Part II. Rigid meromorphic cocycles for orthogonal groups. (Joint with Lennart Gehrmann and Mike Lipnowski.)

The rigid meromorphic cocycles of Part I are cocycles on  $PSL(2,Z[1/p])$, which is contained in the orthogonal group of a ternary quadratic space of signature $(2,1)$, the space of binary quadratic forms with the discriminant form. The notion of rigid meromorphic cocycles can be extended  to  orthogonal groups of arbitrary real signature $(r,s)$.  The resulting objects can be evaluated at ``special points” associated to  tori in the orthogonal group, and these values  lead to non-trivial class invariants in abelian extensions of the associated reflex fields. Whereas the real quadratic field case  arising from quadratic spaces of signature $(2,1)$  is now supported by substantial theoretical evidence, the setting of higher rank orthogonal groups is   more mysterious, and the evidence for it, fragmentary and largely experimental.