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Selmer Groups and Anticyclotomic Zp -extensions

Published online by Cambridge University Press:  12 May 2016

AHMED MATAR
AHMED MATAR*
Affiliation:
Department of Mathematics, University of Bahrain, P.O. Box 32038, Sukhair, Bahrain. e-mail: [email protected]
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Abstract

Let E/Q be an elliptic curve, p a prime and K ∞/K the anticyclotomic Zp -extension of a quadratic imaginary field K satisfying the Heegner hypothesis. In this paper we give a new proof to a theorem of Bertolini which determines the value of the Λ-corank of Selp∞ (E/K ∞) in the case where E has ordinary reduction at p. In the case where E has supersingular reduction at p we make a conjecture about the structure of the module of Heegner points mod p. Assuming this conjecture we give a new proof to a theorem of Ciperiani which determines the value of the Λ-corank of Selp∞ (E/K ∞) in the case where E has supersingular reduction at p.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 161 , Issue 3 , November 2016 , pp. 409 - 433
Copyright
Copyright © Cambridge Philosophical Society 2016 

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References

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] Bourbaki, N. Algébre: Chapitre 8, second edition (Springer, 2012).Google Scholar
[4] 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
[5] Brink, D. Prime decomposition in the anti-cyclotomic extensions. Mathematics of Computation. 76 (2007), no. 260, 2127–2138.CrossRefGoogle Scholar
[6] Ciperiani, M. Tate-Shafarevich groups in anticyclotomic Zp -extensions at supersingular primes. Compositio Math. 145 (2009), 293–308.CrossRefGoogle Scholar
[7] Ciperiani, M. and Wiles, A. Solvable points on genus one cuves. Duke Math. J. 142 (2008), 381–464.Google Scholar
[8] Coates, J. and Greenberg, R. Kummer theory for abelian varieties over local fields. Invent. Math. 124 (1996), 129–174.CrossRefGoogle Scholar
[9] Coates, J. and Sujatha, R. Galois cohomology of elliptic curves. Tata Inst. Fund. Res. Lecture Notes (Narosa Publishing House, 2000).Google Scholar
[10] Cornut, C. Mazur's conjecture on higher Heegner points. Invent. Math. 148 (2002), 495–523.CrossRefGoogle Scholar
[11] Cornut, C. and Vatsal, V. CM points and quarternion algebras. Doc. Math. 10 (2005), 263–309.CrossRefGoogle Scholar
[12] Cornut, C. and Vatsal, V. Nontriviality of Rankin–Selberg L-functions and CM points. Proc. London Math. Soc. Durham Symposium (2004).Google Scholar
[13] Greenberg, R. Iwasawa theory for elliptic curves. Lecture Notes in Math. 1716 (Springer, New York, 1999), pp. 51–144.CrossRefGoogle Scholar
[14] Greenberg, R. Introduction to Iwasawa theory for elliptic curves, IAS/Park City Math. Ser. 9 (Amer. Math Soc. Providence, 2001), pp. 407–464.Google Scholar
[15] Gross, B. Kolyvagin's work on modular elliptic curves. L-functions and arithmetic. London Math. Soc. Lecture Series 153 (1989), 235–256.Google Scholar
[16] Manin, Y. I. Cyclotomic fields and modular curves. Russian Math. Surveys 26 (6) (1971), 7–78.CrossRefGoogle Scholar
[17] Mazur, B. Rational points of abelian varieties with values in towers of number fields. Invent. Math 18 (1972), 183–266.CrossRefGoogle Scholar
[18] Milne, J. S. Arithmetic Duality Theorems, second ed. (BookSurge, LLC, Charleston, SC, 2006).Google Scholar
[19] Neukirch, J., Schmidt, A. and Wingberg, K. Cohomology of Number Fields, second edition, Grundlehren der Mathematischen Wissenschaften 323 (Springer, 2008), xvi+825.CrossRefGoogle Scholar
[20] Perrin-Riou, B. Fonctions L p-adiques, théorie d'Iwasawa et points de Heegner. Bull. Soc. Math. France 115 (1987), 399–456.CrossRefGoogle Scholar
[21] Serre, J.-P. Propriétés galoisiennes des points dordre fini des courbes elliptiques. Invent. Math. 15 (1972), no. 4, 259331.Google Scholar
[22] Silverman, J. The Arithmetic of Elliptic Curves. Grad. Texts in Math. 106 (Springer-Verlag, 1986).CrossRefGoogle Scholar
[23] Vatsal, V. Special values of anticyclotomic L-functions. Duke Math. J. 116 (2003), 219–261.CrossRefGoogle Scholar
[24] Wiles, A. Modular elliptic curves and Fermat's last theorem. Ann. of Math. 141 (2) (1995), 443–551.CrossRefGoogle Scholar