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1. On wavefront sets of global Arthur packets of classical groups: Upper bound

   https://doi.org/10.4171/JEMS/1446  

   Jiang, Dihua; Liu, Baiying (July 2025, Journal of the European Mathematical Society)

   We prove a conjecture of the first named author (2014) on the upper bound Fourier coefficients of automorphic forms in Arthur packets of all classical groups over any number field. This conjecture generalizes the global version of the local temperedL-packet conjecture of Shahidi (1990). Under certain assumption, we also compute the wavefront sets of the unramified unitary dual for split classical groups. 

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2. On the Whittaker range of the generalized metaplectic theta lift

   https://doi.org/10.1016/j.jnt.2023.08.007  

   Friedberg, Solomon; Ginzburg, David (February 2024, Journal of Number Theory)

   The classical theta correspondence, based on the Weil representation, allows one to lift automorphic representations on symplectic groups or their double covers to au- tomorphic representations on special orthogonal groups. It is of interest to vary the orthog- onal group and describe the behavior in this theta tower (the Rallis tower). In prior work, the authors obtained an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups that is based on the tensor product of the Weil representation with another small representation. In this work we study the existence of generic lifts in the resulting theta tower. In the classical case, there are two orthogonal groups that may support a generic lift of an irreducible cuspidal automorphic representation of a symplectic group. We show that in general the Whittaker range consists of r + 1 groups for the lift from the r-fold cover of a symplectic group. We also give a period criterion for the genericity of the lift at each step of the tower. 

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3. Dimensions of automorphic representations, L-functions and liftings

   https://doi.org/10.1007/978-3-030-68506-5_3  

   Friedberg, Solomon; Ginzburg, David (January 2021, Relative Trace Formulas)

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

   The classical theta correspondence establishes a relationship between automorphic representations on special orthogonal groups and automorphic representations on symplectic groups or their double covers. This correspondence is achieved by using as integral kernel a theta series on the metaplectic double cover of a symplectic group that is constructed from the Weil representation. There is also an analogous local correspondence. In this work we present an extension of the classical theta correspondence to higher degree metaplectic covers of symplectic and special orthogonal groups. The key issue here is that for higher degree covers there is no analogue of the Weil representation, and additional ingredients are needed. Our work reflects a broader paradigm: constructions in automorphic forms that work for algebraic groups or their double covers should often extend to higher degree metaplectic covers. 

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4. On the Weak Local Arthur Packets Conjecture for Split Classical Groups

   https://doi.org/10.1093/imrn/rnae098  

   Liu, Baiying; Lo, Chi-Heng (May 2024, International Mathematics Research Notices)

   Abstract Recently, motivated by the theory of real local Arthur packets, making use of the wavefront sets of representations over non-Archimedean local fields $$F$$, Ciubotaru, Mason-Brown, and Okada defined the weak local Arthur packets consisting of certain unipotent representations and conjectured that they are unions of local Arthur packets. In this paper, we prove this conjecture for $$\textrm{Sp}_{2n}(F)$$ and split $$\textrm{SO}_{2n+1}(F)$$ with the assumption of the residue field characteristic of $$F$$ being large. In particular, this implies the unitarity of these unipotent representations. We also discuss the generalization of the weak local Arthur packets beyond unipotent representations, which reveals the close connection with a conjecture of Jiang on the structure of wavefront sets for representations in local Arthur packets. 

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5. Rational cuspidal curves and symplectic fillings

   https://doi.org/10.4310/jsg.241101004405  

   Golla, Marco; Starkston, Laura (October 2024, Journal of Symplectic Geometry)

   A symplectic rational cuspidal curve with positive self-intersection number admits a concave neighborhood, and thus a corresponding contact manifold on the boundary. In this article, we study symplectic fillings of such contact manifolds, providing a complementary perspective to our earlier article on symplectic isotopy classes of rational cuspidal curves. We explore aspects of these symplectic fillings through Stein handlebodies and rational blow-downs. We give examples of such contact manifolds which are identifiable as links of normal surface singularities, other examples which admit no symplectic fillings, and further examples where the fillings can be fully classified. 

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