[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2008.45.6.1635

ALGEBRAIC POINTS ON THE PROJECTIVE LINE

Ih, Su-Ion (DEPARTMENT OF MATHEMATICS UNIVERSITY OF COLORADO)

Publication Information

Journal of the Korean Mathematical Society / v.45, no.6, 2008 , pp. 1635-1646 More about this Journal

Abstract

Schanuel's formula describes the distribution of rational points on projective space. In this paper we will extend it to algebraic points of bounded degree in the case of ${\mathbb{P}}^1$ . The estimate formula will also give an explicit error term which is quite small relative to the leading term. It will also lead to a quasi-asymptotic formula for the number of points of bounded degree on ${\mathbb{P}}^1$ according as the height bound goes to $\infty$ .

Keywords

counting function; height; symmetric product;

Citations & Related Records

Times Cited By Web Of Science : 1 (Related Records In Web of Science)
Times Cited By SCOPUS : 1

Reference
Cited By SCOPUS

1	J. Franke, Y. I. Manin, and Y. Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989), no. 2, 421-435. (Erratum: Invent. Math. 102 (1990), no. 2, 463.)
2	J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 151. Springer-Verlag, New York, 1994
3	V. V. Batyrev and Y. I. Manin, Sur le nombre des points rationnels de hauteur borne des varietes algebriques, Math. Ann. 286 (1990), no. 1-3, 27-43 DOI
4	V. V. Batyrev and Y. Tschinkel, Rational points on some Fano cubic bundles, C. R. Acad. Sci. Paris Ser. I Math. 323 (1996), no. 1, 41-46
5	V. V. Batyrev, Manin's conjecture for toric varieties, J. Algebraic Geom. 7 (1998), no. 1, 15-53
6	M. Hindry and J. H. Silverman, Diophantine Geometry. An Introduction, Graduate Texts in Mathematics, Vol. 201, Springer-Verlag, New York, 2000
7	S. Ih, Height uniformity for algebraic points on curves, Compositio Math. 134 (2002), no. 1, 35-57 DOI ScienceOn
8	S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, New York, 1983
9	D. Masser and J. D. Vaaler, Counting algebraic numbers with large height. II, Trans. Amer. Math. Soc. 359 (2007), no. 1, 427-445 DOI
10	J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1986
11	P. Vojta, Diophantine Approximations and Value Distribution Theory, Lecture Notes in Mathematics, 1239. Springer-Verlag, Berlin, 1987
12	W. M. Schmidt, Northcott's theorem on heights. II. The quadratic case, Acta Arith. 70 (1995), no. 4, 343-375 DOI
13	S. H. Schanuel, Heights in number fields, Bull. Soc. Math. France 107 (1979), no. 4, 433-449
14	W. M. Schmidt, Northcott's theorem on heights. I. A general estimate, Monatsh. Math. 115 (1993), no. 1-2, 169-181 DOI