  We apply Tate's conjecture on algebraic cycles to study the N\'eron-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 N\'eron-Severi groups of these fiber
powers. For fiber squares, we construct unconditionally a set of independent
elements in the N\'eron-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
N\'eron-Severi groups. This is the fiber square analog of the Shioda-Tate
Theorem.
