  A cover of normal varieties is exceptional over a finite field if the map on
points over infinitely many extensions of the field is one-one. A cover over a
number field is exceptional if it is exceptional over infinitely many residue
class fields. The first result: The category of exceptional covers of a normal
variety, Z, over a finite field, F_q, has fiber products, and therefore a
natural Galois group (with permutation representation) limit. This has many
applications to considering Poincare series attached to diophantine questions.
The paper follows three lines:
  * The historical role of the Galois Theoretic property of exceptionality,
first considered by Davenport and Lewis.
  * How the tower structure on the category of exceptional covers of a pair
(Z,F_q) allows forming subtowers that separate known results from unknown
territory.
  * The use of Serre's OIT, especially the GL_2 case, to consider cryptology
periods and functional composition aspects of exceptionality.
  A more extensive html description of the paper is at
http://www.math.uci.edu/~mfried/paplist-ff/exceptTowYFFTA_519.html
