The classification of quadratic forms over local and global fields is well understood. But what about quadratic forms over adele rings? Let G = GL(n,A), where A is the adele ring of a global field F. Let S be the set of symmetric matrices in G. Let G act on S by g * s = g s (transpose g). What are the orbits? Even for n = 1 and F = Q, not every orbit contains a rational point.

$\begingroup$ Dear Jeff, got your reply to my email. The best bet is for you to post your own answer, perhaps with nothing more than the references you found, as I missed the point of your query. That way, your new answer pushes this thread to the front of the "Active" sort. $\endgroup$ – Will Jagy Nov 21 '11 at 20:33
EDIT: I have never seen this one, but it is said to cover some of the same ground as Kneser (1961): A. Weil, Sur la theorie des formes quadratiques (1962).
ORIGINAL: I did not notice this earlier...for what it may be worth, the introduction of adeles into the consideration of quadratic forms may well originate with Martin Kneser, Darstellungsmasse indefiniter quadratische Formen, Math. Z. volume 77 (1961) pp 188194. Also MR0140487.
Similar in spirit, Rainer SchulzePillot, Darstellungsmasse von Spinorgeschlectern tern$\ddot{a}$rer quadratische Fromen, J. Reine Angew. Math. vol. 352 (1984), pp 114132. Also MR758697.

1$\begingroup$ As far as I can tell, these references discuss quadratic forms defined over global fields. Adelic techniques are used to study them. I am interested in quadratic forms defined over adele rings. What are the equivalence classes? For GL(1), we are talking about the idele class group mod squares of ideles. This is a nontrivial quotient. $\endgroup$ – Jeff Hakim Nov 21 '11 at 15:19
