Joseph A. Shalika

Rational points on compactifications of semi-simple groups

We prove Manins conjecture concerning the distribution of rational points of bounded height, and its refinement by Peyre, for wonderful compactifications of semi-simple algebraic groups over number fields. The proof proceeds via the study of the associated height zeta function and its spectral expansion.

The fundamental lemma for the Bessel and Novodvorsky subgroups of GSp(4) II

Abstract We announce, in the case of the group GSp(4), an equality of two local integrals. One is a Kloosterman integral on the Bessel subgroups of GSp(4) and the other is a Kloosterman integral on the Novodvorsky subgroups of GSp(4). We conjecture that Jacquets relative trace formula for GL(2) in [7], where Jacquet has given another proof of Waldspurgers result [9], generalizes to GSp(4). We believe that this approach will lead us to a proof and also a precise formulation of a conjecture of Bocherer [1]. Support for this conjecture may be found in the important paper of Bocherer and Schulze-Pillot [2]. Our result serves as the fundamental lemma for our conjectural relative trace formula for the main relevant double cosets.

On inversion of the bessel and gelfand transforms

We construct the Plancherel measure corresponding to the Bessel model on the split special orthogonal group of odd degree > 5 and the Whit-taker model on a connected split reductive group in general. As an application we prove the inversion formula which expresses the related integral transform in terms of the Satake transform.

Rational Points on Compactifications of Semi-Simple Groups of Rank 1

We explain our approach to the problem of counting rational points of bounded height on equivariant compactifications of semi-simple groups.