Non-split Reductive Groups over Z

The work of Chevalley ([BIBLE], [Chev61]) and Demazure–Grothendieck [SGA3] provides a satisfactory understanding of the reasons for the existence and uniqueness of “Chevalley groups” over Z. These are the reductive group schemes G → Spec(Z) (i.e., smooth affine groups with connected reductive geometric fibers) for which there exists closed subtorus T ⊂ G over Z that is maximal on geometric fibers (and T is necessarily split over Z: see [SGA3, X, 1.2, 5.16] or [Con, B.3.6]). By the Isomorphism Theorem over Z, such groups up to Z-isomorphism correspond to the isomorphism classes of root data. (Many authors require Chevalley groups to be semisimple; this is a matter of convention.) In view of the Existence Theorem over Z and Q, the generic fibers of Chevalley groups are precisely the split connected reductive groups over Q and so are also characterized up to isomorphism by such root data. In particular, if two Chevalley groups over Z have isomorphic Q-fibers then they are isomorphic over Z. The isomorphism result is not as strong as for Néron models of abelian varieties over Q: if G and G ′ are Chevalley groups then it is not true in general that isomorphisms between their Q-fibers extend to isomorphisms over Z. This is already seen for G = G ′ = SL2: AutQ(SL2) = PGL2(Q) but AutZ(SL2) = PGL2(Z). There are semisimple Z-groups that are not Chevalley groups. Loosely speaking, these correspond to semisimple Q-groups with “good reduction” at all primes (but non-split over R; see Remark 1.6). Unlike in the split case, in general there exist non-isomorphic semisimple Z-groups with isomorphic Q-fibers (as we shall see in some explicit examples). The aim of these notes is to discuss the theory and examples related to the Q-groups that arise as generic fibers of reductive Z-groups that are not Chevalley groups. (Using the fact that Pic(Z) = 1, one can show that if GQ is split then so is G ; the proof is sketched in [Con, Exer. 7.3.9].) In §1 we describe the possibilities for the generic fibers of reductive Z-groups, with an emphasis on the case of semisimple groups whose geometric fibers are absolutely simple and simply connected (and we show that case accounts for the rest via direct products and central isogenies). Then in §2 we introduce Coxeter’s order in Cayley’s definite octonion algebra over Q, and we use it in §3 to describe examples over Z. Finally, in §4 we explain how to use mass formulas to prove in some cases that the list of Z-models found in §3 for certain Q-groups is exhaustive.