p-ADIC INTEGRATION AND p-ADIC REGULATORS

AMNON BESSER

X - smooth projective variety of dimension d over a field K A^{i}(X) - group of codimension i algebraic cycles on X j = d − i A_{j}(X) = A^{i}(X)

1. Motivation - the complex Abel-Jacobi map

K = ℂ. H_{B}^{∗}(X, ℤ) - Betti cohomology

cℓ_{B} : A^{i}(X) → H_{B}^{2i}(X, ℤ).

A^{i}(X)_{0} - kernel of cℓ_{B} Complex Abel-Jacobi map:

α_{ℂ} : A_{j}(X)_{0} → [F^{j+1}H_{dR}^{2j+1}(X/ℂ)]^{∗} /H_{2j+1}(X, ℤ).

Construction - Z ∈ A_{j}(X)_{0}, Γ - a topological chain bounding Z. [ω] ∈ H_{dR}^{2j+1}(X/ℂ) represented by the class of a closed 2j + 1 form ω

α_{ℂ}(Z)([ω]) = ∫_{Γ} ω.

2. Coleman’s integration (case j = 0)

K = ℂ_{p} X has good reduction. ω ∈ F^{1}H_{dR}^{1}(X/K) - a closed holomorphic 1-form. Coleman (Annals 85) - to ω corresponds a function f_{ω} : X(ℂ_{p}) → ℂ_{p}, unique up to a constant, locally holomorphic and s.t. df_{ω} = ω Good properties:

1. for G : Y → X, f_{G^{∗}ω} = G^{∗}f_{ω}

2. if Z = ∑ n_{i}x_{i} ∈ A_{0}(X)_{0}, then Z is torsion mod rational equivalence iff

∫_{Z} ω := ∑ n_{i}f_{ω}(x_{i}) (well defined)

is 0 for all ω ∈ F^{1}H_{dR}^{1}(X/K).

3. in the above situation, if ω and Z are defined over a subfield L, then ∫_{Z} ω ∈ L.

3. Generalization. Step I - underlying complexes

K - a finite extension of ℚ_{p}

O_{K} - ring of integers of K, κ - residue field

X - smooth projective \underline{over O_{K}}

X_{K}, X_{s} - generic and special fiber of X.

Construction: complexes of K vector spaces C_{cr} and C_{dR} with following properties:

1. H^{∗}(C_{cr}) = H_{cris}^{∗}(X_{s}, W (κ)) ⊗ K, H^{∗}(C_{dR}) = H_{dR}^{∗}(X_{K}/K);

2. There is a \underline{K-linear} Frobenius ϕ : C_{cr} → C_{cr} inducing Frobenius on H_{cris};

3. There is a descending filtration F^{•}C_{dR} inducing the de Rham filtration on H_{dR};

4. There is a map |_{X_{K}} : C_{cr} → C_{dR} inducing the isomorphism H_{cris}^{∗}(X_{s}, W (κ))⊗ K ≅ H_{dR}^{∗}(X_{K}/K)

Sketch of construction:

Choose an affine covering W_{i} of X with lift of Frob ϕ_{i} : W_{i} → W_{i}.

Put:

U_{i} = (W_{i})_{s}, U_{I} = ⋂_{i∈I} U_{i}

W_{I} = ∏_{i∈I} W_{i}, ϕ_{I} = ∏_{i∈I} ϕ_{i}.

Let T_{I} = the \underline{tube} of U_{I} in W_{I} (some rigid analytic space).

C_{cr} = Tot (⊕_{|I|=j+1}Ω^{i}(T_{I}))

One can take C_{dR} = C_{cr} with appropriate filtration.

4. The finite polynomial cohomology and its properties

Let q = |κ|

Consider polynomials P (t) = ∏_{1}^{deg P} (1 − α_{i}t) ∈ ℚ[t]

p-ADIC INTEGRATION AND p-ADIC REGULATORS 3

Let ¶_{m} be the monoid of such P (t) with all 1/α_{i} having complex absolute value q^{m/2}. Definition 1. H_{f,P} (X, n) is the cohomology of the complex

Cone(C_{cr} ⊕ F^{n}C_{dR} → C_{cr} ⊕ C_{dR})[−1].

