Quasi-genera of quadratic forms

Let f be a quadratic form in n variables (n > 1) with nonzero determinant d. A prime p is said to be exceptional with respect to f if every automorph of f with rational elements, determinant ±1 and denominator prime to 2d has a denominator which is a quadratic residue of p. (Throughout, slight modifications must be made if p = 2.) Except for certain binary forms, each exceptional prime induces a splitting of the genus into two quasi-genera. Building on previous results, necessary and sufficient conditions are given that a prime p be exceptional for n = 2 and n = 3 and necessary conditions for n > 3. It is proved that there are no exceptional primes for n > 4 and only possibly in special cases for n = 4. A connection is shown between representations of integers by certain ternary forms and the existence of quasi-genera. Possible connections with spinor genera are mentioned and a few unanswered questions are posed.