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There is a slight error in given definition. In given definition, one considers points with value in O, whereas G is only defined over K. One should remark: first that one need to consider a model of G over its ring of integers, and consider G(O); then that the notion of arithmetic group is independant of the choice of a model for G.
In Discrete subgroups of semisimple Lie groups Margulis defines S-arithmetic subgroups for algebraic subgroups of GL(N) over number fields. He defines arithmetic subgroups only for fields over Q (S contaning only the unique archmedean place). He notes that such a (I quote) is intrinsic with respect to the K-structure on H.
Milne's Algebraic Groups and Arithmetic Groups defines an arithmetic subgroup of G(Q) as a subgroup commensurable with the stabilizer of a lattice in a faithfull rational representation. Same definition in Borel Reduction Theory for Arithmetic Groups in PSPUM 9 - Algebraic groups and Discontinuous subgroups.
But in N.D. Alan's The problem of the Maximality of Arithmetic groups (in PSPUM9 too) considers subgroups of G(R) or G(C) commensurable with G(Z) (whatever the integral structure). This is especially relevant for algebraic groups with nontrivial center and when working with arithmetic subgroups of Lie groups (not every connected Lie group, even almost simple, is linear algebraic). —Preceding unsigned comment added by 22.214.171.124 (talk) 19:26, 3 April 2009 (UTC)
I rewrote the article including many more topics, bibliography items and links to other pages. Possible things to be added: lattices in solvable groups, Prasad's volume formula (should have its own page), more about arithmetic locally symmetric spaces (systole, Wang's finiteness theorem), adelic theory and automorphic forms (very short w/ links to the relevant articles, in particular strong approximation),... Probably the history section could be better-written. jraimbau 14:01, 16 June 2016 (UTC)
I notice some discrepancies between the description of how the theory was developed, and the years of publication of the referenced articles. For instance when it says "Shortly afterwards the finiteness of covolume was proven in full generality by Borel and Harish-Chandra." But the article referenced in  is from 1961, whereas the developments it is claimed to have come after are from 1965 and 1982.