Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-08T19:29:28.046Z Has data issue: false hasContentIssue false
  • English
  • Français

Independence of points on elliptic curves arising from special points on modular and Shimura curves, II: local results

Part of: Arithmetic problems. Diophantine geometry Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 May 2009

Alexandru Buium  and
Bjorn Poonen
Alexandru Buium
Affiliation:
Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131, USA (email: [email protected])
Bjorn Poonen
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the predecessor to this article, we used global equidistribution theorems to prove that given a correspondence between a modular curve and an elliptic curve A, the intersection of any finite-rank subgroup of A with the set of CM-points of A is finite. In this article we apply local methods, involving the theory of arithmetic differential equations, to prove quantitative versions of a similar statement. The new methods apply also to certain infinite-rank subgroups, as well as to the situation where the set of CM-points is replaced by certain isogeny classes of points on the modular curve. Finally, we prove Shimura-curve analogues of these results.

Keywords

modular curveShimura curveCM pointcanonical liftreciprocity law

MSC classification

Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties 14G20: Local ground fields
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 3 , May 2009 , pp. 566 - 602
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Barcau, M., Isogeny covariant differential modular forms and the space of elliptic curves up to isogeny, Compositio Math. 137 (2003), 237–273.CrossRefGoogle Scholar
[2]Breuil, C., Conrad, B., Diamond, F. and Taylor, R., On the modularity of elliptic curves over ℚ: wild 3-adic exercises, J. Amer. Math. Soc. 14 (2001), 843–939.CrossRefGoogle Scholar
[3]Buium, A., Differential characters of abelian varieties over p-adic fields, Invent. Math. 122 (1995), 309–340.CrossRefGoogle Scholar
[4]Buium, A., Differential characters and characteristic polynomial of Frobenius, J. Reine Angew. Math. 485 (1997), 209–219.Google Scholar
[5]Buium, A., Differential modular forms, J. Reine Angew. Math. 520 (2000), 95–167.Google Scholar
[6]Buium, A., Differential modular forms on Shimura curves, I, Compositio Math. 139 (2003), 197–237.CrossRefGoogle Scholar
[7]Buium, A., Arithmetic differential equations, Mathematical Surveys and Monographs, vol. 118 (Americal Mathematical Society, Providence, RI, 2005).CrossRefGoogle Scholar
[8]Buium, A. and Poonen, B., Independence of points on elliptic curves arising from special points on modular and Shimura curves, I: global results, Duke Math. J., to appear.Google Scholar
[9]Buzzard, K., Integral models of certain Shimura curves, Duke Math. J. 87 (1997), 591–612.CrossRefGoogle Scholar
[10]Chai, C-L., A note on Manin’s theorem of the kernel, Amer. J. Math. 113 (1991), 387–389.CrossRefGoogle Scholar
[11]Conrad, B., The Shimura construction in weight 2 (Appendix to Lectures on Serre’s conjecture by K. Ribet and W. Stein), in Arithmetic algebraic geometry, IAS/Park City Mathematics Series, vol. 9, eds B. Conrad and K. Rubin (American Mathematical Society, Providence, RI, 2001).Google Scholar
[12]Cornut, C., Mazur’s conjecture on higher Heegner points, Invent. Math. 148 (2002), 495–523.CrossRefGoogle Scholar
[13]Diamond, F. and Im, J., Modular forms and modular curves, in Seminar on Fermat’s last theorem, CMS Conf. Proc., vol. 17 (American Mathematical Society, Providence, RI, 1995), 39–133.Google Scholar
[14]Diamond, F. and Shurman, J., A first course in modular forms, Graduate Texts in Mathematics, vol. 228 (Springer, Berlin, 2005).Google Scholar
[15]Dwork, B. and Ogus, A., Canonical liftings of Jacobians, Compositio Math. 58 (1986), 111–131.Google Scholar
[16]Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), 349–366.CrossRefGoogle Scholar
[17]Gross, B., A tameness criterion for Galois representations associated to modular forms mod p, Duke Math. J. 61 (1990), 445–517.CrossRefGoogle Scholar
[18]Gross, B. and Zagier, D., Heegner points and derivatives of L-series, Invent. Math. 84 (1986), 225–320.CrossRefGoogle Scholar
