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1. Entire Theta Operators at Unramified Primes

   https://doi.org/10.1093/imrn/rnab190  

   Eischen, E; Mantovan, E (July 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract Starting with the work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of $$p$$-adic and $$\textrm{mod}\; p$$ modular forms and Galois representations. This paper achieves two main results for theta operators on automorphic forms on PEL-type Shimura varieties: (1) the analytic continuation at unramified primes $$p$$ to the whole Shimura variety of the $$\textrm{mod}\; p$$ reduction of $$p$$-adic Maass–Shimura operators a priori defined only over the $$\mu $$-ordinary locus, and (2) the construction of new $$\textrm{mod}\; p$$ theta operators that do not arise as the $$\textrm{mod}\; p$$ reduction of Maass–Shimura operators. While the main accomplishments of this paper concern the geometry of Shimura varieties and consequences for differential operators, we conclude with applications to Galois representations. Our approach involves a careful analysis of the behavior of Shimura varieties and enables us to obtain more general results than allowed by prior techniques, including for arbitrary signature, vector weights, and unramified primes in CM fields of arbitrary degree. 

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2. Geometric $$L$$-packets of Howe-unramified toral supercuspidal representations

   https://doi.org/10.4171/JEMS/1396  

   Chan, Charlotte; Oi, Masao (March 2025, Journal of the European Mathematical Society)

   We show thatL-packets of toral supercuspidal representations arising from unramified maximal tori ofp-adic groups are realized by Deligne–Lusztig varieties for parahoric subgroups. We prove this by exhibiting a direct comparison between the cohomology of these varieties and algebraic constructions of supercuspidal representations. Our approach is to establish that toral irreducible representations are uniquely determined by the values of their characters on a domain of sufficiently regular elements. This is an analogue of Harish-Chandra’s characterization of real discrete series representations by their characters on regular elements of compact maximal tori, a characterization which Langlands relied on in his construction ofL-packets of these representations. In parallel to the real case, we characterize the members of Kaletha’s toralL-packets by their characters on sufficiently regular elements of elliptic maximal tori. 

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3. Regular Supercuspidal Representations

   Tasho Kaletha (July 2019, Journal of the American Mathematical Society)

   We show that, in good residual characteristic, most supercuspidal representations of a tamely ramified reductive p-adic group G arise from pairs (S,\theta), where S is a tame elliptic maximal torus of G, and \theta is a character of S satisfying a simple root-theoretic property. We then give a new expression for the roots of unity that appear in the Adler-DeBacker-Spice character formula for these supercuspidal representations and use it to show that this formula bears a striking resemblance to the character formula for discrete series representations of real reductive groups. Led by this, we explicitly construct the local Langlands correspondence for these supercuspidal representations and prove stability and endoscopic transfer in the case of toral representations. In large residual characteristic this gives a construction of the local Langlands correspondence for almost all supercuspidal representations of reductive p-adic groups. 

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4. A Tannakian Framework for G-displays and Rapoport–Zink Spaces

   https://doi.org/10.1093/imrn/rnz311  

   Daniels, Patrick (December 2019, International Mathematics Research Notices)

   Abstract We develop a Tannakian framework for group-theoretic analogs of displays, originally introduced by Bültel and Pappas, and further studied by Lau. We use this framework to define Rapoport–Zink functors associated to triples $$(G,\{\mu \},[b])$$, where $$G$$ is a flat affine group scheme over $${\mathbb{Z}}_p$$ and $$\mu$$ is a cocharacter of $$G$$ defined over a finite unramified extension of $${\mathbb{Z}}_p$$. We prove these functors give a quotient stack presented by Witt vector loop groups, thereby showing our definition generalizes the group-theoretic definition of Rapoport–Zink spaces given by Bültel and Pappas. As an application, we prove a special case of a conjecture of Bültel and Pappas by showing their definition coincides with that of Rapoport and Zink in the case of unramified EL-type local Shimura data. 

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5. The cohomology of semi-infinite Deligne–Lusztig varieties

   https://doi.org/10.1515/crelle-2019-0039  

   Chan, Charlotte (January 2020, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes X h {X_{h}} . Boyarchenko’s two conjectures are on the maximality of X h {X_{h}} and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant 1 / n {1/n} in the case h = 2 {h=2} (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of X h {X_{h}} attains its Weil–Deligne bound, so that the cohomology of X h {X_{h}} is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group H c i ⁢ ( X h ) {H_{c}^{i}(X_{h})} are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attached to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence θ ↦ H c i ⁢ ( X h ) ⁢ [ θ ] {\theta\mapsto H_{c}^{i}(X_{h})[\theta]} agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p -adic groups in general. 

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