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1. Quadratic enrichment of the logarithmic derivative of the zeta function

   https://doi.org/10.1090/btran/201  

   Bilu, Margaret; Ho, Wei; Srinivasan, Padmavathi; Vogt, Isabel; Wickelgren, Kirsten (October 2024, Transactions of the American Mathematical Society, Series B)

   We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck–Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure. 

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2. -DISPLAYS AND RAPOPORT–ZINK SPACES

   https://doi.org/10.1017/S1474748018000373  

   Bültel, O.; Pappas, G. (July 2020, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Let $$(G,\unicode[STIX]{x1D707})$$ be a pair of a reductive group $$G$$ over the $$p$$ -adic integers and a minuscule cocharacter $$\unicode[STIX]{x1D707}$$ of $$G$$ defined over an unramified extension. We introduce and study ‘ $$(G,\unicode[STIX]{x1D707})$$ -displays’ which generalize Zink’s Witt vector displays. We use these to define certain Rapoport–Zink formal schemes purely group theoretically, i.e. without $$p$$ -divisible groups. 

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3. Equivariant Witt complexes and twisted topological Hochschild homology

   https://doi.org/10.1016/j.topol.2025.109444  

   Bohmann, Anna Marie; Gerhardt, Teena; Krulewski, Cameron; Petersen, Sarah; Yang, Lucy (May 2025, Topology and its Applications)

   The topological Hochschild homology of a ring (or ring spectrum) R is an S1-spectrum, and the fixed points of THH(R) for subgroups C_n of S1 have been widely studied due to their use in algebraic K-theory computations. Hesselholt and Madsen proved that the fixed points of topological Hochschild homology are closely related to Witt vectors. Further, they defined the notion of a Witt complex, and showed that it captures the algebraic structure of the homotopy groups of the fixed points of THH. Recent work defines a theory of twisted topological Hochschild homology for equivariant rings (or ring spectra) that builds upon Hill, Hopkins and Ravenel's work on equivariant norms. In this paper, we study the algebraic structure of the equivariant homotopy groups of twisted THH. In particular, drawing on the definition of equivariant Witt vectors by Blumberg, Gerhardt, Hill and Lawson, we define an equivariant Witt complex and prove that the equivariant homotopy of twisted THH has this structure. Our definition of equivariant Witt complexes contributes to a growing body of research in the subject of equivariant algebra. 

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4. Relation between two twisted inverse image pseudofunctors in duality theory

   https://doi.org/10.1112/S0010437X14007672  

   Iyengar, Srikanth B.; Lipman, Joseph; Neeman, Amnon (April 2015, Compositio Mathematica)

   Grothendieck duality theory assigns to essentially finite-type maps $$f$$ of noetherian schemes a pseudofunctor $$f^{\times }$$ right-adjoint to $$\mathsf{R}f_{\ast }$$ , and a pseudofunctor $$f^{!}$$ agreeing with $$f^{\times }$$ when $$f$$ is proper, but equal to the usual inverse image $$f^{\ast }$$ when $$f$$ is étale. We define and study a canonical map from the first pseudofunctor to the second. This map behaves well with respect to flat base change, and is taken to an isomorphism by ‘compactly supported’ versions of standard derived functors. Concrete realizations are described, for instance for maps of affine schemes. Applications include proofs of reduction theorems for Hochschild homology and cohomology, and of a remarkable formula for the fundamental class of a flat map of affine schemes. 

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5. The Temperley–Lieb tower and the Weyl algebra

   https://doi.org/10.1112/jlms.70174  

   Harper, Matthew; Samuelson, Peter (August 2024, Journal of the London Mathematical Society)

   We define a monoidal category W and a closely related 2‐category 2Weyl using diagrammatic methods. We show that 2Weyl acts on the category TL of modules over Temperley–Lieb algebras, with its generating 1‐morphisms acting by induction and restriction. The Grothendieck groups of W and a third category we define W^\infty are closely related to the Weyl algebra. We formulate a sense in which K_0(W^\infty) acts asymptotically on K_0(TL). 

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