Syntomic cohomology and p-adic regulators for varieties over p-adic fields.

Jan Nekovar (1) , Wieslawa Niziol (2)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 ENS de Lyon - École normale supérieure de Lyon

We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over p-adic rings extends uniquely to a cohomology theory for varieties over p-adic fields that satisfies h-descent. This new cohomology-syntomic cohomology-is a Bloch-Ogus cohomology theory, admits period map tó etale cohomology, and has a syntomic descent spectral sequence (from an algebraic closure of the given field to the field itself) that is compatible with the Hochschild-Serre spectral sequence on thé etale side and is related to the Bloch-Kato exponential map. In relative dimension zero we recover the potentially semistable Selmer groups and, as an application, we prove that Soulé's ´ etale regulators land in the potentially semistable Selmer groups. Our construction of syntomic cohomology is based on new ideas and techniques developed by Beilinson and Bhatt in their recent work on p-adic comparison theorems.

Domaines: Théorie des nombres [math.NT]; Géométrie algébrique [math.AG]

Jan Nekovar, Wieslawa Niziol. Syntomic cohomology and p-adic regulators for varieties over p-adic fields.. Algebra & Number Theory, 2016. ⟨ensl-01420345⟩