Counting rational points of a Grassmannian

Counting Rational Points on Hypersurfaces

For any n ≥ 2, let F ∈ Z[x1, . . . , xn] be a form of degree d ≥ 2, which produces a geometrically irreducible hypersurface in P. This paper is concerned with the number N(F ; B) of rational points on F = 0 which have height at most B. For any ε > 0 we establish the estimate N(F ; B) = O(B), whenever either n ≤ 5 or the hypersurface is not a union of lines. Here the implied constant depends at ...

Counting rational points of quiver moduli

It is shown that rational points over finite fields of moduli spaces of stable quiver representations are counted by polynomials with integer coefficients. These polynomials are constructed recursively using an identity in the Hall algebra of a quiver.

Counting Lattice Points of Rational Polyhedra

The generating function F (P ) = ∑ α∈P∩ZN xα for a rational polytope P carries all essential information of P . In this paper we show that for any positive integer n, the generating function F (P, n) of nP = {nx : x ∈ P} can be written as F (P, n) = ∑ α∈A Pα(n)x, where A is the set of all vertices of P and each Pα(n) is a certain periodic function of n. The Ehrhart reciprocity law follows autom...

Distribution of Rational Points: A Survey

Counting Rational Points on Algebraic Varieties

In these lectures we will be interested in solutions to Diophantine equations F (x1, . . . , xn) = 0, where F is an absolutely irreducible polynomial with integer coefficients, and the solutions are to satisfy (x1, . . . , xn) ∈ Z. Such an equation represents a hypersurface in A, and we may prefer to talk of integer points on this hypersurface, rather than solutions to the corresponding Diophan...