PDF Arithmetic of Arithmetic Schemes - 広島大学

Arithmetic of Arithmetic Schemes

Kanetomo Sato (Nagoya University)

1

Introduction.

X : a projective smooth variety over a finite field k = F_q p := ch(k)

ζ(X, s) : Hasse -Weil zeta function of X := ^Y

x∈X₀ (1− N(x)^−s)⁻¹,

where X₀ := the set of closed points on X, and N(x) := ](κ(x)).

Then we have the determinant presentation (d := dim(X), ` 6= p):

ζ(X, s) = ^Y2d

i=0 det(1− F^∗·q^−s; Hⁱ_´et(X_k,Q_`))^(−1)i+1 (F^∗ : geometric Frobenius operator)

= P_{`</sub><sup>1</sup>(q<sup>−s</sup>) · P<sub>`}³(q^−s)· · ·P_`^2d−1(q^−s)

P_{`</sub><sup>0</sup>(q<sup>−s</sup>) ·P<sub>`}²(q^−s)· · ·P_{`</sub><sup>2d−2</sup>(q<sup>−s</sup>) · P<sub>`}^2d(q^−s) with P_{`</sub><sup>i</sup>(T) := det(1 − F<sup>∗</sup>·T; H<sub>´et</sub><sup>i</sup> (X<sub>k</sub>,Q<sub>`})).

• ζ(X, s) is a rational function in q^−s with Q-coefficients, because we have Q_`(T) ∩ Q((T)) = Q(T).

• The reciprocal roots of P_{`</sub><sup>i</sup>(T) are algebraic integers whose com- plex absolute values are q<sup>i/2</sup>. Hence P<sub>`}ⁱ(T) is independent of

` 6= p and belongs to Z[T]. (Deligne)

Theorem 1. (Artin-Tate/Milne/Bayer-Neukirch/Schneider) Let n be an integer. Assume the semi-simplicity of F^∗ on

H_´et²ⁿ(X_k,Q<sub>`</sub>) for all ` 6= p, and H²ⁿ_cris(X).

Let ρ_n be the (positive) integer such that ζ(X, s) · (1 − q^n−s)^ρn tends to a non-zero constant as s → n. Then the limit

s→nlim ζ(X, s) · (1− q^n−s)^ρn is written in terms of the cohomology groups

Hⁱ_´et(X,Z(n)) :=^{c Y}

all `

Hⁱ_´et(X,Z_`(n))

with i = 1,2, . . . ,2d + 1, and a certain regulator term.

(Hⁱ_´et(X,Z_`(n))’s will be explained below.) Problem:

◦ Construct {H_´etⁱ (X,Z(n))}^c _i,n for arithmetic schemes X.

◦ Find zeta-value formulae for arithmetic schemes.

Results:

X : regular arithmetic scheme (with some assumption),

• Definition of the coefficient T_r(n)_X ∈ D^b(X_´et,Z/p^rZ) playing the role of Z/p^rZ(n).

• Duality for H_´etⁱ (X,T_r(n)_X) (in case X is proper).

§1 Reveiw on ´etale cohomology groups over a field

• ´etale topology = algebraic analogue of analytic topology on complex manifolds

Example. For a variety X over C and a torsion sheaf F on X_´et, H_´et^∗ (X,F) ' H^∗_an(X(C)^an,F|_X(C)^an).

• ´etale over a field k = finite separable over k Example. (1) For a sheaf F on Spec(k)_´et,

H^∗_´et(Spec(k),F) ' H^∗_Gal(G_k,F(k^sep)).

(2) For a variety X over k and a sheaf F on X_´et, G_k naturally acts on H_´et^∗ (X_k,F|_X

k) and ^∃ spectral sequence E₂^p,q = H^p_Gal(G_k,H_´et^q (X_k,F|_X

k)) =⇒ H^p+q_´et (X,F) (Hochshild-Serre spectral sequence)

Definition 2. Let n be an integer.

(1) X scheme

m positive integer invertible on X

µ_m ´etale sheaf of mth roots of unity on X We define:

Z/mZ(n) :=









µ^⊗n_m (n ≥ 0)

Hom(µ^⊗(−n)_m ,Z/mZ) (n < 0).

(2) k perfect field of characteristic p > 0 X smooth variety over k

W_rΩⁿ_X,log logarithmic part of W_rΩⁿ_X on X_´et (Illusie) Note: • W_rΩⁿ_X,log := 0, if n < 0 or dim(X) < n.

