대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제50권2호
  • /
  • Pages.407-416
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  • 2013
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  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
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ON THE p-PRIMARY PART OF TATE-SHAFAREVICH GROUP OF ELLIPTIC CURVES OVER ℚ WHEN p IS SUPERSINGULAR

  • Kim, Dohyeong (Department of Mathematics Pohang University of Science and Technology)
  • 투고 : 2011.03.09
  • 발행 : 2013.03.31
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초록

Let E be an elliptic curve over $\mathbb{Q}$ and $p$ be a prime of good supersingular reduction for E. Although the Iwasawa theory of E over the cyclotomic ${\mathbb{Z}}_p$-extension of $\mathbb{Q}$ is well known to be fundamentally different from the case of good ordinary reduction at p, we are able to combine the method of our earlier paper with the theory of Kobayashi [5] and Pollack [8], to give an explicit upper bound for the number of copies of ${\mathbb{Q}}_p/{\mathbb{Z}}_p$ occurring in the $p$-primary part of the Tate-Shafarevich group of E over $\mathbb{Q}$.

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참고문헌

  1. G. Chinta, Analytic ranks of elliptic curves over cyclotomic fields, J. Reine Angew. Math. 544 (2002), 13-24.
  2. J. Coates, Z. Liang, and R. Sujatha, The Tate-Shafarevich group for elliptic curves with complex multiplication II, Milan J. Math. 78 (2010), no. 2, 395-416. https://doi.org/10.1007/s00032-010-0127-2
  3. W. Duke, J. B. Friedlander, and H. Iwaniec, Bounds for automorphic L-functions. II, Invent. Math. 115 (1994), no. 2, 219-239. https://doi.org/10.1007/BF01231759
  4. D. Kim, On the Tate-Shafarevich group of elliptic curves over $\mathbb{Q}$, Bull. Korean Math. Soc. 49 (2012), no. 1, 155-163. https://doi.org/10.4134/BKMS.2012.49.1.155
  5. S. Kobayashi, Iwasawa theory for elliptic curves at supersingular primes, Invent. Math. 152 (2003), no. 1, 1-36. https://doi.org/10.1007/s00222-002-0265-4
  6. B. Mazur and P. Swinnerton-Dyer, Arithmetic of Weil Cuves, Invent. Math. 25 (1974), 1-61. https://doi.org/10.1007/BF01389997
  7. B. Mazur, J. Tate, and J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), no. 1, 1-48. https://doi.org/10.1007/BF01388731
  8. R. Pollack, On the p-adic L-function of a modular form at a supersingular prime, Duke Math. J. 118 (2003), no. 3, 523-558. https://doi.org/10.1215/S0012-7094-03-11835-9