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 Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(e) ISSN 0894-0347(p)

     

Integral points on elliptic curves and $ 3$-torsion in class groups


Authors: H. A. Helfgott and A. Venkatesh
Journal: J. Amer. Math. Soc. 19 (2006), 527-550
MSC (2000): Primary 11G05, 11R29; Secondary 14G05, 11R11
Posted: January 19, 2006
MathSciNet review: 2220098
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Abstract | References | Similar Articles | Additional Information

Abstract: We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques and methods based on quasi-orthogonality in the Mordell-Weil lattice. We apply our results to break previous bounds on the number of elliptic curves of given conductor and the size of the $ 3$-torsion part of the class group of a quadratic field. The same ideas can be used to count rational points on curves of higher genus.

References

  • [Ba] Baker, A., The diophantine equation $ y^2 = a x^3 + b x^2 + c x + d$, J. Lond. Math. Soc. 43 (1968), 1-9. MR 0231783 (38:111)
  • [BEG] Brindza, B., Evertse, J.-H., and K. Györy, Bounds for the solutions of some Diophantine equations in terms of discriminants, J. Austral. Math. Soc. Ser. A 51 (1991), no. 1, 8-26. MR 1119684 (92e:11024)
  • [BK] Brumer, A., and K. Kramer, The rank of elliptic curves, Duke Math. J. 44 (1977), 715-743. MR 0457453 (56:15658)
  • [BP] Bombieri, E., and J. Pila, The number of integral points on arcs and ovals, Duke Math. J. 59 (1989), no. 2, 337-357. MR 1016893 (90j:11099)
  • [BS] Brumer, A., and J. H. Silverman, The number of elliptic curves over $ \mathbb{Q}$ with conductor $ N$, Manuscripta Math. 91 (1996), no. 1, 95-102. MR 1404420 (97e:11062)
  • [Bu] Bugeaud, Y., Bounds for the solutions of superelliptic equations, Comp. Math. 107 (1997), 187-219. MR 1458749 (98c:11025)
  • [CS] Conway, J. H., and N. J. A. Sloane, Sphere packings, lattices and groups, Grundlehren der Mathematischen Wissenschaften, 290, Springer-Verlag, New York, 1988.MR 0920369 (89a:11067)
  • [Da] David, S., Points de petite hauteur sur les courbes elliptiques, J. Number Theory 64 (1997), no. 1, 104-129. MR 1450488 (98k:11067)
  • [Du] Duke, W., Bounds for arithmetic multiplicities, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. 1998, extra vol. II, 163-172. MR 1648066 (99j:11042)
  • [DK] Duke, W., and E. Kowalski, A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations; with an appendix by Dinakar Ramakrishnan, Invent. Math. 139 (2000), no. 1, 1-39.MR 1728875 (2001b:11034)
  • [El] Elkies, N.D., Rational points near curves and small $ \vert x^3-y^2 \vert$ via lattice reduction, Algorithmic Number Theory, 33-63, Lecture Notes in Computer Science, 1838, Springer-Verlag, Berlin, 2000. MR 1850598 (2002g:11035)
  • [EV] Ellenberg, J. and A. Venkatesh, On uniform bounds for rational points on non-rational curves. IMRN 35 (2005).
  • [ES] Evertse, J.-H., and J. H. Silverman, Uniform bounds for the number of solutions to $ y^n = f(x)$, Math. Proc. Cambridge Philos. Soc. 100 (1986), no. 2, 237-248. MR 0848850 (87k:11034)
  • [Fo] Fouvry, É, Sur le comportement en moyenne du rang des courbes $ y\sp 2=x\sp 3+k$, Séminaire de Théorie des Nombres, Paris, 1990-91, Progr. Math., 61-84, Birkhäuser Boston, Boston, MA, 1993. MR 1263524 (95b:11057)
  • [GS] Gross, R., and J. H. Silverman, $ S$-integer points on elliptic curves, Pacific J. Math. 167 (1995), no. 2, 263-288. MR 1328329 (96c:11057)
  • [HjHr] Hajdu, L., and T. Herendi, Explicit bounds for the solutions of elliptic equations with rational coefficients, J. Symbolic Comput. 25 (1998), no. 3, 361-366. MR 1615334 (99a:11033)
  • [Ha] Hasse, H., Arithmetische Theorie der kubischen Zahlkörper auf klassenkörper-theoretischer Grundlage, Math. Z. 31 (1930) 565-582. Corrigendum: Math. Z. 31 (1930) 799. MR 1545136
  • [HBR] Heath-Brown, R., The density of rational points on curves and surfaces, Ann. of Math. (2) 155 (2002), no. 2, 553-595. MR 1906595 (2003d:11091)
  • [He] Helfgott, H. A., On the square-free sieve, Acta Arith. 115 (2004), no. 4, 349-402. MR 2099831 (2005h:11211)
  • [Her] Herrmann, E., Bestimmung aller $ S$-ganzen Lösungen auf elliptischen Kurven, Ph.D. thesis, Universität des Saarlandes.
  • [HS] Hindry, M., and J. H. Silverman, Diophantine geometry, Springer-Verlag, New York, 2000. MR 1745599 (2001e:11058)
  • [KL] Kabatjanskii, G. A., and V. I. Levenshtein, Bounds for packings on the sphere and in space, Problemy Peredaci Informacii 14 (1978), no. 1, 3-25. MR 0514023 (58:24018)
