Rigid syntomic cohomology and p-adic polylogarithms

Abstract

The main purpose of this paper is to construct the p-adic realization of the classical polylogarithm following the method of Beilinson and Deligne as explained by Huber and Wildeshaus. A simplicial construction of the p-adic polylogarithm was previously obtained by Somekawa. In this paper, we will give a sheaf theoretic interpretation of this construction. In particular, we will give an interpretation of the p-adic polylogarithm as an object in the p-adic analogue of the category of variation of mixed Hodge structures. We will also calculate the restriction of this object to torsion points, and will prove a result which is compatible with the results of Gros-Kurihara, Gros and Somekawa. IntroductionIn the paper [23], M. Somekawa constructed the p-adic polylogarithm in the syntomic cohomology of a simplicial scheme over the projective line minus three points. This is a p-adic analogue of the elements in absolute Hodge and l-adic cohomology de®ned by Beilinson [2] and is the image by the syntomic regulator of the motivic polylogarithm in motivic cohomology. The purpose of this paper is to develop a theory of p-adic mixed sheaves which is su½cient to give a sheaf theoretic interpretation of the construction by Somekawa. In particular, as in the classical case, the p-adic polylogarithm has an interpretation as an object in the p-adic analogue of the category of variation of mixed Hodge structures on the projective line minus three points. In Theorem 2, which is our main result, we will explicitly determine the shape of the p-adic polylogarithm.As a corollary, we will show that the specialization of this object to the roots of unity can be expressed by the p-adic polylogarithm function de®ned by Coleman. This result is compatible with the result of Somekawa, and is compatible with the results of Gros-Kurihara [14] and Gros [15] on the calculation of the image of the Beilinson element with respect to the syntomic regulator. This gives evidence that the p-adic formalism which we develop in this paper is potentially a powerful tool in the calculation of the image by the syntomic regulator of elements in motivic cohomology.

Syntomic cohomology as a p -adic absolute Hodge cohomology

“…This may be interpreted as a generalization to the case with coefficients of rigid syntomic cohomology. This is also a generalization of absolute syntomic cohomology with coefficients defined in our previous paper [Ba1] to the case when the base field is ramified over Q p . We remark that a cohomology theory with coefficients for proper and smooth schemes has been considered by W. Nizioĺ [Ni].…”

“…Suppose K = K 0 . Then a syntomic datum X and an admissible syntomic coefficient M in the sense of [Ba1] naturally give rise to X and M of this paper. We have a canonical isomorphism…”

“…We remark that a cohomology theory with coefficients for proper and smooth schemes has been considered by W. Nizioĺ [Ni]. The philosophy of this paper played an important role in the papers [Ba1] [Ba2].…”

On the de Rham and $p$-adic realizations of the elliptic polylogarithm for CM elliptic curves

“…Rigid syntomic cohomology. We first briefly recall the theory of filtered overconvergent F -isocrystals (or syntomic coefficients) and rigid syntomic cohomology developed in [Ba1]. Let K be a finite unramified extension of Q p with ring of integers O K and residue field k. We fix an integer q = p m for some m ≥ 1, and we denote by Frob q the Frobenius x → x q on k. We let σ be the extension to O K and K of the Frobenius Frob q on k.…”

“…Suppose we are given a coherent O X K -module M with integrable connection ∇ : M → M ⊗ Ω 1 X K (log D) with logarithmic poles along D. Then as in [Ba1], paragraph before Definition 1.8, we let …”