Equivariance in Tannakian duality and plectic p-adic Hodge theory

For a gerbe G and a finite constant group S acting on it, we use the Grothendieck construction to define a semidirect product gerbe G ⋊ S. We describe such gerbes in terms of Cech cocycles via the cohomology of crossed modules. We characterise a dual form of the Grothendieck construction and show that in the special case of the group S acting on a Tannakian category C, it gives rise to the category C^S of S-equivariant objects of C. If G and C correspond by Tannakian duality, then so do G ⋊ S and C^S. We derive descent for Tannakian categories from a classification of Galois gerbes and give a criterion for a descended category to be neutral. We then study a plectic variant of Fontaine theory for p-adic representations of the local plectic Galois group at p associated to a totally real number field F, when p is inert in F. In particular, we show that the plectic crystalline Fontaine functor is valued in a category of plectic isocrystals. We prove that this category is Tannakian using the theory developed in the first part.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2025

Author dc:creator

• Kofler, Lukas

Advisor dc:contributor.advisor

• Scholl, Anthony

• Algebraic number theory
• Mathematics
• p-adic Hodge theory
• Tannakian duality

Licence

• https://www.repository.cam.ac.uk/bitstreams/8fb52048-6d7a-43c9-a4f7-a2da7dce877a/download
• http://purl.org/NET/rdflicense/allrightsreserved

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.121373

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/389514