×

zbMATH — the first resource for mathematics

Height uniformity for integral points on elliptic curves. (English) Zbl 1118.14026
Summary: We recall the result of D. Abramovich [C. R. Acad. Sci., Paris, Sér. I 321, 755–758 (1995; Zbl 0874.14011)] and its generalization by P. L. Pacelli [Duke Math. J. 88, 77–102 (1997; Zbl 0935.14016)] 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. We 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.

MSC:
11G50 Heights
14G05 Rational points
11G05 Elliptic curves over global fields
11G35 Varieties over global fields
PDF BibTeX Cite
Full Text: DOI
References:
[1] 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). · Zbl 0874.14011
[2] Dan Abramovich, Uniformity of stably integral points on elliptic curves, Invent. Math. 127 (1997), no. 2, 307 – 317. · Zbl 0898.11020
[3] Dan Abramovich and Joe Harris, Abelian varieties and curves in \?_{\?}(\?), Compositio Math. 78 (1991), no. 2, 227 – 238. · Zbl 0748.14010
[4] 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). · Zbl 0679.14008
[5] Gregory S. Call and Joseph H. Silverman, Canonical heights on varieties with morphisms, Compositio Math. 89 (1993), no. 2, 163 – 205. · Zbl 0826.14015
[6] Lucia Caporaso, Joe Harris, and Barry Mazur, Uniformity of rational points, J. Amer. Math. Soc. 10 (1997), no. 1, 1 – 35. · Zbl 0872.14017
[7] Gerd Faltings, Diophantine approximation on abelian varieties, Ann. of Math. (2) 133 (1991), no. 3, 549 – 576. · Zbl 0734.14007
[8] Jens Franke, Yuri I. Manin, and Yuri Tschinkel, Rational points of bounded height on Fano varieties, Invent. Math. 95 (1989), no. 2, 421 – 435. , https://doi.org/10.1007/BF01393904 Jens Franke, Yuri I. Manin, and Yuri Tschinkel, Erratum: ”Rational points of bounded height on Fano varieties” [Invent. Math. 95 (1989), no. 2, 421 – 435; MR0974910 (89m:11060)], Invent. Math. 102 (1990), no. 2, 463. · Zbl 0709.14017
[9] William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. · Zbl 0541.14005
[10] Joe Harris and Joe Silverman, Bielliptic curves and symmetric products, Proc. Amer. Math. Soc. 112 (1991), no. 2, 347 – 356. · Zbl 0727.11023
[11] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001
[12] Marc Hindry, Autour d’une conjecture de Serge Lang, Invent. Math. 94 (1988), no. 3, 575 – 603 (French). · Zbl 0638.14026
[13] Marc Hindry and Joseph H. Silverman, Diophantine geometry, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, 2000. An introduction. · Zbl 0948.11023
[14] 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. · Zbl 0122.38603
[15] Su-Ion Ih, Height uniformity for algebraic points on curves, Compositio Math. 134 (2002), no. 1, 35 – 57. · Zbl 1031.11041
[16] Serge Lang, Fundamentals of Diophantine geometry, Springer-Verlag, New York, 1983. · Zbl 0528.14013
[17] Serge Lang, Number theory. III, Encyclopaedia of Mathematical Sciences, vol. 60, Springer-Verlag, Berlin, 1991. Diophantine geometry. · Zbl 0744.14012
[18] J. S. Milne, Jacobian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 167 – 212.
[19] Charles F. Osgood, A number theoretic-differential equations approach to generalizing Nevanlinna theory, Indian J. Math. 23 (1981), no. 1-3, 1 – 15. · Zbl 0528.30021
[20] Charles F. Osgood, Sometimes effective Thue-Siegel-Roth-Schmidt-Nevanlinna bounds, or better, J. Number Theory 21 (1985), no. 3, 347 – 389. · Zbl 0575.10032
[21] Patricia L. Pacelli, Uniform boundedness for rational points, Duke Math. J. 88 (1997), no. 1, 77 – 102. · Zbl 0935.14016
[22] Patricia L. Pacelli, Uniform bounds for stably integral points on elliptic curves, Proc. Amer. Math. Soc. 127 (1999), no. 9, 2535 – 2546. · Zbl 0935.14017
[23] 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. · Zbl 0676.14005
[24] Jean-Pierre Serre, Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117, Springer-Verlag, New York, 1988. Translated from the French. · Zbl 0703.14001
[25] Joseph H. Silverman, Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math. 342 (1983), 197 – 211. · Zbl 0505.14035
[26] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. · Zbl 0585.14026
[27] Joseph H. Silverman, Rational points on symmetric products of a curve, Amer. J. Math. 113 (1991), no. 3, 471 – 508. · Zbl 0749.14013
[28] Joseph H. Silverman, Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 151, Springer-Verlag, New York, 1994. · Zbl 0911.14015
[29] Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. · Zbl 0609.14011
[30] 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. · Zbl 0749.14018
[31] Paul Vojta, Siegel’s theorem in the compact case, Ann. of Math. (2) 133 (1991), no. 3, 509 – 548. · Zbl 0774.14019
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.