Journal of the Ramanujan Mathematical Society

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Comparing the Corank of Fine Selmer Group and Selmer Group of Elliptic Curves


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1 Department of Mathematics and Statistics, IIT Kanpur, India
     

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Let p be an odd prime, K∞ be a pro-p, p-adic Lie extension of K = ℚ(μp) of dimension two containing the cyclotomic ℤp-extension Kcyc of K and H be the Galois group of K∞/Kcyc. Let Λ(H) be the Iwasawa algebra over H. Given an elliptic curve E defined over ℚ with good and supersingular reduction at p, we compare the Λ(H)-corank of the fine Selmer group of E over K∞ with the Iwasawa λ-invariant of the ±-Selmer group of E over Kcyc. Using this, we find examples of elliptic curves defined over ℚ with good and supersingular reduction at p satisfying pseudo nullity conjecture over K∞.
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  • [CS] J. Coates and R. Sujatha, Fine Selmer group of elliptic curve over p-adic Lie extensions, Math. Ann., 331, 809–839 (2005).

  • [DD] T. Dockchitser and V. Dockchitser, Computations in non commutative Iwasawa theory, Proc. London Math. Soc. (3), 94 (2007) 211–272.

  • [HV] Y. Hachimori and O. Venjakob, Completely faithful Selmer groups over Kummer extensions, Documenta Mathematica, Extra Volume, Kato (2003) 443–478.

  • [K] Shin-ichi Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. math., 152 1–36 (2003).

  • [Ka] K. Kato, p-adic Hodge theory and values of zeta functions of modular curves, Asterisque, 295 (2004) 117–290.

  • [Ku] M. Kurihara, On the Tate Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction I, Invent. Math., 149, 195–224 (2002).

  • [MK] M. Kurihara and R. Pollack, Two p-adic L-functions and rational points on elliptic curves with supersingular reduction. L-functions and Galois representations, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 320 (2007) 300–332,

  • [O] Y. Ochi, A Remark on the Pseudo-nullity conjecture for fine Selmer groups of elliptic curves, Commentary Mathematici Universitatis sancta pauli, 58 no. 1, (2009).

  • [OV] Y. Ochi and O. Venjakob, Structure of Selmer group over p-adic Lie extensions, J. Algebraic Geometry, 11 (2002) 547–580.

  • [P] R. Pollack, On the p-adic L-function of a modular form at a supersingular prime, Duke Mathematical Journal, 118 no. 3, (2003).

  • [P1] R. Pollack, Tables of Iwasawa invariants of elliptic curves, http://math.bu.edu/people/rpollack/Data/curves1-5000.

  • [P2] R. Pollack, The main conjecture for CM elliptic curves at supersingular primes, Annals of Mathematics, 159 (2004) 447–464.

  • [V] O. Venjakob, A non-commutative Weierstrass preparation theorem and applications to Iwasawa theory, Crelle J., 559 (2003) 153–191.

  • [V1] O. Venjakob, On the structure theory of the Iwasawa algebra of a p-adic Lie group, J. Eur. Math. Soc., 4 271–311 (2002).

  • [W] C. Wuthrich, Iwasawa theory of the fine Selmer group Journal of Algebraic Geometry, 16 (2007) 83–108.


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  • Comparing the Corank of Fine Selmer Group and Selmer Group of Elliptic Curves

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Authors

Sudhanshu Shekhar
Department of Mathematics and Statistics, IIT Kanpur, India

Abstract


Let p be an odd prime, K∞ be a pro-p, p-adic Lie extension of K = ℚ(μp) of dimension two containing the cyclotomic ℤp-extension Kcyc of K and H be the Galois group of K∞/Kcyc. Let Λ(H) be the Iwasawa algebra over H. Given an elliptic curve E defined over ℚ with good and supersingular reduction at p, we compare the Λ(H)-corank of the fine Selmer group of E over K∞ with the Iwasawa λ-invariant of the ±-Selmer group of E over Kcyc. Using this, we find examples of elliptic curves defined over ℚ with good and supersingular reduction at p satisfying pseudo nullity conjecture over K∞.

References