0} ζ_{K}(s) = rank (H_{M}^{1}(X, ℤ(1))) H_{M}^{2}(X, ℤ(1)) = Cl(K) : ideal class group of K H_{M}^{1}(X, ℤ(1)) = 𝒪_{K}^{×} : unit group of 𝒪_{K} R_{K} : Dirichlet’s regulator