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In this article we propose a geometric description of Arthur packets for$p$-adic groups using vanishing cycles of perverse sheaves. Our approach isinspired by the 1992 book by Adams, Barbasch and Vogan on the Langlandsclassification of admissible representations of real groups and follows thedirection indicated by Vogan in his 1993 paper on the Langlands correspondence.Using vanishing cycles, we introduce and study a functor from the category ofequivariant perverse sheaves on the moduli space of certain Langlandsparameters to local systems on the regular part of the conormal bundle for thisvariety. In this article we establish the main properties of this functor andshow that it plays the role of microlocalization in the work of Adams, Barbaschand Vogan. We use this to define ABV-packets for pure rational forms of$p$-adic groups and propose a geometric description of the transfercoefficients that appear in Arthur's main local result in the endoscopicclassification of representations. This article includes conjectures modelledon Vogan's work, especially the prediction that Arthur packets are ABV-packetsfor $p$-adic groups. We gather evidence for these conjectures by verifying themin numerous examples.

Arthur packets for $p$-adic groups by way of microlocal vanishing cycles of perverse sheaves, with examples