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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 $p\equiv1\mod N$ outside of a set of primes of density zero. We also prove a strongform 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 $E(N)$ isof finite height.

On Tate's conjecture for elliptic modular surfaces over finite fields