The map defining the cone is

(x, y) → (P (ϕ)x, x|_{X_{K}} − y)

There is a natural map H_{f,P} → H_{f,P Q} Definition 2.

H_{fp}^{i}(X, n) := \begin{matrix}lim \\ -------→ \end{matrix}_{P ∈¶_{i}} H_{f,P}^{i} (X, n)

\underline{Properties}

1. a short exact sequence

0 → H_{dR}^{i−1}(X_{K}/K)/F^{n} → H_{fp}^{i}(X, n)
→ F^{n}H_{dR}^{i}(X_{K}/K) → 0;

(idea: write the sequence of a cone and use the fact that the characteristic polynomial of ϕ on H_{dR}^{i} belongs to ¶_{i})

2. a trace map tr : H_{fp}^{2d+1}(X, d + 1) → K (easy from 1.);

3. a cup product H_{fp}^{i}(X, n) × H_{fp}^{j}(X, m) → H_{fp}^{i+j}(X, n + m) making two diagrams commute:

H_{fp}^{i}(X, n) × H_{fp}^{j}(X, m) ⟶ F^{n}H_{dR}^{i}(X_{K}/K) × F^{m}H_{dR}^{j}(X_{K}/K)
∪↓  ^{∪_{dR}}↓
H_{fp}^{i+j}(X, n + m) ⟶ F^{n+m}H_{dR}^{i+j}(X_{K}/K)

where ∪_{dR} is cup product in de Rham cohomology.

H_{dR}^{i−1}(X_{K}/K)/F^{n} × H_{fp}^{j}(X, m) ⟶ H_{dR}^{i−1}(X_{K}/K)/F^{n} × F^{m}H_{dR}^{j}(X_{K}/K)
↓  ^{∪_{dR}}↓
H_{fp}^{i}(X, n) × H_{fp}^{j}(X, m)  H_{dR}^{i+j−1}(X_{K}/K)/F^{m+n}
∪↓ ↓
H_{fp}^{i+j}(X, n + m)  H_{fp}^{i+j}(X, n + m)

4. Poincaré duality: the pairing

⟨ , ⟩_{fp} : H_{fp}^{i}(X, n) × H_{fp}^{2d+1−i}(X, d + 1 − n) ⟶^{∪}

H_{fp}^{2d+1}(X, d + 1) ⟶^{tr_{X}} K

is perfect.

5. Integration

We are given a de Rham class ω ∈ F^{j+1}H_{dR}^{2j+1}(X_{K}/K).

We choose a pullback ω̃ ∈ H_{fp}^{2j+1}(X, j + 1).

We are given a \underline{smooth} subscheme f : Y → X whose generic fiber has dimension j. Definition 3.

∫_{Y} ω̃ := tr_{Y} (f^{∗}(ω̃)).

The difference between 2 lifts is in H_{dR}^{2j}(X_{K}/K) (constants for j = 0)

Let A^{i}(X_{K})_{0} = Ker (cℓ_{dR} : A^{i}(X_{K}) → H_{dR}^{2i}(X_{K}/K)) Definition 4. Given a cycle Z = ∑ n_{i}Y_{i} ∈ A_{j}(X_{K})_{0}, Y_{i} as in definition 3. Define

∫_{Z} ω := ∑ n_{i} ∫_{Y_{i}} ω̃

for any lift ω̃.

6. Main results

Comparison with Coleman’s integral Theorem 1. If the generic fibers of the Y_{i} are points y_{i} on X_{K}, then for a one form ω on X_{K} we have the equality ∫_{Z} ω = ∫_{z} ω where the second integral is the Coleman integral on the cycle ∑ n_{i}y_{i}.

Computation of p-adic Abel-Jacobi maps

XK̄ = X ×_{spec O_{K}} spec K̄

Étale class map:

cℓ_{p} : A^{i}(X_{K}) → H_{ét}^{2i}(XK̄, ℚ_{p}(i))

Put A^{i}(X_{K})_{0} = Ker cℓ_{p}

p-adic Abel-Jacobi map:

α_{p} : A^{i}(X_{K})_{0} → H_{f}^{1}(K, H_{ét}^{2i−1}(XK̄, ℚ_{p}(i)))

(continuous Galois cohomology).

