Cryptology on Exceptional Covers
be the finite field and φ: X→Y
an Fq cover of normal varieties. We call φ exceptional
if it maps 1-1 on Fqt
points for an infinity of t.
We say φ over Q is exceptional if it is exceptional
mod infinitely many p.
When X=Y, and φ is over Q, we have a map: exceptional p →
period of φ mod p. RSA
cryptography uses x → xk (k odd) and Euler's Theorem gives us its
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 – D-L – would imply a polynomial is exceptional. Both were
(1969). Serre's O(pen) I(mage) T(heorem) produces most remaining
exceptional Q rational
We use the D-L generalization to show exceptional covers (of Y over Fq)
form a category with fiber products:
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
II. Serre's O(pen) I(mage)
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.
Mike Fried, Emeritus UC Irvine 04/15/08