Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T05:13:14.043Z Has data issue: false hasContentIssue false
  • English
  • Français

Tate–Shafarevich groups in anticyclotomic ℤp-extensions at supersingular primes

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  19 February 2009

Mirela Çiperiani
Mirela Çiperiani*
Affiliation:
Mathematics Department, Columbia University, 2990 Broadway, New York, NY 10027, 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.

Let E/ℚ be an elliptic curve and p a prime of supersingular reduction for E. Denote by the anticyclotomic ℤp-extension of an imaginary quadratic field K which satisfies the Heegner hypothesis. Assuming that p splits in K/ℚ, we prove that has trivial Λ-corank and, in the process, also show that and both have Λ-corank two.

Keywords

elliptic curvesupersingular reductionSelmer groupTate–Shafarevich group

MSC classification

Secondary: 11G05: Elliptic curves over global fields
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 2 , March 2009 , pp. 293 - 308
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Bertolini, M., Selmer groups and Heegner points in anticyclotomic ℤp-extensions, Compositio Math. 99 (1995), 153–182.Google Scholar
[2]Bertolini, M. and Darmon, H., Kolyvagin’s descent and Mordell–Weil groups over ring class fields, J. Reine Angew. Math. 412 (1990), 63–74.Google Scholar
[3]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 electronic.CrossRefGoogle Scholar
[4]Çiperiani, M. and Wiles, A., Solvable points on genus one curves, Duke Math. J. 142 (2008), 381–464.Google Scholar
[5]Cornut, C., Mazur’s conjecture on higher Heegner points, Invent. Math. 148 (2002), 495–523.CrossRefGoogle Scholar
[6]Greenberg, R., Introduction to Iwasawa theory for elliptic curves, in Arithmetic algebraic geometry (Park City, Utah, 1999), IAS/Park City Mathematical Series, vol. 9 (American Mathematical Society, Providence, RI, 2001), 407–464.Google Scholar
[7]Gross, B. H., Kolyvagin’s work on modular elliptic curves, in L-functions and arithmetic (Durham, 1989), London Math. Soc. Lecture Note Series, vol. 153 (Cambridge University Press, Cambridge, 1991), 235–256 11G10 (11G40).CrossRefGoogle Scholar
[8]Kato, K., p-adic Hodge theory and values of zeta functions of modular forms, in Cohomologies p-adiques et applications arithmétiques. III, Astérisque, vol. 295 (2004), pp. ix, 117–290.Google Scholar
[9]Kobayashi, S., Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003), 1–36.CrossRefGoogle Scholar
[10]Kurihara, M., On the Tate Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction I, Invent. Math. 149 (2002), 195–224.CrossRefGoogle Scholar
[11]Perrin-Riou, B., Fonctions Lp-adiques, théorie d’Iwasawa et points de Heegner, Bull. Soc. Math. France 115 (1987), 399–456.CrossRefGoogle Scholar
[12]Pollack, R., An algebraic version of a theorem of Kurihara, J. Number Theory 110 (2005), 164–177.CrossRefGoogle Scholar
[13]Rohrlich, D. E., On L-functions of elliptic curves and cyclotomic towers, Invent. Math. 75 (1984), 409–423.CrossRefGoogle Scholar
[14]Rubin, K., On the main conjecture of Iwasawa theory for imaginary quadratic fields, Invent. Math. 93 (1988), 701–713.CrossRefGoogle Scholar
[15]Schneider, P., p-adic height pairings. II, Invent. Math. 79 (1985), 329–374.CrossRefGoogle Scholar
[16]Serre, J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331.CrossRefGoogle Scholar
[17]Vatsal, V., Special values of anticyclotomic L-functions, Duke Math. J. 116 (2003), 219–261.CrossRefGoogle Scholar
[18]Wiles, A., Modular elliptic curves and Fermat’s last theorem, Ann. of Math. (2) 141 (1995), 443–551.CrossRefGoogle Scholar