Reference for conjugacy in unit group of local quaternion algebra

Question

Given $F$ a local non-Archimedean field, and $D$ a quaternion algebra over this field, is there a good reference for finding the conjugacy classes of its group of units $D^{\times}$? I have searched through John Voight's book to no success yet.

Answer 1

Sorry for having not answered this question directly in the book!

As a warmup: over a field $F$, for the quaternion algebra $B=\mathrm{M}_2(F)$, the conjugacy classes in $B^\times=\mathrm{GL}_2(F)$ are also called similarity classes of matrices, and they are understood via canonical forms.

For a division quaternion algebra, the situation is even a bit simpler. See Corollary 7.1.5 (a corollary of the Skolem-Noether theorem): two nonscalar elements in a division quaternion algebra are conjugate if and only if they have the same (necessarily irreducible) reduced characteristic polynomial.

Finally, for a division quaternion algebra over a local field, all separable irreducible polynomials arise: see Corollary 13.4.5. (The situation with inseparable extensions is a bit more delicate--do you need the case $\mathrm{char}(F)=2$?)