Exponential map of Bloch and Kato

exp : H_{dR}^{2i−1}(X_{K}/K)/F^{i} ≅ H_{f}^{1}(K, H_{ét}^{2i−1}(XK̄, ℚ_{p}(i))).

Composition α_{dR} := exp^{−1} ∘ α_{p}

α_{dR} :A_{j}(X_{K})_{0} → H_{dR}^{2i−1}(X_{K}/K)/F^{i}
= Hom_{K} (F^{j+1}H_{dR}^{2j+1}(X_{K}/K), K)

p-ADIC INTEGRATION AND p-ADIC REGULATORS 5

Theorem 2. The following equality holds for ω ∈ F^{j+1}H_{dR}^{2j+1}(X_{K}/K) and Z = ∑ n_{i}Y_{i} with all the Y_{i} smooth:

α_{dR}(Z)(ω) = ∫_{Z} ω

7. Sketch of proof for second theorem

H_{f} cohomology of Nizioł: Similar definition: Replace C_{cr} by C_{cr/K_{0}} - complex computing H_{cris}^{∗}(X_{s}, W (κ)) ⊗ K_{0}, Use semilinear Frobenius ϕ̃ Definition 5. H_{f}^{∗}(X, K(n)) is the cohomology of

Cone(C_{cr/K_{0}} ⊕ F^{n}C_{dR} → C_{cr/K_{0}} ⊕ C_{dR})[−1].

with map given by (x, y) → ((1 − ϕ̃/p^{n})x, x − y) There is a commutative diagram with exact rows

H_{dR}^{2n−1}(X_{K}/K)/F^{n} ⟶ H_{f}^{2n}(X, K(n)) ⟶ F^{n}H_{dR}^{2n}(X_{K}/K)^{ϕ̃=p^{n}}
=↓ ↓ ^{↪}↓
H_{dR}^{2n−1}(X_{K}/K)/F^{n} ⟶ H_{fp}^{2n}(X, n) ⟶ F^{n}H_{dR}^{2n}(X_{K}/K)

Theorem 3. (Nizioł) There is a canonical map l : H_{f}^{∗}(X, K(n)) → H_{ét}^{∗}(X_{K}, ℚ_{p}(n)) and compatible cycle classes

\begin{matrix}A^{i}(X) & ⟶^{cℓ_{f}} & H_{f}^{2i}(X, K(i)) \\ ↓ & & ^{l}↓ \\ A^{i}(X_{K}) & ⟶^{cℓ_{ét}} & H_{ét}^{2i}(X_{K}, ℚ_{p}(i)) \end{matrix}.

The exponential map is given as H_{dR}^{2i−1}(X_{K}/K)/F^{i} → H_{f}^{2i}(X, K(i))

⟶^{l} H_{ét}^{2i}(X_{K}, ℚ_{p}(i)) → H^{1}(K, H_{ét}^{2i−1}(XK̄, ℚ_{p}(i))

Advantage of H_{fp} - by Poincaré duality there is a pushforeward map. Proposition 1. For f : Y ↪ X, Y closed and smooth, Y_{K} of dimension j, cℓ_{f}(Y) viewed inside H_{fp}^{2i}(X, i) is just f_{∗}(1) with 1 ∈ H_{fp}^{0}(X, 0)

8. Further questions, regulators in K_{2}

C - smooth projective curve over O_{K} Étale regulator

r_{ét} : K_{2}(C) → H_{f}^{1}(K, H^{1}(CK̄, ℚ_{p}(2)).

6 AMNON BESSER

Using p-adic periods

r_{dR} : K_{2}(C) → Hom(H_{dR}^{1}(C_{K}/K), K)

Interesting to compare this with the regulator of Coleman and de-Shalit (Inventioness 88) defined on symbols by

r_{p,C}({f, g})(ω) = ∫_{(f)} log(g) · ω.