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1. Perfectoidness via Sen theory and applications to Shimura varieties

   https://doi.org/10.1090/jams/1060  

   He, Tongmu (July 2025, Journal of the American Mathematical Society)

   Sen’s theorem on the ramification of a $p$-adic analytic Galois extension of $p$-adic local fields shows that its perfectoidness is equivalent to the non-vanishing of its arithmetic Sen operator. By developing $p$-adic Hodge theory for general valuation rings, we establish a geometric analogue of Sen’s criterion for any $p$-adic analytic Galois extension of $p$-adic varieties: its (Riemann-Zariski) stalkwise perfectoidness is necessary for the non-vanishing of the geometric Sen operators. As the latter is verified for general Shimura varieties by Pan and Rodríguez Camargo, we obtain the perfectoidness of every completed stalk of general Shimura varieties at infinite level at $p$. As an application, we prove that the integral completed cohomology groups vanish in higher degrees, verifying a conjecture of Calegari-Emerton for general Shimura varieties. 

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2. Finiteness of reductions of Hecke orbits

   https://doi.org/10.4171/JEMS/1395  

   Kisin, Mark; Lam, Yeuk_Hay Joshua; Shankar, Ananth N; Srinivasan, Padmavathi (December 2023, Journal of the European Mathematical Society)

   We prove two finiteness results for reductions of Hecke orbits of abelian varieties over local fields: one in the case of supersingular reduction and one in the case of reductive monodromy. As an application, we show that only finitely many abelian varieties on a fixed isogeny leaf admit CM lifts, which in particular implies that in each fixed dimensiongonly finitely many supersingular abelian varieties admit CM lifts. Combining this with the Kuga–Satake construction, we also show that only finitely many supersingular K3surfaces admit CM lifts. Our tools includep-adic Hodge theory and group-theoretic techniques. 

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3. An extension of the Fukaya–Kato method

   https://doi.org/10.4171/jems/1519  

   Sharifi, Romyar (September 2024, Journal of the European Mathematical Society)

   In a groundbreaking paper, T. Fukaya and K. Kato proved a slight weakening of a conjecture of the author under an assumption that a Kubota–Leopoldtp-adicL-function has no multiple zeros. This article describes a refinement of their method that sheds light on the role of thep-adicL-function. 

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4. 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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5. Dirac geometry II: coherent cohomology

   https://doi.org/10.1017/fms.2024.2  

   Hesselholt, Lars; Pstrągowski, Piotr (January 2024, Forum of Mathematics, Sigma)

   Abstract Dirac rings are commutative algebras in the symmetric monoidal category of$$\mathbb {Z}$$-graded abelian groups with the Koszul sign in the symmetry isomorphism. In the prequel to this paper, we developed the commutative algebra of Dirac rings and defined the category of Dirac schemes. Here, we embed this category in the larger$$\infty $$-category of Dirac stacks, which also contains formal Dirac schemes, and develop the coherent cohomology of Dirac stacks. We apply the general theory to stable homotopy theory and use Quillen’s theorem on complex cobordism and Milnor’s theorem on the dual Steenrod algebra to identify the Dirac stacks corresponding to$$\operatorname {MU}$$and$$\mathbb {F}_p$$in terms of their functors of points. Finally, in an appendix, we develop a rudimentary theory of accessible presheaves. 

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