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1. 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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2. Free Incomplete Tambara Functors are Almost Never Flat

   https://doi.org/10.1093/imrn/rnab361  

   Hill, Michael A.; Mehrle, David; Quigley, James D. (January 2022, International Mathematics Research Notices)

   Abstract Free algebras are always free as modules over the base ring in classical algebra. In equivariant algebra, free incomplete Tambara functors play the role of free algebras and Mackey functors play the role of modules. Surprisingly, free incomplete Tambara functors often fail to be free as Mackey functors. In this paper, we determine for all finite groups conditions under which a free incomplete Tambara functor is free as a Mackey functor. For solvable groups, we show that a free incomplete Tambara functor is flat as a Mackey functor precisely when these conditions hold. Our results imply that free incomplete Tambara functors are almost never flat as Mackey functors. However, we show that after suitable localizations, free incomplete Tambara functors are always free as Mackey functors. 

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3. Cohomological Invariants in Positive Characteristic

   https://doi.org/10.1093/imrn/rnaa321  

   Totaro, Burt (January 2021, International Mathematics Research Notices)

   Abstract We determine the mod $$p$$ cohomological invariants for several affine group schemes $$G$$ in characteristic $$p$$. These are invariants of $$G$$-torsors with values in étale motivic cohomology, or equivalently in Kato’s version of Galois cohomology based on differential forms. In particular, we find the mod 2 cohomological invariants for the symmetric groups and the orthogonal groups in characteristic 2, which Serre computed in characteristic not 2. We also determine all operations on the mod $$p$$ étale motivic cohomology of fields, extending Vial’s computation of the operations on the mod $$p$$ Milnor K-theory of fields. 

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4. The Cohomological Brauer Group of a Torsion $${\mathbb{G}}_{m}$$-Gerbe

   https://doi.org/10.1093/imrn/rnz235  

   Shin, Minseon (November 2019, International Mathematics Research Notices)

   Abstract Let $$S$$ be a scheme and let $$\pi : \mathcal{G} \to S$$ be a $${\mathbb{G}}_{m,S}$$-gerbe corresponding to a torsion class $$[\mathcal{G}]$$ in the cohomological Brauer group $${\operatorname{Br}}^{\prime}(S)$$ of $$S$$. We show that the cohomological Brauer group $${\operatorname{Br}}^{\prime}(\mathcal{G})$$ of $$\mathcal{G}$$ is isomorphic to the quotient of $${\operatorname{Br}}^{\prime}(S)$$ by the subgroup generated by the class $$[\mathcal{G}]$$. This is analogous to a theorem proved by Gabber for Brauer–Severi schemes. 

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5. Why Do God and Humans Punish? Perceived Retributivist Punishment Motives Hinge on Views of the True Self

   https://doi.org/10.1177/01461672231160027  

   Lee, Young-eun; Dunlea, James_P; Heiphetz, Larisa (April 2023, Personality and Social Psychology Bulletin)

   Laypeople often believe that God punishes transgressions; however, their inferences about God’s punishment motives remain unclear. We addressed this topic by asking laypeople to indicate why God punishes. We also examined participants’ inferences about why humans punish to contribute to scholarly conversations regarding the extent to which people may anthropomorphize God’s mind. In Studies 1A to 1C, participants viewed God as less retributive than humans. In Study 2, participants expected God (vs. humans) to view humans’ true selves more positively; this difference mediated participants’ views of God as less retributive than humans. Study 3 manipulated agents’ views of humans’ true selves and examined how such information influenced each agent’s perceived motives. Participants viewed a given agent as less retributive when that agent regarded the true self as good (versus bad). These findings extend scholarship on lay theories of punishment motives and highlight links between religious and moral cognition. 

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