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1. Bounds for the Number of Cohomological Automorphic Representations of GL3/ℚ in the Weight Aspect

   https://doi.org/10.1093/imrn/rnaa048  

   Marshall, Simon ( April 2020 , International Mathematics Research Notices)

   Abstract We prove a power saving over the trivial bound for the number of cohomological cuspidal automorphic representations of fixed level and growing weight on $GL_3/{\mathbb{Q}}$ by adapting the methods of our earlier paper on $GL_2$. 

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2. Shimura varieties for unitary groups and the doubling method

   https://doi.org/10.1007/978-3-030-68506-5_6  

   Harris, Michael ( January 2021 , Relative Trace Formulas)

   Müller, Werner ; Shin, Sug Woo ; Templier, Nicolas (Ed.)

   ThetheoryofGaloisrepresentationsattachedtoautomorphicrepresenta- tions of GL(n) is largely based on the study of the cohomology of Shimura varieties of PEL type attached to unitary similitude groups. The need to keep track of the similitude factor complicates notation while making no difference to the final result. It is more natural to work with Shimura varieties attached to the unitary groups themselves, which do not introduce these unnecessary complications; however, these are of abelian type, not of PEL type, and the Galois representations on their cohomology differ slightly from those obtained from the more familiar Shimura varieties. Results on the critical values of the L-functions of these Galois representations have been established by studying the PEL type Shimura varieties. It is not immediately obvious that the automorphic periods for these varieties are the same as for those attached to unitary groups, which appear more naturally in applications of relative trace formulas, such as the refined Gan-Gross-Prasad conjecture (conjecture of Ichino-Ikeda and N. Harris). The present article reconsiders these critical values, using the Shimura varieties attached to unitary groups, and obtains results that can be used more simply in applications. 

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3. Archimedean zeta integrals for unitary groups

   https://doi.org/10.1515/crelle-2024-0035  

   Eischen, Ellen E ; Liu, Zheng ( June 2024 , Journal für die reine und angewandte Mathematik (Crelles Journal))

   We derive precise formulas for the archimedean Euler factors occurring in certain standard Langlands 𝐿-functions for unitary groups.In the 1980s, Paul Garrett, as well as Ilya Piatetski-Shapiro and Stephen Rallis (independently of Garrett), discovered integral representations of automorphic 𝐿-functions that are Eulerian but, in contrast to the Rankin–Selberg and Langlands–Shahidi methods, do not require that the automorphic representations to which the 𝐿-functions are associated are globally generic.Their approach, thedoubling method, opened the door to a variety of applications that could not be handled by prior methods.For over three decades, though, the integrals occurring in the Euler factors at archimedean places for unitary groups eluded precise computation, except under particular simplifications (such as requiring certain representations to be one-dimensional, as Garrett did in the first major progress on this computation and only prior progress for general signatures).We compute these integrals for holomorphic discrete series of general vector weights for unitary groups of any signature.This has consequences not only for special values of 𝐿-functions in the archimedean setting, but also for 𝑝-adic 𝐿-functions, where the corresponding term had remained open.

    

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4. Vertex algebras of CohFT-type

   Chiara Damiolini, Angela Gibney ( January 2022 , Cambridge University Press, Cambridge,)

   Paolo Aluffi, David Anderson (Ed.)

   Representations of certain vertex algebras, here called of CohFT-type, can be used to construct vector bundles of coinvariants and conformal blocks on moduli spaces of stable curves [DGT2]. We show that such bundles define semisimple cohomological field theories. As an application, we give an expression for their total Chern character in terms of the fusion rules, following the approach and computation in [MOPPZ] for bundles given by integrable modules over affine Lie algebras. It follows that the Chern classes are tautological. Examples and open problems are discussed. 

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5. Vertex algebras of CohFT-type

   Chiara Damiolini, Angela Gibney ( January 2022 , Cambridge Univ. Press, Cambridge, 2022)

   Sam Payne, et al (Ed.)

   Representations of certain vertex algebras, here called of CohFT-type, can be used to construct vector bundles of coinvariants and conformal blocks on moduli spaces of stable curves [DGT2]. We show that such bundles define semisimple cohomological field theories. As an application, we give an expression for their total Chern character in terms of the fusion rules, following the approach and computation in [MOP+2] for bundles given by integrable modules over ane Lie alge- bras. It follows that the Chern classes are tautological. Examples and open problems are discussed. 

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