Kenichi Bannai Abstract. The purpose of this paper is to interpret rigid syntomic cohomology, defined by Amnon Besser [Bes], as a p-adic absolute Hodge cohomology. This is a p-adic analogue of a work of Beilinson [Be1] which interprets Beilinson-Deligne cohomology in terms of absolute Hodge cohomology. In the process, we will define a theory of p-adic absolute Hodge cohomology with coefficients, which may be interpreted as a generalization of rigid syntomic cohomology to the case with coefficients.
MR1799937 (2002c:11068) Bannai, Kenichi(J-TOKYOGM) Rigid syntomic cohomology and -adic polylogarithms. This paper looks very interesting.
MR1809626 (2002c:14035) Besser, Amnon(4-DRHM) Syntomic regulators and -adic integration. I. Rigid syntomic regulators. First paragraph of review: For (possibly non-proper) smooth schemes over a -adic field with good reduction, the author constructs a theory of rigid syntomic cohomology, which is a -adic analogue of the Deligne cohomology. This theory is based on P. Berthelot’s rigid cohomology [Invent. Math. 128 (1997), no. 2, 329377; MR1440308 (98j:14023)], in the same way as the Deligne cohomology is based on the de Rham/Hodge cohomology. He also constructs a syntomic regulator from -theory to his rigid syntomic cohomology, which is an analogue of Beilinson’s regulator. Probably one may expect to formulate a -adic version of the Beilinson conjecture, which should predict a relation between special values of -adic -functions and his syntomic regulator. The author obtains a result in this direction for CM elliptic curves [Part II, Israel J. Math. 120 (2000), part B, 335359; MR1809627 (2002c:14036); see the following review].
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