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.

Let K=Q(√(-q)), where q is any prime number congruent to 7 modulo 8, and let O be the ring of integers of K. The prime 2 splits in K, say 2O=pp∗, and there is a unique Z_2-extension K_∞ of K which is unramified outside p. Let H be the Hilbert class field of K, and write H_∞=HK_∞. Let M(H_∞) be the maximal abelian 2-extension of H_∞ which is unramified outside the primes above p, and put X(H_∞)=Gal(M(H_∞)/H_∞). We prove that X(H_∞) is always a finitely generated Z_2-module, by an elliptic analogue of Sinnott’s cyclotomic argument. We then use this result to prove for the first time the weak p-adic Leopoldt conjecture for the compositum J_∞ of K_∞ with arbitrary quadratic extensions J of H. We also prove some new cases of the finite generation of the Mordell–Weil group E(J_∞) modulo torsion of certain elliptic curves E with complex multiplication by O.

In the present paper, we prove the 2-part of Birch and Swinnerton-Dyer conjecture for an explicit infinite family of rank 0 quadratic twists of the modular elliptic curve χ_0(14), using an explicit form of the Waldspurger formula. We also give an explicit infinite family of rank 1 quadratic twists of χ_0(14) whose Tate–Shafarevich group is of odd cardinality.

We investigate Sarnak's M\"obius Disjointness Conjecture through asymptotically periodic functions. It is shown that Sarnak's conjecture for rigid dynamical systems is equivalent to the disjointness of M\"obius from asymptotically periodic functions. We give sufficient conditions and a partial answer to the later one. As an application, we show that Sarnak's conjecture holds for a class of rigid dynamical systems, which improves an earlier result of Kanigowski-Lema{\'{n}}czyk-Radziwi{\l}{\l}.

8(1):187-187