Syntomic cohomology

General Introduction
From Hannu's thesis, section 2.2.4: "Syntomic cohomology is an “arithmetic” version of crystalline cohomology, whose purpose in padic Hodge theory is to bridge the gap between etale and crystalline cohomology. It was originally defined by Fontaine and Messing in [FM87] using “syntomic topology”, but soon thereafter an easier approach, based on “syntomic complexes”, was found by Kato [Kat87]. Our definition, taken from [Tsu00], follows this approach."
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Online References
There should be a paper by Nekovar with a useful appendix, see K0212
A useful thesis
Look also at the thesis of Hannu
See Galois Cohomology of Fontaine rings, in Fontaine theory folder
Things by Besser, see his webpage.
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Paper References
Bannai: Syntomic cohomology as a padic absolute Hodge cohomology. Abstract. The purpose of this paper is to interpret rigid syntomic cohomology, defined by Amnon Besser [Bes], as a padic absolute Hodge cohomology. This is a padic analogue of a work of Beilinson [Be1] which interprets BeilinsonDeligne cohomology in terms of absolute Hodge cohomology. In the process, we will define a theory of padic absolute Hodge cohomology with coefficients, which may be interpreted as a generalization of rigid syntomic cohomology to the case with coefficients.
Kato and Messing: Syntomic cohomology and $p$adic étale cohomology. Tohoku 1992.
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Definition
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Properties
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Standard theorems
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Open Problems
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Connections to Number Theory
Articles on syntomic regulators:
MR1031903 (91e:11070) Gros, Michel(FRENNBisomorphism) Régulateurs syntomiques et valeurs de fonctions $L$ $p$adiques. I. (French) [Syntomic regulators and values of $p$adic $L$functions. I] With an appendix by Masato Kurihara. Invent. Math. 99 (1990), no. 2, 293320.
Also part II by Gros, in Inventiones.
MR1632798 (99f:11151) Kolster, Manfred(3MMAS); Nguyen Quang Do, Thong(FFRANM) Syntomic regulators and special values of $p$adic $L$functions. (English summary)
MR1728549 (2001d:11070) Besser, Amnon(ILBGUN); Deninger, Christopher(DMUNSisomorphism) $p$adic Mahler measures. J. Reine Angew. Math. 517 (1999), 1950.
MR1909217 (2003e:11070) Bannai, Kenichi(JTOKYOGM) On the $p$adic realization of elliptic polylogarithms for CMelliptic curves. (English summary) Duke Math. J. 113 (2002), no. 2, 193236.
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Computations and Examples
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History and Applications
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Some Research Articles
The syntomic regulator for Ktheory of fields, by Amnon Besser and Rob de Jeu: K0523
arXiv:0910.4436 Cohomologie syntomique: liens avec les cohomologies étale et rigide from arXiv Front: math.AG by JeanYves Etesse Syntomic cohomology here defined yields a link between rigid cohomology and etale cohomology, viewing the last one as the fixed points under Frobenius of the former one. Let V be a complete discrete valuation ring, with perfect residue field k = V/m of characteristic p > 0 and fraction field K of characteristic 0. Having defined syntomic cohomology with compact supports of an abelian sheaf G on a kscheme X, we show that it coincides with etale cohomology with compact supports when G is a lisse sheaf. If moreover the convergent Fisocrystal associated to G comes from an overconvergent isocrystal E, then the rigid cohomology of E expresses as a limit of syntomic cohomologies: then the etale cohomology with compact supports of G is the fixed points of Frobenius acting on the rigid cohomology of E.
arXiv:1003.2810 Cyclotomic complexes from arXiv Front: math.AT by D. Kaledin We construct a triangulated category of cyclotomic complexes, a homological counterpart of cyclotomic spectra of Bokstedt and Madsen. We also construct a version of the Topological Cyclic Homology functor TC for cyclotomic complexes, and an equivariant homology functor from cycloctomic spectra to cyclotomic complexes which commutes with TC. Then on the other hand, we prove that the category of cyclotomic complexes is essentially a twisted 2periodic derived category of the category of filtered Dieudonne modules of Fontaine and Lafaille. We also show that under some mild conditions, the functor TC on cyclotomic complexes is the syntomic cohomology functor.
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Other Information
In MR2275605 (review of Niziol ICM talk) it says that syntomic cohomology has a motivic description, due to work of Geisser, and assuming the BK conjecture.
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