Let Fq be the finite field and : XY an Fq cover of normal varieties. We call exceptional if it maps 1-1 on Fqt points for an infinity of t. We say over Q is exceptional if it is exceptional mod infinitely many p. When X=Y, and is over Q, we have a map: exceptional p period of mod p. RSA cryptography uses x xk (k odd) and Euler's Theorem gives us its periods.
We give a paragraph of history: Schur (1921) posed a list of all Q exceptional polynomials. This inspired Davenport and Lewis (1961) to propose that a geometric property C D-L C would imply a polynomial is exceptional. Both were right (1969). Serre's O(pen) I(mage) T(heorem) produces most remaining exceptional Q rational functions (1977).
We use the D-L generalization to show exceptional covers (of Y over Fq) form a category with fiber products: the (Y,Fq) exceptional tower. Using that we can generate subtowers that connect the tower to two famous results.
I. Denef-Loeser-Nicaise motives: They attach a "motivic Poincare series" to any problem over Q. A generalization of exceptional covers produces (we say Weil) relations among Poincare series over (Y,Fq). The easiest converse question is this: If the zeta functions of X and P1 have a special Weil relation, is X an exceptional cover?
II. Serre's O(pen) I(mage) T(heorem): Rational functions from the OIT generate two (P1,Fq) exceptional subtower. The C(omplex) M(ultiplication) part of the OIT produces exceptional covers. We see their periods from the CM analog of Euler's Theorem. Periods of the subtower from the G(eneral) L(inear) part of the OIT give our most serious challenge.