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I'm looking to learn about integral structures (or models?) on classical algebraic groups.

To begin with I have been learning about algebraic groups, quadratic forms and lattices. And also looking at this paper by Jonathan Hanke called "Algorithms for computing maximal lattices in bilinear (and quadratic) spaces over number fields" available here http://arxiv.org/pdf/1208.2481.pdf

The thing is I want to learn about these integral structures but can't seem to find a good place to learn about them.

Thanks

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    $\begingroup$ See also J.S. Milne: "Algebraic Groups, Lie Groups, and their Arithmetic Subgroups". $\endgroup$
    – Dietrich Burde
    Commented Nov 7, 2013 at 19:13
  • $\begingroup$ Borel has a book (in French) about arithmetic groups that is very illuminating. The work of Bruhat and Tits also provides a wealth of integral structures for studying the theory over non-archimedean local fields. Perhaps if you say more about what you want to do with such integral structures it would clarify what references would be useful to you (e.g., presumably "SGA3, volumes II and III" is not what you're looking for). $\endgroup$
    – Marguax
    Commented Nov 8, 2013 at 2:04
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    $\begingroup$ Look at the book by Rapinchuk and Platonov: Algebraic Groups and Number Theory. $\endgroup$
    – Venkataramana
    Commented Nov 8, 2013 at 3:10

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not sure what you mean by good. The standard books on quadratic forms over number fields are in Hanke's references. I would add that Hanke studied under Shimura, so you should take a look at shimura_2010 and shimura_2012, as the standard interplay is quadratic forms and modular forms. Note that the book that defined notation for a generation is O'Meara, in Hanke's references. I'm not sure Hanke mentions Kitaoka.

For other classical groups, Grove is unusual in including characteristic 2 in full detail.

Finally, i never got interested in using number fields. So i like Cassels, Rational Quadratic Forms.

Added in proof: take a look at THIS and THIS, maybe you will like something.

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You may want to look at

A. Borel, S. Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962) 485–535. MR0147566 (26 #5081)

B.H. Gross, Groups over Z, Invent. Math. 124 (1996) 263–279. MR1369418 (96m:20075)

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