CRYSTALLINE SHEAVES, SYNTOMIC COHOMOLOGY AND p-ADIC POLYLOGARITHMS

In [BD92] (see also [HW98]), A. A. Beilinson and P. Deligne constructed the motivic polylogarithmic sheaf on PQ\{0, 1,∞}. Its specializations at primitive d-th roots of unity give the Beilinson’s elements of H M(Q(μd),Q(m)) = K2m−1(Q(μd))⊗ Q (m ≥ 1), whose images under the regulator maps to Deligne cohomology are the values of m-th polylogarithmic functions at primitive d-th roots of unity. The polylogarithmic functions appear as the (complex) period functions. In these notes, we will show that the following two p-adic realizations correspond to each other via the theory of crystalline sheaves by G. Faltings [Fal89] V f). One is the realization in the category of smooth Qp-sheaves on (PQp\{0, 1,∞})ét. Its specializations at primitive d-th roots of unity give the images of the Beilinson’s elements in H(Qp(μd),Qp(m)) = Ext1RepQp (GK)(Qp,Qp(m)) (m ≥ 1) under the regulator maps. It is known that they also coincide with the Soulé’s cyclotomic elements. The other is the realization in the category of (log) filtered convergent F -isocrystals on P1Zp endowed with the log structure associated to the divisor {0, 1,∞}. In [Ban00a], K. Bannai constructed the realization in the category of filtered overconvergent F -isocrystals on PZp\{0, 1,∞} using rigid syntomic cohomology, gave an explicit description of it in terms of p-adic polylogarithmic functions, and then proved that the specializations of the crystalline polylogarithmic sheaf at primitive d-th roots of unity for d prime to p give the values of p-adic polylogarithmic functions at primitive d-th roots of unity. His construction and calculation also work for log filtered convergent F -isocrystals and log syntomic cohomology, and we prefer the log version to the overconvergent one because we have the theory of crystalline sheaves by G. Faltings for the former. We will see that some functions which live in a big ring Bcrys and satisfy the same differential equations as the complex polylogarithmic functions, appear as the p-adic period functions of these p-adic realizations (Proposition 6.8). By combining the result of K. Bannai explained above with our comparison theorem, we immediately obtain a new proof of the following fact: The images of the Beilinson’s elements in H(Qp(μd),Qp(m)) (m ≥ 1) under the regulator maps for d prime to p coincide with the images of the values of m-th p-adic polylogarithmic functions at primitive d-th roots of unity under the map: