p-ADIC INTEGRATION AND p-ADIC REGULATORS

AMNON BESSER

Let K be a finite extension of the field ℚ_{p} of p-adic numbers and let O_{K} be the ring of integers of K. Let X be a projective smooth scheme over O_{K} and let X_{K} denote its generic fiber and XK̄, X_{ℂ_{p}} extensions of scalars to the algebraic closure K̄ of K, respectively its completion ℂ_{p}. At the beginning of the 80’s Coleman began developing a theory of p-adic integration of one-forms on X_{K}. In particular, his theory allows one to associate to any holomorphic one form ω on X_{K} a primitive f_{ω} which is uniquely defined up to a constant. The function f_{ω} is a locally analytic function on X_{ℂ_{p}} and satisfies the relation df_{ω} = ω. It is also Galois equivariant. Given a zero cycle z = ∑ n_{i}y_{i} on X_{K} (i.e., y_{i} ∈ X(K̄) and z is Gal(K̄/K) invariant) whose degree ∑ n_{i} is 0, the expression ∫_{z} ω := ∑ n_{i}f_{ω}(y_{i}) is well defined. It can be shown to be related to the albanese image of z.

The aim of this work is to generalize the theory of Coleman to integration of de Rham cohomology classes of odd degree ω ∈ F^{j+1}H_{dR}^{2j+1}(X_{K}/K) (so Coleman’s integral is the case j = 0). We will concentrate on one of two possible approaches which is based on a new cohomology theory, the finite polynomial cohomology of X.

The new cohomology theory, denoted H_{fp}^{i}(X, n), possesses a number of very convenient properties:

1. It sits in 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;

2. As an easy consequence of 1. it has a trace map tr : H_{fp}^{2d+1}(X, d + 1) → K, where d is the dimension of X_{K};

3. It has a cup product which is compatible in a natural way with the exact sequence of 1. and it satisfied Poincaré duality;

4. There is a natural map from syntomic cohomology into finite polynomial cohomology.

The generalization of Coleman’s integral is now as follows: Take a de Rham class ω ∈ F^{j+1}H_{dR}^{2j+1}(X_{K}/K) and choose a pullback ω̃ ∈ H_{fp}^{2j+1}(X, j + 1). Given a smooth subscheme f : Y → X whose generic fiber has dimension j, we define ∫_{Y} ω̃ to be tr_{Y} (f^{∗}(ω̃)). Given a cycle Z = ∑ n_{i}Y_{i} with Y_{i} as above and such that Z is homologically equivalent to 0 on X_{K}, it is not hard to show that ∫_{Z} ω := ∑ n_{i} ∫_{Y_{i}} ω̃ depends only on ω and not on the particular lift.

We can show that this integral coincides with the Coleman integral in the case j = 0:

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}.

28 AMNON BESSER

Suppose Z as above (j arbitrary). Using the theory of p-adic periods and the étale Abel-Jacobi map we can define an Abel-Jacobi map into de-Rham cohomology

α_{dR}(Z) ∈ Hom_{K}(F^{j+1}H_{dR}^{2j+1}(X_{K}/K), K)

Our main result is that this map can be computed using p-adic integration: Theorem 2. The following equality holds for ω ∈ F^{j+1}H_{dR}^{2j+1}(X_{K}/K):

α_{dR}(Z)(ω) = ∫_{Z} ω