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1. On the K-theory of division algebras over local fields

   https://doi.org/10.1007/s00222-019-00909-x  

   Hesselholt, Lars; Larsen, Michael; Lindenstrauss, Ayelet (July 2019, Inventiones mathematicae)

   Let K be a complete discrete valuation field with finite residue field of characteristic p, and let D be a central division algebra over K of finite index d. Thirty years ago, Suslin and Yufryakov showed that for all prime numbers ℓ different from p and integers j≥1 , there exists a "reduced norm" isomorphism of ℓ-adic K-groups Nrd_{D/K}:K_j(D,Z_ℓ)→K_j(K,Z_ℓ) such that d⋅Nrd_{D/K} is equal to the norm homomorphism N_{D/K}. The purpose of this paper is to prove the analogous result for the p-adic K-groups. To do so, we employ the cyclotomic trace map to topological cyclic homology and show that there exists a "reduced trace" equivalence Trd_{A/S}:THH(A|D,Z_p)→THH(S|K,Z_p) between two p-complete cyclotomic spectra associated with D and K, respectively. Interestingly, we show that if p divides d, then it is not possible to choose said equivalence such that, as maps of cyclotomic spectra, d⋅Trd_{A/S} agrees with the trace Tr_{A/S}, although this is possible as maps of spectra with T-action 

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2. Overconvergent modular forms are highest-weight vectors in the Hodge-Tate weight zero part of completed cohomology

   https://doi.org/10.1017/fms.2021.16  

   Howe, Sean (January 2021, Forum of Mathematics, Sigma)

   null (Ed.)

   Abstract We construct a $$(\mathfrak {gl}_2, B(\mathbb {Q}_p))$$ and Hecke-equivariant cup product pairing between overconvergent modular forms and the local cohomology at $$0$$ of a sheaf on $$\mathbb {P}^1$$ , landing in the compactly supported completed $$\mathbb {C}_p$$ -cohomology of the modular curve. The local cohomology group is a highest-weight Verma module, and the cup product is non-trivial on a highest-weight vector for any overconvergent modular form of infinitesimal weight not equal to $$1$$ . For classical weight $$k\geq 2$$ , the Verma has an algebraic quotient $$H^1(\mathbb {P}^1, \mathcal {O}(-k))$$ , and on classical forms, the pairing factors through this quotient, giving a geometric description of ‘half’ of the locally algebraic vectors in completed cohomology; the other half is described by a pairing with the roles of $H^1$ and $H^0$ reversed between the modular curve and $$\mathbb {P}^1$$ . Under minor assumptions, we deduce a conjecture of Gouvea on the Hodge-Tate-Sen weights of Galois representations attached to overconvergent modular forms. Our main results are essentially a strict subset of those obtained independently by Lue Pan, but the perspective here is different, and the proofs are short and use simple tools: a Mayer-Vietoris cover, a cup product, and a boundary map in group cohomology. 

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3. Algebraic stability of meromorphic maps descended from Thurston’s pullback maps

   https://doi.org/10.1090/tran/8221  

   Ramadas, Rohini (January 2021, Transactions of the American Mathematical Society)

   null (Ed.)

   Let ϕ : S 2 → S 2 \phi :S^2 \to S^2 be an orientation-preserving branched covering whose post-critical set has finite cardinality n n . If ϕ \phi has a fully ramified periodic point p ∞ p_{\infty } and satisfies certain additional conditions, then, by work of Koch, ϕ \phi induces a meromorphic self-map R ϕ R_{\phi } on the moduli space M 0 , n \mathcal {M}_{0,n} ; R ϕ R_{\phi } descends from Thurston’s pullback map on Teichmüller space. Here, we relate the dynamics of R ϕ R_{\phi } on M 0 , n \mathcal {M}_{0,n} to the dynamics of ϕ \phi on S 2 S^2 . Let ℓ \ell be the length of the periodic cycle in which the fully ramified point p ∞ p_{\infty } lies; we show that R ϕ R_{\phi } is algebraically stable on the heavy-light Hassett space corresponding to ℓ \ell heavy marked points and ( n − ℓ ) (n-\ell ) light points. 

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4. Cup products on curves over finite fields

   Frauke M. Bleher and Ted Chinburg (January 2021, arXivorg)

   In this paper we compute cup products of elements of the first étale cohomology group of μℓ over a smooth projective geometrically irreducible curve C over a finite field k when ℓ is a prime and #k≡1 mod ℓ. Over the algebraic closure of k, such products are values of the Weil pairing on the ℓ-torsion of the Jacobian of C. Over k, such cup products are more subtle due to the fact that they naturally take values in Pic(C)⊗Zμ~ℓ rather than in the group μ~ℓ of ℓth roots of unity in k∗. 

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5. Local types of (Γ,G)$$(\Gamma ,G)$$‐bundles and parahoric group schemes

   https://doi.org/10.1112/plms.12544  

   Damiolini, Chiara; Hong, Jiuzu (June 2023, Proceedings of the London Mathematical Society)

   Abstract Let be a simple algebraic group over an algebraically closed field . Let be a finite group acting on . We classify and compute the local types of ‐bundles on a smooth projective ‐curve in terms of the first nonabelian group cohomology of the stabilizer groups at the tamely ramified points with coefficients in . When , we prove that any generically simply connected parahoric Bruhat–Tits group scheme can arise from a ‐bundle. We also prove a local version of this theorem, that is, parahoric group schemes over the formal disc arise from constant group schemes via tamely ramified coverings. 

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