On the Néron-severi Groups of Fibered Varieties

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 X0(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.