Kneser-tits for a Rank 1 Form of E6 (after Veldkamp)

We prove the Kneser-Tits Conjecture for groups of index E 6,1 using an argument inspired by a 1968 paper by Veldkamp. The notion of simple for an algebraic group is different from the notion of simple for abstract groups. Recall that an abstract group Γ is projectively simple if Γ/Z(Γ) is simple as an abstract group. For a given field k, the Kneser-Tits Conjecture asserts: For every simply connected quasi-simple kisotropic algebraic group G, the abstract group G(k) is projectively simple. A good survey of the conjecture is given in [PR, §7.2]. Here are a few highlights. Many cases of the conjecture for classical groups are part of “geometric algebra” as in the books by E. Artin and J. Dieudonné. The conjecture holds for k algebraically closed, for the real numbers (E. Cartan), and for non-archimedean locally compact fields (V.P. Platonov). It fails wildly if the simply connected hypothesis is dropped. Some groups of inner type An provide counterexamples to the conjecture; these amount to central division algebras with nontrivial SK1. In order to prove the conjecture for a particular field k, G. Prasad and M.S. Raghunathan showed that it suffices to consider the groups of k-rank 1. For k a number field, no counterexamples are known. In order to prove the conjecture in that case, it remains only to prove it for groups with the following Tits indexes: 2E29 6,1 r r r r r r ®­ § ¥ ¦ r r r r r r ®­ f 2E35 6,1 (The conjecture has long been known for the classical groups, cf. [PR, p. 410]. The trialitarian groups are treated in [Pra].) We remark that when k is a totally imaginary number field or a global function field, the two indexes displayed above do not occur [Gi, p. 315, Th. 9b], hence the conjecture is proved in that case. The conjecture is still open for number fields with real embeddings, like the rational numbers. In fact, one of the two “open” cases was—essentially—settled in 1968. The purpose of this paper is to give a proof of that case, i.e., to prove the following theorem. Date: 20 February 2006. 2000 Mathematics Subject Classification. 20G15.