On the Density of Unnormalized Tamagawa Numbers of Orthogonal Groups I

An Arithmetical Invariant of Orbits of Affine Actions and Its Application to Similarity Classes of Quadratic Spaces

“…This has already been done by Hayasaka and Yukie [6], and so we shall merely have to establish some notation and then recall the necessary facts from [6]. Suppose that v is a finite place of k. Let H v denote the unique quaternion division k v -algebra, M v the maximal order in H v , and L v the unique unramified quadratic extension of k v .…”

“…The completeness and non-redundancy of the list is contained in [6,Proposition 4.16] for n ≥ 4. The n = 2 case is well known, and is stated in the discussion surrounding [6,Eq. (4.5)].…”

“…(4.5)]. [6,Proposition 5.23] contains the optimality claim. Note that, when applied to a non-singular integral form, optimality coincides with what Hayasaka and Yukie labeled as Condition 5.7.…”

“…The method of proof is the same as that used by Hayasaka and Yukie to prove [6,Proposition 5.23]. The core of the proof is to prove more in the case where n = 2, and we first do this.…”

“…The work of Hayasaka and Yukie [6,7] and Yukie [21,22] on the space of quadratic forms with the similarity action is important for our work in Sec. 3, and absolutely critical for our work in Sec.…”