On the ℓ-adic cohomology of Jacobian elliptic surfaces over finite fields

Abstract

For a Jacobian elliptic surface $S_0$ over a finite field $k$ and a prime $\ell$ different from the characteristic of $k$, the points of period ${\ell }^r$ on the smooth fibers of $S_0$ yield, for each $r\in {\mathbb{Z}}_{\ge 0}$, a smooth projective curve $C_r$ over $k$ by taking Zariski closure in $S_0$ and normalization. We consider the restriction map in $\ell$-adic étale cohomology $H^2(S_0,{\mathbb{Z}}_{\ell }(1\left)\right)\to H^2({\bigsqcup }_{r\ge 0}{C}_r,{\mathbb{Z}}_{\ell }(1\left)\right)={\prod }_{r\ge 0}{H}^2(C_r,{\mathbb{Z}}_{\ell }(1\left)\right)$. By using an earlier result of ours we prove that, except for at most a finite number of such primes $\ell$, this map is faithful on the submodule $F^1{H}^2(S_0,{\mathbb{Z}}_{\ell }(1){)}^0$ of those classes vanishing on the geometric fibers and on the zero section of $S_0$, and that it gives an isomorphism between this submodule and the subgroup of $Pic\left({\bigsqcup }_{r\ge 0}{C}_r\right)={\prod }_{r\ge 0}Pic\left(C_r\right)$ of primitive elements in the sense of Serre.