• W_rΩⁿ_X,log is a flat Z/p^rZ-sheaf and

0 → W_r−1Ωⁿ_X,log → W_rΩⁿ_X,log → W₁Ωⁿ_X,log → 0 (exact).

• W₁Ωⁿ_X,log ' Ωⁿ_X,log := Im(dlog : (O_X^×)^⊗n −→ Ωⁿ_X/k).

Then we define

Z/p^rZ(n) := W_rΩⁿ_X,log[−n].

(3) In the situation of (2), the group H_´etⁱ (X,Z(n)) is defined as^c lim←−

(m,p)=1

Hⁱ_´et(X,Z/mZ(n)) × lim←−

r≥1

Hⁱ_´et(X,Z/p^rZ(n)).

Remark. For a projective smooth variety X over a finite field k, the group H_´etⁱ (X,Z(n)) is finite for^c i 6= 2n,2n + 1, and the group H_´et²ⁿ(X,Z(n))^c _tors is finite. The group H²ⁿ⁺¹_´et (X,Z(n))^c _tors is finite as well under the semi-simplicity assumption in Theorem 1.

In what follows, we restrict our attentions to finite coefficient cases and review duality facts for varieties over finite fields.

X : smooth variety over a finite field k d := dim(X)

Theorem 3. (Poincar´e-Pontryagin duality)

Let m be a positive integer prime to ch(k). Then:

(1) ^∃ canonical trace map H^2d+1_c (X, µ^⊗d_m ) ' Z/mZ.

(2) For a constructible Z/mZ-sheaf F on X_´et, the pairing Hⁱ_c(X,F) × Ext^2d+1−i_X,Z/mZ(F, µ^⊗d_m ) −→ Z/mZ

is a non-degenerate pairing of finite Z/mZ-modules, for ^∀i.

Here, F constructible := ^∃ finite partition X = ∪_j Z_j

by locally closed subschemes such that F|_Zj is locally constant with finite stalks Corollary 4. For any integers n and i, the pairing

Hⁱ_c(X, µ^⊗n_m ) × H_´et^2d+1−i(X, µ^⊗d−n_m ) −→ Z/mZ is a non-degenerate pairing of finite Z/mZ-modules.

Proof. We have

Ext^∗_X,Z/mZ(µ^⊗n_m , µ^⊗d_m ) ' Ext^∗_X,Z/mZ(Z/mZ, µ^⊗d−n_m ) ' H^∗_´et(X, µ^⊗d−n_m ).

Theorem 5. (Milne duality)

Let r be a positive integer. Suppose X to be proper. Then:

(1) ^∃ canonical trace map H_´et^d+1(X, W_rΩ^d_X,log) ' Z/p^rZ.

(2) For any integers n and i, the pairing

H_´etⁱ (X, W_rΩⁿ_X,log) × H^d+1−i_´et (X, W_rΩ^d−n_X,log) −→ Z/p^rZ is a non-degenerate pairing of finite Z/p^rZ-modules.

Key point: Construction of the trace map

(then reduced to the case r = 1 and the Serre duality by the theory of Cartier operators).

Theorem 6. (Moser)

(1) ^∃ canonical trace map H^d+1_c (X, W_rΩ^d_X,log) ' Z/p^rZ.

(2) For a constructible Z/p^rZ-sheaf F on X_´et, the pairing Hⁱ_c(X,F) × Ext^d+1−i_X,Z/prZ(F, W_rΩ^d_X,log) −→ Z/p^rZ

is a non-degenerate pairing of finite Z/p^rZ-modules, for ^∀i.

Note:

• Theorem 5 (2) does not immediately imply Theorem 6.

• Theorem 6 recovers Theorem 5 (2) only in case n = 0.

Combining Theorem 3 – Theorem 6, we obtain:

Fact 7.

Let m be a positive integer. Then:

(1) ^∃ canonical trace map H^2d+1_c (X,Z/mZ(d)) ' Z/mZ.

(2) For a constructible Z/mZ-sheaf F on X_´et, the pairing Hⁱ_c(X,F) × Ext^2d+1−i_X,Z/prZ(F,Z/mZ(d)) −→ Z/mZ

is a non-degenerate pairing of finite Z/mZ-modules, for ^∀i.

(3) In case X is proper, for any integers n and i, the pairing Hⁱ_´et(X,Z/mZ(n)) × H^2d+1−i_´et (X,Z/mZ(d − n)) −→ Z/mZ

is a non-degenerate pairing of finite Z/mZ-modules.

