The place of exceptional covers among all diophantine relations

Abstract

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