  Following D. Ramakrishnan, we explain how L. Lafforgue's modularity theorem
and an analytic theorem of H. Jacquet and J. Shalika can be applied to prove
the following result related to the Tate Conjecture: for a smooth, projective,
geometrically-connected variety defined over a global function field, the
algebraic rank is less than or equal to the analytic rank.
  Also discussed is the analogous (open) question for number fields and an easy
extension of Lafforgue's theorem to remove the "finite-order character"
assumption. All results are likely "known to the experts", but don't appear to
be written down.
