We define and study a candidate for ‘arithmetic’ cohomology theory with values in crystalline p-adic smooth sheaves. We show that it injects into arithmetic étale cohomology and prove a duality result.

The space of elliptic virtual representations of a p-adic group is endowed with a natural inner product EP(,), defined analytically by Kazhdan and homologically by Schneider–Stuhler. Arthur has computed EP in terms of analytic R-groups. For Iwahori spherical representations, we show that EP can also be expressed in terms of a corresponding inner product on space of elliptic virtual representations of Weyl groups. This leads to an explicit description of both elliptic representation theories, in terms of the Kazhdan–Lusztig and Springer correspondences

Let $\Bbb V$ be n-dimensional complex vector space. The aim of this paper is to give an elementary proof of the isomorphism ($\cal O$t$\Bbb _V$∧[n] ≃ $\cal O$t$\Bbb _V$*, which quantizes the Fourier–Sato transform of the conic sheaf $\cal O$t$\Bbb _V$ of temperate holomorphic functions

Let L/K be a finite Galois extension of number fields. We use complexes arising from the étale cohomology of $\Bbb Z$ on open subschemes of Spec $\cal O$L to define a canonical element of the relative algebraic K-group K0($\Bbb Z$[Gal(L/K)], $\Bbb R$. We establish some basic properties of this element, and then use it to reinterpret and refine conjectures of Stark, of Chinburg and of Gruenberg, Ritter and Weiss. Our results precisely explain the connection between these conjectures and the seminal work of Bloch and Kato concerning Tamagawa numbers. This provides significant new insight into these important conjectures and also allows one to use powerful techniques from arithmetic algebraic geometry to obtain new evidence in their favour.