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By Skip Garibaldi


Abstract. The paper “Weakly commensurable arithmetic groups and isospectral locally symmetric spaces ” by Prasad and Rapinchuk left unresolved a question regarding algebraic groups of type D4 over a number field. We settle the question by proving an isomorphism criterion for such groups over number fields. Along the way, we also consider the question of existence of outer automorphisms of semisimple algebraic groups over arbitrary fields. The remarkable paper [PrR09] by Prasad and Rapinchuk addressed— amongotherthings—thequestionof whentwo weakly commensurablearithmetic subgroups of absolutely almost simple algebraic groups are necessarily commensurable. They settled this question in ibid., except for the cases where the algebraic groups have type D2n for n ≥ 2. Later, in [PrR10, Th. 9.1], they settled the question for types D2n with n ≥ 3, leaving open only the D4 case. Here, we settle the D4 case and also give a new proof of their result from [PrR10]. Specifically, we prove: Theorem 1. Let G1 and G2 be adjoint groups of type D2n for some n ≥ 2 over a global field K of characteristic ̸ = 2, such that G1 and G2 have the same quasi-split type—i.e., the smallest Galois extension of K over which G1 is of inner type is the same as for G2. If there exists a maximal torus Ti in Gi for i = 1 and 2 such that (1) there is a Ksep-isomorphism φ: G1 → G2 whose restriction to T1 is a K-isomorphism T1 → T2; and (2) there is a finite set V of places of K such that: (a) For all v ̸ ∈ V, G1 and G2 are quasi-split over Kv. (b) For all v ∈ V, (Ti)Kv contains a maximal Kv-split subtorus in (Gi)Kv; then G1 and G2 are isomorphic over K. The hypotheses are what one obtains by assuming the existence of weakly commensurablearithmeticsubgroups,seeforexampleTheorems1and6and Remark 4.4 in [PrR09]. (Bruce Allison gave an isomorphism criterion with very different hypotheses in [A, Th. 7.7].) The simply connected covers of G1 and G2 in the theorem are the K-forms of Spin 4n mentioned in the title of the paper. Note that these groups can be trialitarian, i.e., of type 3 D4 or 6 D4

Year: 2011
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