Question

The context for this question is from page 284 - 287 of Berger's paper: http://pdn.sciencedirect.com/science?_ob=MiamiImageURL&_cid=272332&_user=209810&_pii=S0021869398976785&_check=y&_origin=article&_zone=toolbar&_coverDate=1999--01&view=c&originContentFamily=serial&wchp=dGLbVlS-zSkzS&md5=2dd8e0d714d264cf7c4acdd9ec58ac84&pid=1-s2.0-S0021869398976785-main.pdf

Particularly, in his assumption at the top of page 287, he says that "From now on, assume that our map $\pi_\mathfrak{p}$ surjects onto $\text{PU}_2(\zeta,\mathcal{O}_K/\mathfrak{p})\cong \text{PSL}_2(\mathbb{F}_q)$, that $q$ is odd, and that $(6,k) = 1$, where $k = \sharp\langle -\zeta\rangle$."

I'm guessing that he's assuming the conditions in proposition 2 (from the previous page) to be true, so that $\pi_\mathfrak{p}$ surjects onto $\text{PU}_2(\zeta,\mathcal{O}_K/\mathfrak{p}) = \text{PSU}_2(\mathcal{O}_K/\mathfrak{p})$, and that he's claiming that the latter group is isomorphic to $\text{PSL}_2(\mathbb{F}_q)$.

Is this generally true?

Also, on page 284, where he gives the matrix $H$ for the hermitian form, he claims that $H\in GL_{n-1}(\mathbb{Z}[t,t^{-1}])$, but the matrix he gives obviously does not lie in that group.

Where might I find a good book on unitary matrices over finite fields?

thanks,

• will

Answer 1

Here is a more bare hands explanation. Let $\phi$ be the field automorphism of ${\rm SL}_n(q^2)$ that acts by applying $x \mapsto x^q$ to the matrix entries. Let $\gamma$ be the graph automorphism that maps matrices $A$ to their inverse-tranpose $A^{- \mathrm{T}}$. Then ${\rm SL}_n(q)$ is the subgroup of ${\rm SL}_n(q^2)$ that is centralized by $\phi$, whereas the group ${\rm SU}_n(q^2)$ (which is confusingly often denoted by ${\rm SU}_n(q)$) that fixes the identity matrix as unitary form is the subgroup of ${\rm SL}_n(q^2)$ that is centralized by $\phi\gamma$.

The automorphism $\gamma$ is outer for $n>2$, but when $n=2$ it is inner and acts in the same way as conjugation by the matrix $\left( \begin{array}{rr}0&1\\ -1&0\end{array} \right)$. It turns out in this case that $\phi$ and $\phi\gamma$ are conjugate in the automorphism group of ${\rm SL}_2(q^2)$ by (the projective image of) an element $g \in {\rm GL}_2(q^2)$, and hence that ${\rm SL}_2(q)$ is conjugate to ${\rm SU}_2(q^2)$ in ${\rm GL}_2(q^2)$. With a bit of calculation on the back of an envelope, we find that $g = \left( \begin{array}{rr}a&b\\ c&d\end{array} \right)$, where $b = -t^qa^q$ and $d= -t^qc^q$ for some field element $t$ with $t^{q+1} = -1$.

Answer 2

It is a fairly standard result that $SU(2,q^2)$ and $SL(2,q)$ are isomorphic, see e.g. II.8.8. in Huppert's Endliche Gruppen. I would expect that it is also in the third volume of The Classification of the Finite Simple Groups by Gorenstein-Lyons-Solomon, but I don't have the volume at hand right now.