Height uniformity for integral points on elliptic curves

Ih, Su-ion

doi:10.1090/S0002-9947-05-03760-8

Height uniformity for integral points on elliptic curves
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Trans. Amer. Math. Soc. 358 (2006), 1657-1675 Request permission

Abstract:

We recall the result of D. Abramovich and its generalization by P. Pacelli on the uniformity for stably integral points on elliptic curves. It says that the Lang-Vojta conjecture on the distribution of integral points on a variety of logarithmic general type implies the uniformity for the numbers of stably integral points on elliptic curves. In this paper we will investigate its analogue for their heights under the assumption of the Vojta conjecture. Basically, we will show that the Vojta conjecture gives a naturally expected simple uniformity for their heights.

References

Dan Abramovich, Uniformité des points rationnels des courbes algébriques sur les extensions quadratiques et cubiques, C. R. Acad. Sci. Paris Sér. I Math. 321 (1995), no. 6, 755–758 (French, with English and French summaries). MR 1354720
Dan Abramovich, Uniformity of stably integral points on elliptic curves, Invent. Math. 127 (1997), no. 2, 307–317. MR 1427620, DOI 10.1007/s002220050121
Dan Abramovich and Joe Harris, Abelian varieties and curves in $W_d(C)$, Compositio Math. 78 (1991), no. 2, 227–238. MR 1104789
V. V. Batyrev and Yu. I. Manin, Sur le nombre des points rationnels de hauteur borné des variétés algébriques, Math. Ann. 286 (1990), no. 1-3, 27–43 (French). MR 1032922, DOI 10.1007/BF01453564
Gregory S. Call and Joseph H. Silverman, Canonical heights on varieties with morphisms, Compositio Math. 89 (1993), no. 2, 163–205. MR 1255693
Lucia Caporaso, Joe Harris, and Barry Mazur, Uniformity of rational points, J. Amer. Math. Soc. 10 (1997), no. 1, 1–35. MR 1325796, DOI 10.1090/S0894-0347-97-00195-1
Gerd Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), no. 3, 549–576. MR 1109353, DOI 10.2307/2944319
Jens Franke, Yuri I. Manin, and Yuri Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989), no. 2, 421–435. MR 974910, DOI 10.1007/BF01393904
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
Joe Harris and Joe Silverman, Bielliptic curves and symmetric products, Proc. Amer. Math. Soc. 112 (1991), no. 2, 347–356. MR 1055774, DOI 10.1090/S0002-9939-1991-1055774-0
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
Marc Hindry, Autour d’une conjecture de Serge Lang, Invent. Math. 94 (1988), no. 3, 575–603 (French). MR 969244, DOI 10.1007/BF01394276
Marc Hindry and Joseph H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. MR 1745599, DOI 10.1007/978-1-4612-1210-2
Heisuke Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero. I, II, Ann. of Math. (2) 79 (1964), 109–203; ibid. (2) 79 (1964), 205–326. MR 0199184, DOI 10.2307/1970547
Su-Ion Ih, Height uniformity for algebraic points on curves, Compositio Math. 134 (2002), no. 1, 35–57. MR 1931961, DOI 10.1023/A:1020246809487
Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. MR 715605, DOI 10.1007/978-1-4757-1810-2
Serge Lang, Number theory. III, Encyclopaedia of Mathematical Sciences, vol. 60, Springer-Verlag, Berlin, 1991. Diophantine geometry. MR 1112552, DOI 10.1007/978-3-642-58227-1
J. S. Milne, Jacobian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 167–212. MR 861976
Charles F. Osgood, A number theoretic-differential equations approach to generalizing Nevanlinna theory, Indian J. Math. 23 (1981), no. 1-3, 1–15. MR 722894
Charles F. Osgood, Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better, J. Number Theory 21 (1985), no. 3, 347–389. MR 814011, DOI 10.1016/0022-314X(85)90061-7
Patricia L. Pacelli, Uniform boundedness for rational points, Duke Math. J. 88 (1997), no. 1, 77–102. MR 1448017, DOI 10.1215/S0012-7094-97-08803-7
Patricia L. Pacelli, Uniform bounds for stably integral points on elliptic curves, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2535–2546. MR 1676288, DOI 10.1090/S0002-9939-99-05429-5
Jean-Pierre Serre, Lectures on the Mordell-Weil theorem, Aspects of Mathematics, E15, Friedr. Vieweg & Sohn, Braunschweig, 1989. Translated from the French and edited by Martin Brown from notes by Michel Waldschmidt. MR 1002324, DOI 10.1007/978-3-663-14060-3
Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. MR 918564, DOI 10.1007/978-1-4612-1035-1
Joseph H. Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197–211. MR 703488, DOI 10.1515/crll.1983.342.197
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Joseph H. Silverman, Rational points on symmetric products of a curve, Amer. J. Math. 113 (1991), no. 3, 471–508. MR 1109348, DOI 10.2307/2374836
Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. MR 1312368, DOI 10.1007/978-1-4612-0851-8
Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. MR 883451, DOI 10.1007/BFb0072989
Paul Vojta, Arithmetic discriminants and quadratic points on curves, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 359–376. MR 1085268, DOI 10.1007/978-1-4612-0457-2_{1}7
Paul Vojta, Siegel’s theorem in the compact case, Ann. of Math. (2) 133 (1991), no. 3, 509–548. MR 1109352, DOI 10.2307/2944318