These notes reproduce the contents of lectures given at the Tata Institute
in January and February 1967, with some details added which had not
been given in the lectures. The main result is the Hasse principle for
the one-dimensional Galois cohomology of simply connected classical
groups over number fields. For most groups, this result is closely related
to other types of Hasse principle. Some of these are well known, in
particular those for quadratic forms. Two less well known cases are:
i) Hermitian forms over a division algebra with an involution of the
second kind; here the result is connected with (but not equivalent to) a
theorem of Landherr. The simplified proof of Landherr’s theorem, given
in §5.5, has been obtained independently by T. Springer;
ii) Skew-hermitian forms over a quaternion division algebra; here
a proof by T. Springer, different from the one given in the lectures in
§5.10, is reproduced as an appendix.
I wish to thank the Tata Institute for its hospitality and P. Jothilingam
for taking notes and filling in some of the details. |