### The modular tower of curves over a finite field.

I want to investigate the tower of modular curves $X(\ell^n) \to X(1)$ over a finite field $\mathbb F_q$ with $\ell \neq 0 0 \in \mathbb F_q$ and pullbacks of this tower by maps $C \to X(1)$.

In particular, I want to investigate the characteristic polynomial of the Frobenius on the $\ell$-adic cohomology. If I knew defining equations for $X(n)$ in terms of the parameter $t$ on $X(1)$, I guess I could do it but I am not sure...