[19]Gross, B., Kohnen, W. and Zagier, D., Heegner points and derivatives of L-series II, Math. Ann. 278 (1987), 497–562.CrossRefGoogle Scholar
[20]Hurlburt, C., Isogeny covariant differential modular forms modulo p, Compositio Math. 128 (2001), 17–34.CrossRefGoogle Scholar
[21]Katz, N., p-adic properties of modular schemes and modular forms, Lecture Notes in Mathematics, vol. 350 (Springer, Berlin, 1973), 69–190.Google Scholar
[22]Katz, N., Serre-Tate local moduli, Lecture Notes in Mathematics, vol. 868 (Springer, Berlin, 1981), 138–202.Google Scholar
[23]Kolyvagin, V. A., Finiteness of E(ℚ) and SH(E,ℚ) for a subclass of Weil elliptic curves, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), 522–540.Google Scholar
[24]Lang, S., Introduction to modular forms (Springer, Heidelberg, 1976).Google Scholar
[25]Manin, Yu. I., Algebraic curves over fields with differentiation, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 737–756.Google Scholar
[26]Manin, Yu. I., Rational points of algebraic curves over function fields, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 1395–1440.Google Scholar
[27]Mazur, B., Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18 (1972), 183–266.CrossRefGoogle Scholar
[28]Mazur, B., Modular curves and arithmetic, in Proceedings of the International Congress of Mathematicians (Warsaw, 1983) (PWN, 1984), 185–211.Google Scholar
[29]Messing, W., The crystals associated to Barsotti-Tate groups, Lecture Notes in Mathematics, vol. 264 (Springer, Berlin, 1972).CrossRefGoogle Scholar
[30]Mumford, D., Abelian varieties (Oxford University Press, Oxford, 1970).Google Scholar
[31]Nekovár, J. and Schappacher, N., On the asymptotic behaviour of Heegner points, Turkish J. Math. 23 (1999), 549–556.Google Scholar
[32]Ogus, A., Hodge cycles and crystalline cohomology, in Hodge cycles, motives and Shimura varieties, Lecture Notes in Mathematics, vol. 900, eds P. Deligne, J. Milne, A. Ogus and K. Shih (Springer, Berlin, 1982), 357–414.CrossRefGoogle Scholar
[33]Pink, R., A combination of the conjectures of Mordell-Lang and André-Oort, in Geometric methods in algebra and number theory, Progress in Mathematics, vol. 235 (Birkhäuser, Basel, 2005), 251–282.CrossRefGoogle Scholar
[34]Poonen, B., Mordell-Lang plus Bogomolov, Invent. Math. 137 (1999), 413–425.CrossRefGoogle Scholar
[35]Rosen, M. and Silverman, J. H., On the independence of Heegner points associated to distinct quadratic imaginary fields, J. Number Theory 127 (2007), 10–36.CrossRefGoogle Scholar
[36]Serre, J.-P., Complex multiplication, in Algebraic number theory (Proc. instructional conf.) (Brighton, 1965) (Thompson, Washington, DC, 1967), 292–296.Google Scholar
[37]Serre, J.-P., Propriétés galoisennes des point d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331.CrossRefGoogle Scholar
[38]Serre, J.-P., Formes modulaires et fonctions zéta p-adiques, Lecture Notes in Mathematics, vol. 350 (Springer, Berlin, 1973).CrossRefGoogle Scholar
[39]Serre, J.-P., Algebraic groups and class fields, Graduate Texts in Mathematics, vol. 117 (Springer, Heidelberg, New York, 1988).CrossRefGoogle Scholar
[40]Serre, J.-P., Topics in Galois theory (Jones and Bartlett, Boston, 1992).Google Scholar
[41]Shimura, G., Introduction to the arithmetic theory of automorphic functions (Princeton University Press, Princeton, NJ, 1971).Google Scholar
[42]Silverman, J. H., Wieferich criterion and the abc conjecture, J. Number Theory 30 (1988), 226–237.CrossRefGoogle Scholar
[43]Silverman, J. H., Hecke points on modular curves, Duke Math. J. 60 (1990), 401–423.CrossRefGoogle Scholar
[44]Taylor, R. and Wiles, A., Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995), 553–572.CrossRefGoogle Scholar
[45]Vatsal, V., Uniform distribution of Heegner points, Invent. Math. 148 (2002), 1–48.CrossRefGoogle Scholar
[46]Voloch, J. F., Elliptic Wieferich primes, J. Number Theory 81 (2000), 205–209.CrossRefGoogle Scholar
[47]Wiles, A., Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), 443–551.CrossRefGoogle Scholar
[48]Zhang, S., Heights of Heegner points on Shimura curves, Ann. of Math. (2) 153 (2001), 27–147.CrossRefGoogle Scholar