On the Neron-Severi groups of fibered varieties

Abstract

We apply Tate's conjecture on algebraic cycles to study the Néron-Severi groups of varieties fibered over a curve. This is inspired by the work of Rosen and Silverman, who carry out such an analysis to derive a formula for the rank of the group of sections of an elliptic surface. For a semistable fibered surface, under Tate's conjecture we derive a formula for the rank of the group of sections of the associated Jacobian fibration. For fiber powers of a semistable elliptic fibration $E --> C$, under Tate's conjecture we give a recursive formula for the rank of the Néron-Severi groups of these fiber powers. For fiber squares, we construct unconditionally a set of independent elements in the Néron-Severi groups. When $E --> C$ is the universal elliptic curve over the modular curve $X_0(M)/\Q$, we apply the Selberg trace formula to verify our recursive formula in the case of fiber squares. This gives an analytic proof of Tate's conjecture for such fiber squares over $\Q$, and it shows that the independent elements we constructed in fact form a basis of the Néron-Severi groups. This is the fiber square analog of the Shioda-Tate Theorem.