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

Abstract

For $N\ge 3$, we show Tate's conjecture for the elliptic modular surface $E(N)$ of level $N$ over ${\mathbb{F}}_p$ for a prime $p$ satisfying $p\equiv 1\mathrm{mod}N$ 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.

16 pages. Some improvements to v2