The p -part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large

Remke Kloosterman[1]

  • [1] Institute for Mathematics and Computer Science (IWI) University of Groningen P.O. Box 800 NL-9700 AV Groningen, The Netherlands Current address: Institut für Geometrie Universität Hannover Welfengarten 1 D-30167 Hannover, Germany

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 3, page 787-800
  • ISSN: 1246-7405

Abstract

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In this paper we show that for every prime p ≥ 5 the dimension of the p -torsion in the Tate-Shafarevich group of E / K can be arbitrarily large, where E is an elliptic curve defined over a number field K , with [ K : ℚ ] bounded by a constant depending only on p . From this we deduce that the dimension of the p -torsion in the Tate-Shafarevich group of A / ℚ can be arbitrarily large, where A is an abelian variety, with dim A bounded by a constant depending only on p .

How to cite

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Kloosterman, Remke. "The $p$-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large." Journal de Théorie des Nombres de Bordeaux 17.3 (2005): 787-800. <http://eudml.org/doc/249416>.

@article{Kloosterman2005,
abstract = {In this paper we show that for every prime $p\ge 5$ the dimension of the $p$-torsion in the Tate-Shafarevich group of $E/K$ can be arbitrarily large, where $E$ is an elliptic curve defined over a number field $K$, with $[K:\mathbb\{Q\}]$ bounded by a constant depending only on $p$. From this we deduce that the dimension of the $p$-torsion in the Tate-Shafarevich group of $A/\mathbb\{Q\}$ can be arbitrarily large, where $A$ is an abelian variety, with $\dim A$ bounded by a constant depending only on $p$.},
affiliation = {Institute for Mathematics and Computer Science (IWI) University of Groningen P.O. Box 800 NL-9700 AV Groningen, The Netherlands Current address: Institut für Geometrie Universität Hannover Welfengarten 1 D-30167 Hannover, Germany},
author = {Kloosterman, Remke},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Tate-Shafarevich group; elliptic curve; abelian variety},
language = {eng},
number = {3},
pages = {787-800},
publisher = {Université Bordeaux 1},
title = {The $p$-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large},
url = {http://eudml.org/doc/249416},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Kloosterman, Remke
TI - The $p$-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 3
SP - 787
EP - 800
AB - In this paper we show that for every prime $p\ge 5$ the dimension of the $p$-torsion in the Tate-Shafarevich group of $E/K$ can be arbitrarily large, where $E$ is an elliptic curve defined over a number field $K$, with $[K:\mathbb{Q}]$ bounded by a constant depending only on $p$. From this we deduce that the dimension of the $p$-torsion in the Tate-Shafarevich group of $A/\mathbb{Q}$ can be arbitrarily large, where $A$ is an abelian variety, with $\dim A$ bounded by a constant depending only on $p$.
LA - eng
KW - Tate-Shafarevich group; elliptic curve; abelian variety
UR - http://eudml.org/doc/249416
ER -

References

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  1. R. Bölling, Die Ordnung der Schafarewitsch-Tate Gruppe kann beliebig groß werden. Math. Nachr. 67 (1975), 157–179. Zbl0314.14008MR384812
  2. J.W.S. Cassels, Arithmetic on Curves of Genus 1 (VI). The Tate-Šafarevič group can be arbitrarily large. J. Reine Angew. Math. 214/215 (1964), 65–70. Zbl0236.14012MR162800
  3. J.W.S. Cassels, Arithmetic on curves of genus 1 (VIII). On the conjectures of Birch and Swinnerton-Dyer. J. Reine Angew. Math. 217 (1965), 180–189. Zbl0241.14017MR179169
  4. T. Fisher, On 5 and 7 descents for elliptic curves. PhD Thesis, Camebridge, 2000. 
  5. T. Fisher, Some examples of 5 and 7 descent for elliptic curves over ℚ . J. Eur. Math. Soc. 3 (2001), 169–201. Zbl1007.11031MR1831874
  6. H. Halberstam, H.-E. Richert, Sieve Methods. Academic Press, London, 1974. Zbl0298.10026MR424730
  7. R. Kloosterman, E.F. Schaefer, Selmer groups of elliptic curves that can be arbitrarily large. J. Number Theory 99 (2003), 148–163. Zbl1074.11032MR1957249
  8. K. Kramer, A family of semistable elliptic curves with large Tate-Shafarevich groups. Proc. Amer. Math. Soc. 89 (1983), 379–386. Zbl0567.14018MR715850
  9. B. Mazur, A. Wiles, Class fields of abelian extensions of ℚ . Invent. Math. 76 (1984), 179–330. Zbl0545.12005MR742853
  10. J. S. Milne, On the arithmetic of abelian varieties. Invent. Math. 17 (1972), 177–190. Zbl0249.14012MR330174
  11. B. Poonen, E.F. Schaefer, Explicit descent for Jacobians of cyclic covers of the projective line. J. Reine Angew. Math. 488 (1997), 141–188. Zbl0888.11023MR1465369
  12. D.E. Rohrlich, Modular Curves, Hecke Correspondences, and L -functions. In Modular forms and Fermat’s last theorem (Boston, MA, 1995), 41–100, Springer, New York, 1997. Zbl0897.11019MR1638476
  13. J.-P. Serre, Local fields. Graduate Texts in Mathematics 67, Springer-Verlag, New York-Berlin, 1979. Zbl0423.12016MR554237
  14. J.-P. Serre, Lectures on the Mordell-Weil theorem. Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1989. Zbl0676.14005MR1757192
  15. E.F. Schaefer, Class groups and Selmer groups. J. Number Theory 56 (1996), 79–114. Zbl0859.11034MR1370197
  16. E.F. Schaefer, M. Stoll, How to do a p -descent on an elliptic curve. Preprint, 2001. Zbl1119.11029
  17. G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton Univ. Press, Princeton, 1971. Zbl0872.11023MR314766
  18. J. Silverman, The Arithmetic of Elliptic Curves. GTM 106, Springer-Verlag, New York, 1986. Zbl0585.14026MR817210
  19. P. Stevenhagen, H.W. Lenstra, Jr, Chebotarëv and his density theorem. Math. Intelligencer 18 (1996), 26–37. Zbl0885.11005MR1395088
  20. J. Vélu, Courbes elliptiques munies d’un sous-groupe Z / n Z × μ n . Bull. Soc. Math. France Mém. No. 57, 1978. Zbl0433.14029MR507751
  21. L.C. Washington, Galois cohomology. Modular forms and Fermat’s last theorem (Boston, MA, 1995), 101–120, Springer, New York, 1997. Zbl0928.12003MR1638477

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