On the distribution of points in projective space of bounded height.pdfVIP

ON THE DISTRIBUTION OF POINTS IN PROJECTIVE SPACEOF BOUNDED HEIGHTKWOK-KWONG CHOIAbstract. In this manuscript, we consider the uniform distribution of pointsin compact metric spaces. We assume that there exists a probability measureon the Borel subsets of the space which is invariant under a suitable groupof isometries. In this setting we prove the analogue of the Weyls Criterionand the Erdos-Turan inequality by using orthogonal polynomials associatedwith the space and the measure. In particular, we discuss the special case ofprojective space over completions of number elds in some detail. An invariantmeasure in these projective spaces is introduced and the explicit formulas forthe orthogonal polynomials in this case are given. Finally, using the analogousErdos-Turan inequality, we prove that the set of all projective points overthe number eld with bounded Arakelov height is uniformly distributed withrespect to the invariant measure as the bound increases.1. IntroductionLet k be an algebraic number eld, v a place of k and kv the completion of k withrespect to v. Let k
kv be an absolute value from v which extends the Euclideanabsolute value on kv if vj1 and the usual p-adic absolute value if vjp. We also usea second absolute value determined byj
jv := k
k dvdv ;where d = [k : Q] and dv = [kv : Qv ]. We note that the product formula holdsfor the absolute values j
jv . We extend both absolute values to a norm on nitedimensional vector spaces over kv as follows. For any column vector = 0BBB@ 01...N1 1CCCAin kNv , de ne kkv := 8:fPN1j=0 kjk2vg1=2 if vj1,max0jN1 kjkv if v -1, (1.1)and jjv := kk dvdv : (1.2)in both the in nite and nite cases.1991 Mathematics Subject Classi cation. Primary 11J61, 11J71, 11K60.The author was supported by NSF Grant DMS 9304580.1 2 KWOK-KWONG CHOILet PN1(kv) denote the N -dimensional projective space over kv and write[0; 1;


; N1] for the homogeneous coordinates of a generic element in PN1(kv).We let t