  For $N\geq 3$, we show Tate's conjecture for the elliptic modular surface
$E(N)$ of level $N$ over $\mathbb{F}_p$ for a prime $p$ satisfying $p\equiv
1\mod N$ outside of a set of primes of density zero. We also prove a strong
form of Tate's conjecture for $E(N)$ over any finite field of characteristic
$p$ prime to $N$ under the assumption that the formal Brauer group of $E(N)$ is
of finite height.
