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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. 

Let $$\BA$$ be the ring of adeles of a number field $$k$$ and $$\pi$$ be an irreducible cuspidal automorphic representation of $$\GL_n(\BA)$$. In Jiang and Luo (Pac J Math 318:339–374. https://doi.org/10.2140/pjm.2022.318.339, 2022, Pac J Math 326: 301–372. https://doi.org/10.2140/pjm.2023.326.301, 2023), the authors introduced $$\pi$$-Schwartz space $$\CS_\pi(\BA^\times)$$ and $$\pi$$-Fourier transform $$\CF_{\pi,\psi}$$ with a non-trivial additive character $$\psi$$ of $$k\bs\BA$$, proved the associated Poisson summation formula over $$\BA^\times$$, based on the Godement-Jacquet theory for the standard $$L$$-functions $$L(s,\pi)$$, and provided interesting applications. In this paper, in addition to the further development of the local theory, we found two global applications. First, we find a Poisson summation formula proof of the Voronoi summation formula for $$\GL_n$$ over a number field, which was first proved by A. Ichino and N. Templier (Am J Math 135:65–101. https://doi.org/10.1353/ajm.2013.0005, 2013, Theorem 1). Then we introduce the notion of the Godement-Jacquet kernels $$H_{\pi,s}$$ and their dual kernels $$K_{\pi,s}$$ for any irreducible cuspidal automorphic representation $$\pi$$ of $$\GL_n(\BA)$$ and show in Theorems \ref{thm:H=FK} and \ref{thm:CTh-pi} that $$H_{\pi,s}$$ and $$K_{\pi,1-s}$$ are related by the nonlinear $$\pi_\infty$$-Fourier transform if and only if $$s\in\BC$$ is a zero of $$L_f(s,\pi_f)=0$$, the finite part of the standard automorphic $$L$$-function $$L(s,\pi)$$, which are the $$(\GL_n,\pi)$$-versions of Clozel (J Number Theory 261: 252–298 https://doi.org/10.1016/j.jnt.2024.02.018, 2024, Theorem 1.1), where the Tate kernel with $n=1$ and $$\pi$$ the trivial character are considered. 

We prove the uniqueness of the Ginzburg–Rallis models over p-adic local fields of characteristic zero, which completes the local uniqueness problem for the Ginzburg–Rallis models, starting from the work of Nien (Models of representations of general linear groups over p-adic fields, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)-University of Minnesota, 2006) that proves the non-split case, and the work of Jiang et al. (Trans Am Math Soc 363(5): 2763–2802, 2011) that proves the general case over Archimedean local fields. Our proof extends the strategy of [16] to the p-adic case with the help of the refined structure of the wavefront sets of z-finite distributions as developed by Aizenbud et al. (Adv Math 285:1376–1414,2015). 

Abstract Let G be a group and let H be a subgroup of G . The classical branching rule (or symmetry breaking) asks: For an irreducible representation π of G ,determine the occurrence of an irreducible representation σ of H in the restriction of π to H . The reciprocal branching problem of this classical branching problemis to ask: For an irreducible representation σ of H , find an irreducible representation π of G such that σ occurs in the restrictionof π to H . For automorphic representations of classical groups, the branching problem has been addressed by the well-known global Gan–Gross–Prasad conjecture.In this paper, we investigate the reciprocal branching problem for automorphic representationsof special orthogonal groups using the twisted automorphic descent method as developed in [13]. The method may be applied toother classical groups as well. 

In [2], J. Arthur classifies the automorphic discrete spectrum of symplectic groups up to global Arthur packets. We continue with our investigation of Fourier coefficients and their im- plication to the structure of the cuspidal spectrum for symplectic groups ([16] and [20]). As result, we obtain certain characteri- zation and construction of small cuspidal automorphic represen- tations and gain a better understanding of global Arthur packets and of the structure of local unramified components of the cusp- idal spectrum, which has impacts to the generalized Ramanujan problem as posted by P. Sarnak in [43].