  • [KT] Kotov, S. V., and L. A. Trelina, $ S$-ganze Punkte auf elliptischen Kurven, J. Reine Angew. Math. 306 (1979), 28-41. MR 0524645 (80b:10024)
  • [La] Lang, S., Elliptic Curves: Diophantine Analysis, Springer-Verlag, 1978.MR 0518817 (81b:10009)
  • [Le] Levenshtein, V. I., Universal bounds for codes and designs, Handbook of coding theory, North-Holland, Amsterdam, Vol. I, 499-648. MR 1667942
  • [Ma] Mazur, B., Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183-266. MR 0444670 (56:3020)
  • [Me] Merel, L., Bornes pour la torsion des courbes elliptiques sur les corps de nombres, Invent. Math. 124 (1996), no. 1-3, 437-449. MR 1369424 (96i:11057)
  • [Mu] Mumford, D., A remark on Mordell's conjecture, Amer. J. Math. 87 (1965), 1007-1016.MR 0186624 (32:4083)
  • [Mur] Murty, M. R., Exponents of class groups of quadratic fields, Topics in number theory, 229-239, Kluwer Academic, Dordrecht, 1997. MR 1691322 (2000b:11123)
  • [Nek] Nekovár, J., Class numbers of quadratic fields and Shimura's correspondence, Math. Ann. 287 (1990), no. 4, 577-594. MR 1066816 (91k:11051)
  • [Pi] Pierce, L. B., The 3-part of class numbers of quadratic fields, J. London Math. Soc. (2) 71 (2005), no. 3, 579-598. MR 2132372
  • [Pin] Pintér, A., On the magnitude of integer points on elliptic curves, Bull. Austral. Math. Soc. 52 (1995), no. 2, 195-199. MR 1348477 (96k:11070)
  • [Schm] Schmidt, W., Integer points on curves of genus $ 1$, Compositio Math. 81 (1992), 33-59. MR 1145607 (93e:11076)
  • [Sch] Scholz, A., Über die Beziehung der Klassenzahlen quadratischer Körper zueinander, J. Reine Angew. Math. 166 (1932), 201-203.
  • [Se] Serre, J.-P., Lectures on the Mordell-Weil theorem, 3rd ed., Vieweg, Braunschweig, 1997. MR 1757192 (2000m:11049)
  • [Se2] Serre, J.-P., Local fields, Springer-Verlag, New York-Berlin, 1979. MR 0554237 (82e:12016)
  • [Shi] Shintani, T., On Dirichlet series whose coefficients are class numbers of integral binary cubic forms, J. Math. Soc. Japan 24 (1972), 132-188. MR 0289428 (44:6619)
  • [Si] Siegel, C. L., Lectures on the geometry of numbers, notes by B. Friedman, rewritten by K. Chandrasekharan with the assistance of R. Suter, Springer-Verlag, Berlin, 1989. MR 1020761 (91d:11070)
  • [Sil] Silverman, J. H., Advanced topics in the arithmetic of elliptic curves, Springer-Verlag, New York, 1994. MR 1312368 (96b:11074)
  • [Sil2] Silverman, J. H., The arithmetic of elliptic curves, Springer-Verlag, New York, 1985. MR 0817210 (87g:11070)
  • [Sil3] Silverman, J. H., The difference between the Weil height and the canonical height on elliptic curves, Math. Comp. 55 (1990), 723-743. MR 1035944 (91d:11063)
  • [Sil4] Silverman, J. H., Lower bound for the canonical height on elliptic curves, Duke Math. J. 48 (1981), no. 3, 633-648. MR 0630588 (82k:14043)
  • [Sil5] Silverman, J. H., Lower bounds for height functions, Duke Math. J. 51 (1984), no. 2, 395-403. MR 0747871 (87d:11039)
  • [Sil6] Silverman, J. H., A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves, J. Reine Angew. Math. 378 (1987), 60-100. MR 0895285 (89g:11047)
  • [So] Soundararajan, K., Divisibility of class numbers of imaginary quadratic fields, J. London Math. Soc. (2) 61 (2000), no. 3, 681-690. MR 1766097 (2001i:11128)
  • [Wo] Wong, S., Automorphic forms on $ {\rm GL}(2)$ and the rank of class groups, J. Reine Angew. Math. 515 (1999), 125-153. MR 1717617 (2000g:11042)
  • [Wo2] Wong, S., On the rank of ideal class groups, Number theory (Ottawa, ON, 1996), 377-383, CRM Proc. Lecture Notes, 19, Amer. Math. Soc., Providence, RI, 1999. MR 1684617 (2000k:11126)

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Additional Information

H. A. Helfgott
Affiliation: Department of Mathematics, Yale University, New Haven, Connecticut 06520
Address at time of publication: Département de mathématiques et de statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal QC; H3C 3J7, Canada
Email: helfgott@dms.umontreal.ca

A. Venkatesh
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139--4307
Address at time of publication: Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540
Email: akshay@ias.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-06-00515-7
PII: S 0894-0347(06)00515-7
Keywords: Class groups, elliptic curves, integral points.
Received by editor(s): May 21, 2004
Posted: January 19, 2006
Additional Notes: The second author was supported in part by NSF grant DMS-0245606.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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