20 Pages
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# COUNTING POINTS OF HOMOGENEOUS VARIETIES OVER FINITE FIELDS

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Niveau: Supérieur, Doctorat, Bac+8
COUNTING POINTS OF HOMOGENEOUS VARIETIES OVER FINITE FIELDS MICHEL BRION AND EMMANUEL PEYRE Abstract. Let X be an algebraic variety over a finite field Fq, 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 Fqn is a polynomial function of qn with integer coefficients. Moreover, the shifted polynomials, where qn is formally replaced with qn + 1, have non-negative coefficients. 1. Introduction and statement of the results Given an algebraic variety X over a finite field k = Fq, one may consider the points of X which are rational over an arbitrary finite field extension Fqn . The number of these points is given by Grothendieck's trace formula, (1.1) |X(Fqn)| = ∑ i≥0 (?1)i Tr ( F n, H ic(X) ) , where F denotes the Frobenius endomorphism of Xk¯ and H i c(X) stands for the ith -adic cohomology group of Xk¯ with proper supports, being a prime not dividing q (see e.g. [De77, Thm. 3.2,p. 86]). Moreover, by celebrated results of Deligne (see [De74, De80]), each eigenvalue ? of F acting on H ic(X) is an algebraic number, and all the complex conjugates of ? have absolute value q w 2 for some non-negative integer w ≤ i, with equality

• then gx

• a?g?x ?

• quotient group

• group

• algebraic variety

• any connected algebraic

• varieties over

• over finite

• connected linear

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