On Tate’s conjecture for the elliptic modular surface of level $N$ over a prime field of characteristic $1\ \protect \mathrm{mod}\ N$

Abstract

Assuming partial semisimplicity of Frobenius, we show Tate’s conjecture for the reduction of the elliptic modular surface E(N) of level N at a prime p satisfying p≡1modN and show that the Mordell–Weil rank is zero in this case. This extends a result of Shioda to N>4. Furthermore, we show that for every number field L partial semisimplicity holds for the reductions of E(N) L at a set of places of density 1.