/7-adic Curvature and a Conjecture of Serre

1. In this note we announce a vanishing theorem for the cohomology of discrete subgroups of p-adic groups. The methods and results bear a striking analogy with the real case (see Matsushima [4]). In particular, we define "p-adic curvature" for the p-adic symmetric spaces of Bruhat-Tits (see [1]). As in the real case, we then reduce the proof of our vanishing theorem to the assertion that the minimum eigenvalues of certain p-adic curvature transformations are sufficiently large. This last condition can then be verified for "sufficiently large" residue class fields. Before giving a more detailed description of our results we introduce some notation. Thus let Z denote the ring of rational integers and Q, R, and Cthe fields of rational, real and complex numbers, respectively. For a prime p, Qp will denote the p-adic completion of Q. More generally kv will denote a nondiscrete, totally disconnected, and locally compact (commutative) field. Let G denote a simply-connected, linear algebraic group defined and simple over kv and let Gkv denote the fc^-rational points of G Let VQ denote a finite-dimensional vector space over Q, T an abstract group, and p : T -• Aut VQ a representation. Let H (T, p) denote the fth EilenbergMac Lane group of T with respect to p. If V = Q, and p is the trivial representation we write H(T, Q) in place of H(T, p). By a uniform lattice in Gkv we mean a discrete subgroup F a Gkv such that GkJT is compact.