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1. Drinfeld’s Lemma for F -isocrystals, I

   https://doi.org/10.1093/imrn/rnae039  

   Kedlaya, Kiran S (March 2024, International Mathematics Research Notices)

   Abstract We prove that in either the convergent or overconvergent setting, an absolutely irreducible $$F$$-isocrystal on the absolute product of two or more smooth schemes over perfect fields of characteristic $$p$$, further equipped with actions of the partial Frobenius maps, is an external product of $$F$$-isocrystals over the multiplicands. The corresponding statement for lisse $$\overline{{\mathbb{Q}}}_{\ell }$$-sheaves, for $$\ell \neq p$$ a prime, is a consequence of Drinfeld’s lemma on the fundamental groups of absolute products of schemes in characteristic $$p$$. The latter plays a key role in V. Lafforgue’s approach to the Langlands correspondence for reductive groups with $$\ell $$-adic coefficients; the $$p$$-adic analogue will be considered in subsequent work with Daxin Xu. 

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2. Automorphy lifting with adequate image

   https://doi.org/10.1017/fms.2023.3  

   Miagkov, Konstantin; Thorne, Jack A. (January 2023, Forum of Mathematics, Sigma)

   Abstract Let F be a CM number field. We generalise existing automorphy lifting theorems for regular residually irreducible p -adic Galois representations over F by relaxing the big image assumption on the residual representation. 

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3. Potential automorphy for $$GL_n$$

   https://doi.org/10.1007/s00222-022-01161-6  

   Qian, Lie (September 2022, Inventiones mathematicae)

   We prove potential automorphy results for a single Galois representation 𝐺𝐹→𝐺𝐿𝑛(ℚ⎯⎯⎯⎯⎯𝑙) where F is a CM number field. The strategy is to use the p, q switch trick to go between the p-adic and q-adic realisation of a certain variant of the Dwork motive. We choose this variant to break self-duality shape of the motives, but not the Hodge-Tate weights. Another key result to prove is that certain p-adic representations we choose that come from the Dwork motives is ordinarily automorphic. One input is the automorphy lifting theorem in Allen et al.: (Potential automorphy over CM fields, Cornell University, New York 2018) . 

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4. Syntomic complexes and p -adic étale Tate twists

   https://doi.org/10.1017/fmp.2022.21  

   Bhatt, Bhargav; Mathew, Akhil (January 2023, Forum of Mathematics, Pi)

   Abstract The primary goal of this paper is to identify syntomic complexes with the p -adic étale Tate twists of Geisser–Sato–Schneider on regular p -torsion-free schemes. Our methods apply naturally to a broader class of schemes that we call ‘ F -smooth’. The F -smoothness of regular schemes leads to new results on the absolute prismatic cohomology of regular schemes. 

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5. G -valued crystalline deformation rings in the Fontaine–Laffaille range

   https://doi.org/10.1112/S0010437X23007297  

   Booher, Jeremy; Levin, Brandon (August 2023, Compositio Mathematica)

   Let$$G$$be a split reductive group over the ring of integers in a$$p$$-adic field with residue field$$\mathbf {F}$$. Fix a representation$$\overline {\rho }$$of the absolute Galois group of an unramified extension of$$\mathbf {Q}_p$$, valued in$$G(\mathbf {F})$$. We study the crystalline deformation ring for$$\overline {\rho }$$with a fixed$$p$$-adic Hodge type that satisfies an analog of the Fontaine–Laffaille condition for$$G$$-valued representations. In particular, we give a root theoretic condition on the$$p$$-adic Hodge type which ensures that the crystalline deformation ring is formally smooth. Our result improves on all known results for classical groups not of type A and provides the first such results for exceptional groups. 

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