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Equivariance in Tannakian duality and plectic p-adic Hodge theory https://doi.org/10.17863/CAM.121373 Kofler, Lukas Scholl, Anthony 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. University of Cambridge 2025-03-21 Thesis Doctoral Doctor of Philosophy (PhD) eng Metheringham Scholarship https://www.repository.cam.ac.uk/handle/1810/389514 https://www.repository.cam.ac.uk/bitstreams/20e23853-078c-4f51-adc7-b188c7ef8657/download 57e861d612a9b9237679aa7d3ac9cb32 https://www.repository.cam.ac.uk/bitstreams/8fb52048-6d7a-43c9-a4f7-a2da7dce877a/download 87eda9de84448d1f82354d60eee3eb5f http://purl.org/NET/rdflicense/allrightsreserved Algebraic number theory Mathematics p-adic Hodge theory Tannakian duality