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1. The slice spectral sequence for a motivic analogue of the connective K(1)$K(1)$‐local sphere

   https://doi.org/10.1112/plms.70006  

   Kong, Hana Jia; Quigley, J D (November 2024, Proceedings of the London Mathematical Society)

   Abstract We compute the 2‐adic effective slice spectral sequence (ESSS) for the motivic stable homotopy groups of , a motivic analogue of the connective ‐local sphere over prime fields of characteristic not two. Together with the analogous computation over algebraically closed fields, this yields information about the motivic ‐local sphere over arbitrary base fields of characteristic not two. To compute the spectral sequence, we prove several results that may be of independent interest. We describe the ‐differentials in the slice spectral sequence in terms of the motivic Steenrod operations over general base fields, building on analogous results of Ananyevskiy, Röndigs, and Østvær for the very effective cover of Hermitian K‐theory. We also explicitly describe the coefficients of certain motivic Eilenberg–MacLane spectra and compute the ESSS for the very effective cover of Hermitian K‐theory over prime fields. 

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2. Period Relations and Special Values of Rankin-Selberg L-Functions

   https://doi.org/10.1007/978-3-319-59728-7_9  

   Harris, Michael; Lin, Jie (January 2017, Representation Theory, Number Theory, and Invariant Theory)

   ThisisasurveyofrecentworkonvaluesofRankin-SelbergL-functions of pairs of cohomological automorphic representations that are critical in Deligne’s sense. The base field is assumed to be a CM field. Deligne’s conjecture is stated in the language of motives over Q, and express the critical values, up to rational factors, as determinants of certain periods of algebraic differentials on a projective algebraic variety over homology classes. The results that can be proved by automorphic methods express certain critical values as (twisted) period integrals of automorphic forms. Using Langlands functoriality between cohomological automorphic repre- sentations of unitary groups, which can be identified with the de Rham cohomology of Shimura varieties, and cohomological automorphic representations of GL.n/, the automorphic periods can be interpreted as motivic periods. We report on recent results of the two authors, of the first-named author with Grobner, and of Guerberoff. 

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3. Generalized Bockstein maps and Massey products

   https://doi.org/10.1017/fms.2022.103  

   Lam, Yeuk Hay; Liu, Yuan; Sharifi, Romyar; Wake, Preston; Wang, Jiuya (January 2023, Forum of Mathematics, Sigma)

   Abstract Given a profinite group G of finite p -cohomological dimension and a pro- p quotient H of G by a closed normal subgroup N , we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H . We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H , we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H . We apply our study to prove lower bounds on the p -ranks of class groups of certain nonabelian extensions of $$\mathbb {Q}$$ and to give a new proof of the vanishing of Massey triple products in Galois cohomology. 

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4. Automorphic cohomology, motivic cohomology, and the adjoint L-function

   Prasanna, Kartik A.; Venkatesh, Akshay (July 2021, Asterisque)

   We propose a relationship between the cohomology of arithmetic groups, and the motivic cohomology of certain (Langlands-)attached motives. The motivic cohomology group in question is that related, by Beilinson’s conjecture, to the adjoint L-function at s=1. We present evidence for the conjecture using the theory of periods of automorphic forms, and using analytic torsion. 

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5. Hochschild homology of mod-𝑝 motivic cohomology over algebraically closed fields

   https://doi.org/10.1090/cams/36  

   Dundas, Bjørn; Hill, Michael; Ormsby, Kyle; Østvær, Paul (September 2024, Communications of the American Mathematical Society)

   We perform Hochschild homology calculations in the algebro-geometric setting of motives over algebraically closed fields. The homotopy ring of motivic Hochschild homology contains torsion classes that arise from the mod-p motivic Steenrod algebra and generating functions defined on the natural numbers with finite non-empty support. Under Betti realization, we recover Bökstedt’s calculation of the topological Hochschild homology of finite prime fields. 

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