Orders of quaternion algebras with involution

Abstract

Abstract. We introduce the notion of maximal orders over quaternion algebras with orthogonal involution and give a classification over local and global fields. Over local fields, we show that there is a correspondence between maximal and/or modular lattices and orders closed under the involution.

Asymptotic Density of Apollonian-Type Packings

“…Finally, the discriminant of an orthogonal involution ; on a quaternion algebra H over F is the set discp;q " α 2 pF ˆq2 , where α is any non-zero element in H ´. Unlike, the others, this is not an ideal, but this is easily fixed-we write ιpdiscp;qq for the R-ideal generated by discp;q X R. With these conventions, we get an easy way to characterize maximal ;-orders: if K is a global or local field (such as an algebraic number field or one of its localizations), o K is its ring of integers, H is a quaternion algebra over K, and O is a ;-order of H, then O is maximal if and only if discrdpOq " discpHq X ιpdiscp;qq [18]. If I is a Z-ideal, we shall write |I| to denote its smallest non-negative generator.…”

Is Unreasonable Slightness a General Phenomenon?

“…where O 1 is a maximal ;-order containing O. But volpR 4 {O 1 q " lcm p|discpHq|, nrmpξqq 4 using the computation of 16volpR 4 {O 1 q 2 done previously by the author [8]. Using this bound, we can restrict ourselves to looking at all definite, rational quaternion algebra H with discriminant no more than 64π ă 202, and all orthogonal involution ; such that nrmpξq ă 202-there are only finitely many of both.…”

“…Using this bound, we can restrict ourselves to looking at all definite, rational quaternion algebra H with discriminant no more than 64π ă 202, and all orthogonal involution ; such that nrmpξq ă 202-there are only finitely many of both. The maximal ;-orders O 1 of any such quaternion algebra with involution will be orders of the form O 2 X O 2 ; with discriminant |discpHq|nrmpξq, where O 2 is a maximal order of H [8]-there are only finitely many of these as well. Finally, any order O with covering radius 1 must be a sub-;-order of such an O 1 possessing a basis of elements in O 1 of length no more than 4-once again, there are only finitely many choices.…”

The Twisted Euclidean Algorithm: Applications to Number Theory and Geometry

“…Theorem 4.1 (Theorem 1.1 of [She17]). Let H be a quaternion algebra over a local or global field F with characteristic not 2, with orthogonal involution ‡.…”