Remark. For a map f : X → X⁰ of smooth varieties over k,

∃ a canonical pull-back map

f^∗Z/mZ(n)_X⁰ −→ Z/mZ(n)_X.

It is an isomorphism in the following cases (but not, otherwise):

• f is ´etale,

• (m,ch(k)) = 1,

• n ≤ 0,

• max{dim(X),dim(X⁰)} < n.

§2 Construction of p-adic ´etale Tate twists p : prime number, r : positive integer A : algebraic integer ring or p-adic integer ring

X : integral regular scheme flat of finite type over Spec(A), satisfying:

(∗) X is smooth or a semistable family over Spec(A) around the fibers of characteristic p.

For n ≥ 0, we want K ∈ D^b(X_´et,Z/p^rZ) satisfying (T1)–(T4) (=variant of Beilinson-Lichtenbaum axioms for ‘Z(n)’ on X_´et):

T1: ^∃ t : K|_V ' µ^⊗n_pr , where V := X[1/p].

T2: K is concentrated in [0, n].

T3: For a locally closed regular subscheme i : Z → X of characteristic p, we have a Gysin isomorphism:

Gysⁿ_i : W_rΩ^n−c_Z,log[−n − c] −−→^' τ_≤n+cRi^!K (c := codim_X(Z)) in D^b(Z_´et,Z/p^rZ), where W_rΩ^n−c_Z,log := 0 if n < c.

T4: A compatibility property between Gysin maps and boundary maps of Galois cohomology groups.

Precise statement of T4:

Consider points y and x on X satisfying

ch(x) = p, x ∈ {y} and codim_X(x) = codim_X(y) + 1.

Put c := codim_X(x). (Note that ch(y) = 0 or p.)

Then the following diagram commutes up to a sign depending only on (ch(y), c):

H^n−c+1_´et (y,Z/p^rZ(n − c+ 1)) −−→^∂val H^n−c_´et (x,Z/p^rZ(n − c))

Gysⁿ_iy



 y



 yGysⁿ_ix

H^n+c−1_y,´et (Spec(O_X,y),K) −−→^δloc H^n+c_x,´et(Spec(O_X,x),K), where

∂^val : boundary map of Galois cohomology groups (Kato) δ^loc : boundary map of a localization sequence

i_x : x → Spec(O_X,x) i_y : y → Spec(O_X,y)

Gysⁿ_ix : Gysin map coming from (T3) Gysⁿ_iy :

















Gysin map coming from (T3), if ch(y) = p, Gysin map coming from (T1) and

Deligne’s cycle class, if ch(y) = 0.

Theorem I. For a fixed n ≥ 0, ^∃! a pair of K ∈ D^b(X_´et,Z/p^rZ) and

t : K |_V ' µ^⊗n_pr

that satisfies T2–T4 as above.

Definition. (originally due to Schneider in the smooth case) For n ≥ 0, fix a pair (K, t) as in Theorem 1,

and define T_r(n)_X := K (∈ D^b(X_´et,Z/p^rZ)).

For n < 0, define T_r(n)_X := j_! Hom

Ã

µ^⊗(−n)_pr ,Z/p^rZ

!

. Remark:

• For n > d := dim(X), t induces T_r(n)_X ' Rj_∗µ^⊗n_pr .

• If X → Spec(A) is smooth around Y(:= union of the fibers of characteristic p) and p ≥ n + 2, then T_r(n)_X|_Y is isomorphic to the syntomic complex ‘S_r(n)’ of Fontaine-Messing (by a result of Kurihara).

• T_r(n)_X is not a log syntomic complex of Kato-Tsuji for 1 ≤ n ≤ dim(X), because the latter object is rather τ_≤nRj_∗µ^⊗n_pr .

• For the ´etale sheafification Z(n)^´et_X of the cycle complex defining higher Chow groups, one can hope Z(n)^´et_X ⊗^L Z/p^rZ ' T_r(n)_X. However, to prove this, we need to show T2 and T3 for the left hand side.

Theorem II. (Product and Contravariant functoriality) (1) For integers m and n, ^∃! a morphism

T_r(m)_X ⊗^L T_r(n)_X −→ T_r(m + n)_X.

in D⁻(X_´et,Z/p^rZ) extending the natural isomorphism µ^⊗m_pr ⊗ µ^⊗n_pr −→' µ^⊗m+n_pr on V = X[1/p].

(2) For a map f : X → X⁰ of Spec(A)-schemes satisfying (∗), ^∃!

a morphism

res^f : f^∗T_r(n)_X⁰ −→ T_r(n)_X.

in D⁻(X_´et,Z/p^rZ) extending the natural isomorphism g^∗µ^⊗n_pr,V⁰ −→' µ^⊗n_pr,V on V.

where g denotes V → V⁰ := X⁰[1/p].

