Denote the finite field of order q by F_q. By an
exceptional cover f: X -> Y over F_q we mean one where
f: X(F_{ q^t}) -> Y(F_{ q^t}) is one-one and onto for infinitely many t.
A Galois characterization of exceptionality produces a canonical tower of exceptional covers. Cryptology applications benefit from using subtowers of this tower. The literature has considered the values of t for which an exceptional cover is not exceptional. We give an example of where that appeared in a paper of Katz in 1981, and another in a paper of mine that calls for a charactization of median value curves. This talk is a complement to that I'm giving at Cal Tech on Thursday