Counting points of homogeneous varieties over finite fields

Counting Points on Varieties over Finite Fields Related to a Conjecture of Kontsevich

2 . Points over Finite Fields and the Weil Conjectures

In this chapter we will relate the topology of smooth projective varieties over the complex numbers with counting points over finite fields, via the Weil conjectures. If X is a variety defined over a finite field Fq, one can count its points over the various finite extensions of Fq; denote Nm = |X(Fmq )| (for instance, if X ⊂ AnFq is affine, given by equations f1, . . . , fk, then Nm = |{x ∈ Fm...

Using zeta functions to factor polynomials over finite fields

In 2005, Kayal suggested that Schoof’s algorithm for counting points on elliptic curves over finite fields might yield an approach to factor polynomials over finite fields in deterministic polynomial time. We present an exposition of his idea and then explain details of a generalization involving Pila’s algorithm for abelian varieties.

Rational Points and Coxeter Group Actions on the Cohomology of Toric Varieties

We derive a simple formula for the action of a finite crystallographic Coxeter group on the cohomology of its associated complex toric variety, using the method of counting rational points over finite fields, and the Hodge structure of the cohomology. Various applications are given, including the determination of the graded multiplicity of the reflection representation.

Counting points on varieties over finite fields of small characteristic

We present a deterministic polynomial time algorithm for computing the zeta function of an arbitrary variety of fixed dimension over a finite field of small characteristic. One consequence of this result is an efficient method for computing the order of the group of rational points on the Jacobian of a smooth geometrically connected projective curve over a finite field of small characteristic. ...