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"In this article we propose a geometric description of Arthur packets for padic groups using vanishing cycles of perverse sheaves. Our approach is inspired by the 1992 book by Adams, Barbasch and Vogan on the Langlands classification of admissible representations of real groups and follows the direction indicated by Vogan in his 1993 paper on the Langlands correspondence. Using vanishing cycles, we introduce and study a functor from the category of equivariant perverse sheaves on the moduli space of certain Langlands parameters to local systems on the regular part of the conormal bundle for this variety. In this article we establish the main properties of this functor and show that it plays the role of microlocalization in the work of Adams, Barbasch and Vogan. We use this to define ABV-packets for pure rational forms of p-adic groups and propose a geometric description of the transfer coefficients that appear in Arthur's main local result in the endoscopic classification of representations. This article includes conjectures modelled on Vogan's work, including the prediction that Arthur packets are ABV-packets for p-adic groups. We gather evidence for these conjectures by verifying them in numerous examples"--

p-adic groups. --- Representations of groups. --- Sheaf theory. --- Number theory. --- Number theory -- Discontinuous groups and automorphic forms -- Representation-theoretic methods; automorphic representations over local and global fields. --- Topological groups, Lie groups -- Lie groups -- Representations of Lie and linear algebraic groups over local fields. --- Partial differential equations -- General topics -- Microlocal methods; methods of sheaf theory and homological algebra in PDE. --- Several complex variables and analytic spaces -- Singularities -- Deformations of singularities; vanishing cycles.