On optimal embeddings and trees

Abstract

We apply the theory of Bruhat-Tits trees to the study of optimal embeddings of two and three dimensional commutative orders into quaternion algebras. Specifically, we determine how many conjugacy classes of global Eichler orders in a quaternion algebra yield optimal representations of such orders. This completes the previous work by C. Maclachlan, who considered only Eichler orders of square free level and integral domains as sub-orders. The same technique is used in the second part of this work to compute local embedding numbers, extending previous results by J. Brzezinski.

On some branches of the Bruhat–Tits tree

“…More precisely, in this work we no longer require the orthogonality condition. Our second result extends the embedding number computations in [6] to the case of orders contained in fields, the only orders of non-maximal rank that failed to be considered in our previous work.…”

“…Many combinatorial properties of the branch can be described in terms of two invariants. The stem length l, i.e., the length of w, and the depth p. The computation of local embedding numbers [6] or representation fields for global orders [2], [3], reduces to determining these invariants. This was done explicitly in [4], for orders generated by a pair of orthogonal pure quaternions as in (1).…”

“…Let us begin by recalling the method employed in [6] to compute embedding numbers for an order Ω spanning an algebra isomorphic to k × k. Essentially, it depends on the following observations: Figure 8. The minimal graph containing four different given vertices, and ends beyond them.…”

Class number formulas for the norm one group of totally definite quaternion algebras

“…The interest in selectivity is rekindled by the influential paper of Chinburg and Friedman [16], who studied selectivity for maximal orders in quatenion algebras. Their work inspired a series of generalization by Guo and Qin [21], Maclachlan [30], Linowitz [29], Arenas-Carmona [3,4,5], Arenas et al [2], and the list is too extensive to exhaust here. In a different setting, the spinor class (genus) field is introduced by Estes and Hsia [20] and used to study representations of spinor genera of quadratic lattices by many authors: Hisa, Shao and Xu [23], Hisa [22], Chan and Xu [14], Xu [47], etc.…”

“…Thanks to the uniqueness, each M(Ó) is symmetric in the sense of [37, §1]. It is shown in [37,Corollary,§2,p. 130] that when p ≡ 1 (mod 4), the map M establishes a one-to-one correspondence between the maximal orders of D p,∞ and the symmetric maximal orders of D. The proof hinges on the fact that p is the only prime ramified in F , so it applies to the case p = 2 as well.…”