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.

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. 