Remark. The morphism res^f is an isomorphism in the following cases (but not, otherwise):

• f is ´etale,

• (m,ch(k)) = 1,

• n ≤ 0,

• max{dim(X),dim(X⁰)} < n.

§3 Duality for arithmetic schemes A : algebraic integer ring

Theorem. (Artin-Verdier duality)

(1) ^∃ canonical trace map H³_c(Spec(A),G_m) ' Q/Z.

(2) For a constructible sheaf F on Spec(A)_´et, the pairing Hⁱ_c(Spec(A),F) × Ext³⁻ⁱ_Spec(A)(F,G_m) −→ Q/Z

is a non-degenerate pairing of finite abelian groups, for ^∀i.

Remark.

• This duality is deduced from the Tate duality theorems for global fields and local fields.

• Generalized to ‘Z-constructible’ sheaves. (Deninger)

• Generalized to two-dimensional arithmetic schemes, replacingG_m with a modified version of Lichtenbaum complex Z(2). (Spieß)

Corollary. (Duality outside of characteristic p)

Let V be an integral regular scheme which is flat separated of finite type over Spec(A[1/p]). Put d := dim(V). Then:

(1) ^∃ canonical trace map H^2d+1_c (V, µ^⊗d_pr ) ' Z/p^rZ.

(2) For a constructible Z/p^rZ-sheaf F on V_´et, the pairing Hⁱ_c(V,F) × Ext^2d+1−i_V,Z/prZ(F, µ^⊗d_pr ) −→ Z/p^rZ

is a non-degenerate pairing of finite Z/p^rZ-modules, for ^∀i.

Proof. • The case d = 1: By the Kummer sequence µ_p^r −→ G_m −→ G_m −→ µ_p^r[1], reduced to the Artin-Verdier duality.

• The general case: f : V → Spec(A[1/p]) structure map.

We have a canonical isomorphism:

tr^f : µ^⊗d_pr [2(d − 1)] −−→^' Rf^!µ_p^r in D⁺(V_´et,Z/p^rZ) (Poincar´e duality + absolute purity (Thomason-Gabber)).

Hence by Deligne’s duality

Rf_∗RHom_V,Z/p^r_Z(F, Rf^!µ_p^r) = RHomSpec(A[1/p]),Z/p^rZ(Rf_!F, µ_p^r), reduced to the 1-dimensional case.

A remains to be global. Let X → Spec(A) be as in §2.

Suppose X to be separated, and put d := dim(X).

Theorem III. (Jannsen - Saito - S)

(1) ^∃ canonical trace map H^2d+1_c (X,T_r(d)_X) ' Z/p^rZ.

(2) For a constructible Z/p^rZ-sheaf F on X_´et, the pairing Hⁱ_c(X,F) × Ext^2d+1−i_X,Z/prZ(F,T_r(d)_X) −→ Z/p^rZ

is a non-degenerate pairing of finite Z/p^rZ-modules, for ^∀i.

Proof. • The case d = 1: By the Kummer sequence T_r(1)_X −→ G_m −→ G_m −→ T_r(1)_X[1], reduced to the Artin-Verdier duality.

• The general case: Glue the relative trace map for X[1/p] → Spec(A[1/p]) with those of fibers of characteristic p, and prove the isomorphism in D⁺(X_´et,Z/p^rZ):

tr^f : T_r(d)_X[2(d − 1)] −−→^' Rf^!T_r(1)_Spec(A).

Then by the same argument as the previous corollary, reduced to the case X = Spec(A).

Theorem IV. (S)

Suppose that X is proper over Spec(A). Then for any integers n and i, the pairing (coming from Theorems II and III (1))

Hⁱ_c(X,T_r(n)_X) × H^2d+1−i_´et (X,T_r(d −n)_X) −→ Z/p^rZ is a non-degenerate pairing of finite Z/p^rZ-modules.

Proof. V := X[1/p], Y := union of fibers of characteristic p.

The case n < 0: the previous corollary.

The case n ≥ 0: By the exact sequences

· · · → Hⁱ_c(X, j_!µ^⊗n_pr ) → Hⁱ_c(X,T_r(n)_X) → Hⁱ_´et(Y,T_r(n)_X|_Y) → · · · ,

· · · →Hⁱ_Y,´et⁰ (X,T_r(d−n)_X) →Hⁱ_´et⁰(X,T_r(n)_X) →H_´etⁱ⁰(V, µ^⊗d−n_pr ) → · · ·

with i⁰ := 2d + 1 − i and by the previous corollary, reduced to the following theorem:

Theorem V. (S)

Suppose that A is local and that X is proper over Spec(A).

