Let $F$ be a local non-archimedean field, let $F^\times$ be its group of units, let $\mathfrak{O}_F$ be its integers, let $\mathfrak{p}_F$ be the unique maximal ideal of $\mathfrak{O}_F$, let $\varpi_F$ be the prime element that generates $\mathfrak{p}_F$, let $\mathbb{F}_q$ be the residue field $\mathfrak{O}_F / \mathfrak{p}_F$, and assume $2$ does not divide $q$. Let $U_F ^n$ denote the subgroup $1+\mathfrak{p}_F^n$ of $F^\times$ for $n \geq 1$, and let $U_F^0$ denote $\mathfrak{O}_F^\times$, the group of units of $\mathfrak{O}_F$.
Let $E_r$ be a (tamely) ramified quadratic extension of $F$, and let $E_u$ be an unramified quadratic extension of $F$. Let $D$ be a quaternion division algebra over $F$. The notation in the previous paragraph carries over in an obvious way to the settings of $E_r$, $E_u$, and $D$.
If my understanding is correct,
- $F^\times \cong \mathbb{Z} \times U_F^0 \cong \mathbb{Z} \times \mathbb{F}_q^\times \times U_F^1 $
- $E_u^\times \cong \mathbb{Z} \times U_{E_u}^0 \cong \mathbb{Z} \times \mathbb{F}_{q^2}^\times \times U_{E_u}^1 $, and $\varpi_F \mathfrak{O}_{E_u} = \varpi_{E_u} \mathfrak{O}_{E_u}$
- $E_r^\times \cong \mathbb{Z} \times U_{E_r}^0 \cong \mathbb{Z} \times \mathbb{F}_{q}^\times \times U_{E_r}^1 $, and $\varpi_F \mathfrak{O}_{E_r} = \varpi_{E_r}^2 \mathfrak{O}_{E_r}$
- $D$ is noncommutative, generated over $F$ by $\alpha$ and $\pi$, where $F[\alpha]$ is unramified and $F[\pi]$ is ramified; the set consisting of 0 and the powers of $\alpha$ is a complete set of residues for $\mathfrak{O}_D / \mathfrak{p}_D = \mathbb{F}_{q^2}$; $\pi$ is a prime element of $D$ (so sticking with the aforementioned notation, we could denote it $\varpi_D$), which satisfies $\pi^2 \mathfrak{O}_D = \varpi_F \mathfrak{O}_D$; and $E_r$ and $E_u$ embed in $D$.
Question:
How can we calculate the the indices $\left[ D^\times : E_u^\times U_D^0 \right]$ and $\left[ D^\times : E_r^\times U_D^1 \right]$?
I suspect that $\left[ D^\times : E_u^\times U_D^0 \right]$ should be $2$, the same as $\left[ D^\times : F^\times U_D^0 \right]$; and I don't know about $\left[ D^\times : E_r^\times U_D^1 \right]$ -- maybe $q+1$? (Those guesses are based on some computations I saw in a paper by Corwin, Moy, and Sally involving division algebras of higher dimensions.) It seems like it should be possible to compute these indices somehow using homomorphisms arising from the valuation on $D$, the reduced norm, and the reduced trace, but I don't know how. Admittedly, I need to do more exercises involving quaternion division algebras over local fields to get a better grasp of the situation... In the meantime, any hints or solutions would be much appreciated!
