The p-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large
Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 787-800.

Nous montrons dans ce papier que pour chaque nombre premier p5, la dimension de la partie de p-torsion du groupe de Tate et Shafarevich, Ш(E/K), peut être arbitrairement grande, où E est une courbe elliptique définie sur un corps de nombres K de degré borné par une constante dépendant seulement de p. En utilisant ce résultat, nous obtenons aussi que la partie de p-torsion du Ш(A/) peut être arbitrairement grande, pour des variétées abéliennes A de dimension bornée par une constante dépendant seulement de p.

In this paper we show that for every prime p5 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:] 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/ can be arbitrarily large, where A is an abelian variety, with dimA bounded by a constant depending only on p.

Publié le :
DOI : https://doi.org/10.5802/jtnb.521
Mots clés : Tate-Shafarevich group, elliptic curve, abelian variety
@article{JTNB_2005__17_3_787_0,
     author = {Remke Kloosterman},
     title = {The $p$-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {787--800},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {3},
     year = {2005},
     doi = {10.5802/jtnb.521},
     mrnumber = {2212126},
     zbl = {1153.11313},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.521/}
}
Remke Kloosterman. The $p$-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large. Journal de Théorie des Nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 787-800. doi : 10.5802/jtnb.521. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.521/

[1] R. Bölling, Die Ordnung der Schafarewitsch-Tate Gruppe kann beliebig groß werden. Math. Nachr. 67 (1975), 157–179. | MR 384812 | Zbl 0314.14008

[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. | EuDML 150606 | MR 162800 | Zbl 0236.14012

[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. | EuDML 150666 | MR 179169 | Zbl 0241.14017

[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 . J. Eur. Math. Soc. 3 (2001), 169–201. | EuDML 277443 | MR 1831874 | Zbl 1007.11031

[6] H. Halberstam, H.-E. Richert, Sieve Methods. Academic Press, London, 1974. | MR 424730 | Zbl 0298.10026

[7] R. Kloosterman, E.F. Schaefer, Selmer groups of elliptic curves that can be arbitrarily large. J. Number Theory 99 (2003), 148–163. | MR 1957249 | Zbl 1074.11032

[8] K. Kramer, A family of semistable elliptic curves with large Tate-Shafarevich groups. Proc. Amer. Math. Soc. 89 (1983), 379–386. | MR 715850 | Zbl 0567.14018

[9] B. Mazur, A. Wiles, Class fields of abelian extensions of . Invent. Math. 76 (1984), 179–330. | EuDML 143124 | MR 742853 | Zbl 0545.12005

[10] J. S. Milne, On the arithmetic of abelian varieties. Invent. Math. 17 (1972), 177–190. | MR 330174 | Zbl 0249.14012

[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. | MR 1465369 | Zbl 0888.11023

[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. | MR 1638476 | Zbl 0897.11019

[13] J.-P. Serre, Local fields. Graduate Texts in Mathematics 67, Springer-Verlag, New York-Berlin, 1979. | MR 554237 | Zbl 0423.12016

[14] J.-P. Serre, Lectures on the Mordell-Weil theorem. Aspects of Mathematics, Friedr. Vieweg & Sohn, Braunschweig, 1989. | MR 1757192 | Zbl 0676.14005

[15] E.F. Schaefer, Class groups and Selmer groups. J. Number Theory 56 (1996), 79–114. | MR 1370197 | Zbl 0859.11034

[16] E.F. Schaefer, M. Stoll, How to do a p-descent on an elliptic curve. Preprint, 2001.

[17] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Princeton Univ. Press, Princeton, 1971. | MR 314766 | Zbl 0872.11023

[18] J. Silverman, The Arithmetic of Elliptic Curves. GTM 106, Springer-Verlag, New York, 1986. | MR 817210 | Zbl 0585.14026

[19] P. Stevenhagen, H.W. Lenstra, Jr, Chebotarëv and his density theorem. Math. Intelligencer 18 (1996), 26–37. | MR 1395088 | Zbl 0885.11005

[20] J. Vélu, Courbes elliptiques munies d’un sous-groupe Z/nZ×μ n . Bull. Soc. Math. France Mém. No. 57, 1978. | Numdam | MR 507751 | Zbl 0433.14029

[21] L.C. Washington, Galois cohomology. Modular forms and Fermat’s last theorem (Boston, MA, 1995), 101–120, Springer, New York, 1997. | MR 1638477 | Zbl 0928.12003