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 Fq
with q elements, an Fq cover φ: X→Y
of normal varieties is
exceptional if it maps
one-one on Fqt 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 Fq) 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 Fq)
with φ1 and
φ2 having the same
image on Fqt 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 → xk for some odd integer
k. Euler's Theorem tells how
the periods of xk
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,Fq)
have a universal effect on all Poincare series over (Y,Fq). We say they produce Weil relations. The simplest of
these holds for X in the
exceptional tower of (Pn,Fq).
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,Fq) 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