Title: Poincare series coming from Cryptology questions about Exceptional Towers

Abstract: Schur (1921) posed how to describe all polynomials with rational (Q) coefficients that give  one-one mappings on infinitely many residue class fields. Davenport and Lewis (1961) asked if a polynomial map over Q, with a particular geometric property, would automatically have Schur's one-one mapping property. Both conjectures were right (1969).

Given a finite field F_q with q elements, an F_q cover φ: X→Y of normal varieties is exceptional if it maps one-one on F_q^t points for infinitely many t.  We use the Davenport-Lewis name "exceptional" because,  equivalently, a version of their geometric property holds for φ.

Exceptional covers of Y  (over F_q) form a category with fiber products having a uniquely defined arithmetic monodromy group. Pr-exceptional covers (pr ⇔ possibly reducible) form a large generalization. They include  Davenport pairs: φ₁: X₁→Y and φ₂: X₂→Y (over F_q) with  φ₁ and φ₂ having the same image on F_q^t points for infinitely many t.

This talk explores uses for pr-exceptional covers. Number field versions consider, say, covers over Q having  infinitely many primes with exceptional reductions. Exceptional covers φ: Pⁿ →Pⁿ  over Q have uses in cryptology because you can iterate φ and consider its period as a function of the prime of exceptional reduction. In Rivest-Shamir-Addleman, the exceptional covers are f: x → x^k for some odd integer k. Euler's Theorem tells how the periods of x^k  change with exceptional primes.

This  connects exceptional covers of Pⁿ to two well-known results in arithmetic geometry.

I. Denef-Loeser-Nicaise motives: They attach a "motivic Poincare series" to any diophantine problem over a number field.  Certain Davenport pairs over (Y,F_q) have a universal effect on all Poincare series over  (Y,F_q). We say they produce Weil relations. The simplest of these holds for X in the exceptional tower of (Pⁿ,F_q). How about the converse: If the zeta function for X satisfies this Weil relation, does X appear as an exceptional cover?

II. Serre's Open Image Theorem (OIT): The fiber product on the exceptional tower of  (Y,F_q) lets us  focus on particular subtowers.  For Y=P¹, we consider two subtowers generated by exceptional rational functions (X=P¹ also). Describing the first is equivalent to the C(omplex) M(ultiplication) part of Serre's OIT. Describing the second is equivalent to the G(eneral) L(inear) part of Serre's OIT. How do explicitness problems in the OIT  translate to exceptional covers?

Mike Fried, UC Irvine and MSU-Billings 01/17/07