On the Classification of Rank 1 Groups over Non-archimedean Local Fields

We outline the classification of K-rank 1 groups over non-archimedean local fields K up to strict isogeny, as in [Ti1] and [Ti2]. We outline the classification of absolutely simple algebraic groups over non-archimedean local fields, up to strict isogeny. This is classical, and accounts of it have been written by Tits ([Ti1], [Ti2]) and Satake ([Sa]). Tits compiled tables of ‘admissible indices’ from which groups can be constructed. Here we describe Tits’ tables for groups of relative rank 1 over non-archimedean fields. Our references are [Ti1], [Ti2], and [PR] (Ch 2). The author would like to thank participants of the ‘Junior Number Theory Seminar’ at Harvard in 1997/1998. This paper is a summary of the subject matter studied in the seminar. In particular the author would like to thank J. Lansky, D. Pollack and A. Silverberg for many discussions about p-adic groups at a workshop at Harvard in the Spring of 1998. The non-archimedean local fields K have been classified. If Char(K) = 0, then K = Qp or a finite extension of Qp. If Char(K) = p > 0, then K is isomorphic to Fq((t−1)), the field of formal Laurent series in one variable over Fq, where q is a power of p. In each case, K has finite residue class field. Let K be a non-archimedean local field, let K be its algebraic closure, and let