On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior

Prasad, Gopal ; Rapinchuk, Andrei S. (2006) On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior Advances in Mathematics, 207 (2). pp. 646-660. ISSN 0001-8708

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Official URL: http://linkinghub.elsevier.com/retrieve/pii/S00018...

Related URL: http://dx.doi.org/10.1016/j.aim.2006.01.001

Abstract

This note is a follow-up on the paper [A. Borel, G. Harder, Existence of discrete cocompact subgroups of reductive groups over local fields, J. Reine Angew. Math. 298 (1978) 53–64] of A. Borel and G. Harder in which they proved the existence of a cocompact lattice in the group of rational points of a connected semi-simple algebraic group over a local field of characteristic zero by constructing an appropriate form of the semi-simple group over a number field and considering a suitable S-arithmetic subgroup. Some years ago A. Lubotzky initiated a program to study the subgroup growth of arithmetic subgroups, the current stage of which focuses on "counting" (more precisely, determining the asymptotics of) the number of lattices of bounded covolume (the finiteness of this number was established in [A. Borel, G. Prasad, Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 119–171; Addendum: Publ. Math. Inst. Hautes Études Sci. 71 (1990) 173–177] using the formula for the covolume developed in [G. Prasad, Volumes of S-arithmetic quotients of semi-simple groups, Publ. Math. Inst. Hautes Études Sci. 69 (1989) 91–117]). Work on this program led M. Belolipetsky and A. Lubotzky to ask questions about the existence of isotropic forms of semi-simple groups over number fields with prescribed local behavior. In this paper we will answer these questions. A question of similar nature also arose in the work [D. Morris, Real representations of semisimple Lie algebras have Q-forms, in: Proc. Internat. Conf. on Algebraic Groups and Arithmetic, December 17–22, 2001, TIFR, Mumbai, 2001, pp. 469–490] of D. Morris (Witte) on a completely different topic. We will answer that question too.

Item Type:Article
Source:Copyright of this article belongs to Elsevier Science.
Keywords:Forms of Semi-Simple Algebraic Groups; Galois Cohomology
ID Code:35795
Deposited On:17 Apr 2011 14:58
Last Modified:02 Jun 2011 08:35

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