The Exceptional Tower of a variety over
a finite field
Let Y be a normal (absolutely
irreducible) variety over a finite field Fq.
An exceptional cover X → Y over Fq
is a cover of normal varieties with the map on Fqt
points one-one for infinitely many t.
Exceptional covers of (Y,Fq)
form a category with fiber products. So there is a well-defined
I will outline its construction, and describe in some subtowers. The
most well-known subtower for (P1,Fq)
is given by the collection of maps P1x→ P1z by x → xn,
with (n,q-1)=1. The
exceptional tower of any (Y,Fq)
is a natural place for cryptography. There are also versions over
number fields, and both have their applications. I'll say something
about the following two.
Cryptography questions in the number field case naturally extend
questions on Serre's Open Image Theorem (action of an absolute Galois
group on elliptic curve division points; especially his Tchebotareff
Exceptional covers generalize to strong Davenport pairs. These give
universal relations among Poincare series defined by diophantine
problems over finite fields. For understanding Chow motives we would
like to know if all relations come from strong Davenport pairs.
Example: If Poincare series of a curve over Fq
has its t-th coefficient
equal to qt+1 for infinitely many t, is there a chain of exceptional
correspondences from the curve to P1?