this question relates to the beautiful construction of expander graphs using Cayley graphs of $PGL_2(\mathbb{F}_q)$, as exposited by Davidoff-Sarnak-Valette in their book, *Elementary Number Theory, Group Theory and Ramanujan Graphs*.

I have heard rumors of a different proof of the main result -- that the Cayley graphs $\Gamma(p,q)$ constructed from $PGL_2(\mathbb{F}_q)$ and a generating set $S_p$ have the Ramanujan bound on their largest real eigenvalue -- which relies on the following series of ideas.

One defines the Ihara Zeta function of the graph, and shows that the expander property for a graph (this is an inequality on its largest real eigenvalue) is equivalent to the Riemann hypothesis for the Ihara Zeta function, i.e. the assertion that it has poles only along $z=q^{1/2}$.

One identifies the Ihara Zeta function for $\Gamma(p,q)$ with the Zeta function for some curve over $\mathbb{F}_q$, presumably it is something double coset space $H\backslash PGL_2(\mathbb{F}_q) / H,$ for some canonical subgroup $H$. Herein lies my question!

One then applies the Weil conjectures/Riemann hypothesis for curves over finite fields, to determine that the $\Gamma(p,q)$ have the claimed Ramanujan property.

The problem is that I can't find a clear exposition anywhere of step 2, i.e. a simple statement of exactly which curve has its Zeta function equal that of the Ihara Zeta function, and how to prove that.