In this article we propose a geometric description of Arthur packets for p-adic 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, especially the prediction that Arthur packets are ABV-packets for p-adic groups. We gather evidence for these conjectures by verifying them in numerous examples.

We study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp. of semi-stable, reduction under tame ramification hypotheses.

We prove that the image of period map is algebraic, as conjectured by Griffiths.

We study the set of connected components of certain unions of affine Deligne-Lusztig varieties arising from the study of Shimura varieties. We determine the set of connected components for basic $\s $-conjugacy classes. As an application, we verify the Axioms in\cite {HR} for certain PEL type Shimura varieties. We also show that for any nonbasic $\s $-conjugacy class in a residually split group, the set of connected components is" controlled" by the set of straight elements associated to the $\s $-conjugacy class, together with the obstruction from the corresponding Levi subgroup. Combined with\cite {Zhou}, this allows one to verify in the residually split case, the description of the mod- p isogeny classes on Shimura varieties conjectured by Langland and Rapoport. Along the way, we determine the Picard group of the Witt vector affine Grassmannian of\cite {BS} and\cite {Zhu} which is of independent interest.

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