(1) ^∃ canonical trace map H^2d+1_Y,´et (X,T_r(d)_X) ' Z/p^rZ.

(2) For any integers n ≥ 0 and i, the pairing

H_´etⁱ (X,T_r(n)_X) × H^2d+1−i_Y,´et (X,T_r(d − n)_X) −→ Z/p^rZ is a non-degenerate pairing of finite Z/p^rZ-modules.

Proof. (smooth case, for simplicity)

The case n > d : Tate duality for K := Frac(A) and Poincar´e duality for V_K = X ⊗_A K The case n = 0 : Milne duality for Y

The case 0 < n ≤ d : Reduced to the case r = 1 and ζ_p ∈ A.

Let ι : Y → X be the canonical map.

By the Bloch-Kato theorem on R^qj_∗µ^⊗q_p , the cohomology sheaves of Z/pZ(n) and Rι^!Z/pZ(d − n) are extensions of

Ω^∗_Y, dΩ^∗_Y, Ω^∗_Y,log.

A key step is to show that the induced pairing on those pieces are the same as the wedge product of differential forms up to a non-zero constant (explicit formula).

§4 Consequences of duality results

Let p, A and X be as in Theorems IV or V. Put d := dim(X).

Corollary.

Suppose that A is local. Put V := X[1/p] = X ⊗_A Frac(A).

Define the ‘unramified part’ Hⁱ_ur(V, µ^⊗n_pr ) as

Ker^µH_´etⁱ (V, µ^⊗n_pr ) −→ Hⁱ⁺¹_Y,´et(X,T_r(n)_X)^¶. Then under the non-degenerate pairing

H_´etⁱ (V, µ^⊗n_pr ) × H^2d−i_´et (V, µ^⊗d−n_pr ) −→ Z/p^rZ,

the subgroups Hⁱ_ur(V, µ^⊗n_pr ) and H^2d−i_ur (V, µ^⊗d−n_pr ) are exact annihi- lators of each other.

Corollary. (Lichtenbaum)

Suppose that A is local and d = 2. (i.e., V is a curve.) Then ^∃ canonical non-degenerate pairing of finite groups:

Pic(V)/p^r × _p^rBr(V) −→ Z/p^rZ.

Proof. Follows from the previous corollary and the equality Pic(V)/p^r = H²_ur(V, µ_p^r) ( ⊂ H_´et² (V, µ_p^r) ).

Corollary. (Tate/Cassels/Saito)

Suppose that A is global, d = 2 and p ≥ 3.

(i.e., X is an arithmetic surface.)

Define Br(X)_p-cotor := Br(X){p}/(Br(X){p})_Div (finite group).

Then ^∃ canonical non-degenerate alternating pairing

h , i : Br(X)_p-cotor × Br(X)_p-cotor −→ Q_p/Z_p. In particular, the order of Br(X)_p-cotor is a square number.

Proof. Decompose the non-degenerate pairing (Theorem IV) H_´et² (X,T_Qp/Zp(1)) × H³_´et(X,T_Zp(1)) −→ H⁵_´et(X,T_Qp/Zp(2))



 y'

Q_p/Z_p and compute signs of cup products

§5 Euler-Poincar´e characteristics Let A, X and p be as in Theorem IV.

Problem.

◦ For fixed i and n, is dim_QpHⁱ_´et(X,T_Qp(n)_X) independent of p?

Concerning this problem, we have the following weaker result:

Theorem VI. Put

χ(X,T_Qp(n)) := ^X∞

i=0 (−1)ⁱ · dim_Qp Hⁱ_´et(X,T_Qp(n)_X), χ_D(X_/R,R(n)) := ^X∞

i=0 (−1)ⁱ · dim_R Hⁱ_D(X_/R,R(n)), where H^∗_D(X_/R,R(n)) denotes the real Deligne cohomology of X_C/Z := X ⊗_Z C:

H^∗_D(X_/R,R(n)) := H^∗_an(X_C/Z(C)^an,R(n)_D)^Gal(C/R). Then we have

χ(X,T_Qp(n)) = χ_D(X_/R,R(n)).

In particular, χ(X,T_Qp(n)) is independent of p for which p-adic

´etale Tate twists are defined.