Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ALGEBRAIC POINTS ON THE PROJECTIVE LINE

  • Ih, Su-Ion (DEPARTMENT OF MATHEMATICS UNIVERSITY OF COLORADO)
  • Published : 2008.11.01
Download PDF
⟨ Previous Next ⟩

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

References

  1. 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 https://doi.org/10.1007/BF01453564
  2. 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
  3. V. V. Batyrev, Manin's conjecture for toric varieties, J. Algebraic Geom. 7 (1998), no. 1, 15-53
  4. 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.)
  5. M. Hindry and J. H. Silverman, Diophantine Geometry. An Introduction, Graduate Texts in Mathematics, Vol. 201, Springer-Verlag, New York, 2000
  6. S. Ih, Height uniformity for algebraic points on curves, Compositio Math. 134 (2002), no. 1, 35-57 https://doi.org/10.1023/A:1020246809487
  7. S. Lang, Fundamentals of Diophantine Geometry, Springer-Verlag, New York, 1983
  8. D. Masser and J. D. Vaaler, Counting algebraic numbers with large height. II, Trans. Amer. Math. Soc. 359 (2007), no. 1, 427-445 https://doi.org/10.1090/S0002-9947-06-04115-8
  9. S. H. Schanuel, Heights in number fields, Bull. Soc. Math. France 107 (1979), no. 4, 433-449
  10. W. M. Schmidt, Northcott's theorem on heights. I. A general estimate, Monatsh. Math. 115 (1993), no. 1-2, 169-181 https://doi.org/10.1007/BF01311215
  11. W. M. Schmidt, Northcott's theorem on heights. II. The quadratic case, Acta Arith. 70 (1995), no. 4, 343-375 https://doi.org/10.4064/aa-70-4-343-375
  12. J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1986
  13. J. H. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 151. Springer-Verlag, New York, 1994
  14. P. Vojta, Diophantine Approximations and Value Distribution Theory, Lecture Notes in Mathematics, 1239. Springer-Verlag, Berlin, 1987

Cited by

  1. COUNTING MULTISECTIONS IN CONIC BUNDLES OVER A CURVE DEFINED OVER 𝔽q vol.07, pp.06, 2011, https://doi.org/10.1142/S179304211100485X