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.

• 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. Nov 21 '11 at 20:33

Similar in spirit, Rainer Schulze-Pillot, Darstellungsmasse von Spinorgeschlectern tern$\ddot{a}$rer quadratische Fromen, J. Reine Angew. Math. vol. 352 (1984), pp 114-132. Also MR758697.