Counting points of homogeneous varieties over finite fields

Abstract

Abstract. Let X be an algebraic variety over a finite field F q , homogeneous under a linear algebraic group. We show that there exists an integer N such that for any positive integer n in a fixed residue class mod N , the number of rational points of X over F q n is a polynomial function of q n with integer coefficients. Moreover, the shifted polynomials, where q n is formally replaced with q n + 1, have non-negative coefficients.

Multidimensional Exact Classes, Smooth Approximation and Bounded 4-Types

“…Sketch of proof. 6 We first show that there cannot exist infinitely many pairwise elementarily inequivalent Lie coordinatizable L-structures with the same skeletal type, where a skeletal type is, roughly speaking, a full description of the Lie coordinatizing tree structure in an extended language L sk ; see Section 4.2 of [13] for the full definition. So, for a contradiction, suppose that there are in fact infinitely many such L-structures {N i : i < } with the same skeletal type S. Working in L sk , by a judicious choice of ultrafilter we can take a non-principal ultraproduct N * of the N i such that N * ≡ N i for all i < .…”

“…The work of Brion and Peyre in [6] would be a good starting point for research into this question, as it suggests that algebraic varieties homogeneous under a linear algebraic group may provide a generic example of a polynomial exact class.…”

Multidimensional exact classes, smooth approximation and bounded 4-types

“…Sketch of proof. 6 We first show that there cannot exist infinitely many pairwise elementarily inequivalent Lie coordinatisable L-structures with the same skeletal type, where a skeletal type is, roughly speaking, a full description of the Lie coordinatising tree structure in an extended language L sk ; see § 4.2 of [14] for the full definition. So, for a contradiction, suppose that there are in fact infinitely many such L-structures {N i : i < ω} with the same skeletal type S. Working in L sk , by a judicious choice of ultrafilter we can take a non-principal ultraproduct N * of the N i such that N * ̸ ≡ N i for all i < ω.…”

Notions of Purity and the Cohomology of Quiver Moduli

“…However, the notion of strong purity of [5] differs from the above notion (again, for the constant local system): rather than requiring that all eigenvalues of F acting on H * et (X, Q l ) are integer powers of q, it requires that each eigenvalue α of F acting on H i et (X, Q l ) satisfies α n = q in/2 for some positive integer n.…”

“…We then apply these results to study the cohomology of quiver moduli. We prove that a natural stratification of the space of representations of a quiver with a fixed dimension vector is equivariantly perfect and from it deduce that each of the l-adic cohomology groups of the quiver moduli space is strongly pure.In [5], the first-named author and Peyre studied in detail the properties of the counting function n → |X(F q n )|, when X is a homogeneous variety under a linear algebraic group (all defined over F q ). For this, they introduced the notion of a weakly pure variety X, by requesting that all eigenvalues of F in H * c (X, Q l ) are of the form ζq j , where ζ is a root of unity, and j a nonnegative integer.…”