Syntomic cohomology and regulators for varieties over p-adic fields

Abstract

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 to étale 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 the étale side. In relative dimension zero we recover the potentially semistable Selmer groups and, as an application, we prove that Soulé's étale 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

Final version; to appear in Algebra and Number Theory

References in corpus (2)

• Finite polynomial cohomology for general varieties
• The rigid syntomic ring spectrum

Cited by in corpus (16)

• Rankin--Eisenstein classes and explicit reciprocity laws
• Euler systems for Hilbert modular surfaces
• The universal $p$-adic Gross-Zagier formula
• Finite polynomial cohomology for general varieties
• Anticyclotomic Euler systems for unitary groups
• Morphismes de périodes et cohomologie syntomique
• Geometric syntomic cohomology and vector bundles on the Fargues-Fontaine curve
• Log Rigid Syntomic Cohomology for Strictly Semistable Schemes
• A new proof of a vanishing result due to Berthelot, Esnault, and Rülling
• Syntomic cohomology and $p$-adic motivic cohomology
• The Galois group of the category of mixed Hodge-Tate structures
• $(\phi ,\Gamma )$-modules de de Rham et fonctions $L$ $p$-adiques
• A universal Euler system for GSp(4)
• Milnor K-theory, F-isocrystals and Syntomic Regulators
• Theta cycles and the Beilinson--Bloch--Kato conjectures
• A Conjecture of Flach and Morin