The period-index obstruction for elliptic curves.

*(English)*Zbl 1033.11029
J. Number Theory 95, No. 2, 329-339 (2002); erratum ibid. 109, No. 2, 390 (2004).

Consider an elliptic curve \(E\) defined over a field \(K\) with absolute Galois group \(G_K\). The elements of \(H^1(G_K,E)\) can be interpreted as isomorphism classes of principal homogeneous spaces \(C\) of \(E\). The order \(n\) of such an element \(C\) in \(H^1(G_K,E)\) is called its period; the index of \(C\) is the smallest positive integer \(d\) such that there exists a \(K\)-rational line bundle of degree \(d\) on \(C\). It is known that \(n | d\) and that they have the same prime factors [S. Lang and J. Tate, Am. J. Math. 80, 659–684 (1958; Zbl 0097.36203)]; moreover, if \(K\) is a local field, then we have \(n = d\) by results of S. Lichtenbaum [Am. J. Math. 90, 1209–1223 (1968; Zbl 0187.18602)].

In this article, the author constructs a map \(\text{Ob}\) from \(H^1(G_K,E[n])\) to the Brauer group \(Br(K)\) whose properties are then used to study the relation between the period \(n\) and the order \(d\) of principal homogeneous spaces over fields \(K\) containing the \(n\)-torsion points \(E[n]\) of \(E\). Almost trivial consequences of the existence of \(\text{Ob}\) are the inequality \(d \leq n^2\), or J. W. S. Cassels’ result [J. Lond. Math. Soc. 38, 244–248 (1963; Zbl 0113.03701)] that, over global fields \(K\), we have \(n=d\) for elements of the Tate-Shafarevich group of \(E\). In addition, the author shows that the function \(\text{Ob}\) is quadratic on the \({\mathbb Z}\)-module \(H^1(G_K,E[n])\), and the works out relations with Hilbert symbols and the Tate pairing.

In this article, the author constructs a map \(\text{Ob}\) from \(H^1(G_K,E[n])\) to the Brauer group \(Br(K)\) whose properties are then used to study the relation between the period \(n\) and the order \(d\) of principal homogeneous spaces over fields \(K\) containing the \(n\)-torsion points \(E[n]\) of \(E\). Almost trivial consequences of the existence of \(\text{Ob}\) are the inequality \(d \leq n^2\), or J. W. S. Cassels’ result [J. Lond. Math. Soc. 38, 244–248 (1963; Zbl 0113.03701)] that, over global fields \(K\), we have \(n=d\) for elements of the Tate-Shafarevich group of \(E\). In addition, the author shows that the function \(\text{Ob}\) is quadratic on the \({\mathbb Z}\)-module \(H^1(G_K,E[n])\), and the works out relations with Hilbert symbols and the Tate pairing.

Reviewer: Franz Lemmermeyer (Bilkent)

##### MSC:

11G05 | Elliptic curves over global fields |

##### Keywords:

Weil-Chatelet group; Tate-Shafarevich group; torsors; period-index obstruction; elliptic curves
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DOI

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##### References:

[1] | Cassels, J.W.S., Arithmetic on curves of genus 1, V. two counterexamples, J. London math. soc., 38, 244-248, (1963) · Zbl 0113.03701 |

[2] | Farb, B.; Dennis, R.K., “noncommutative algebra,”, graduate texts in mathematics, (1993), Springer-Verlag New York |

[3] | K. Hulek, “Projective Geometry of Elliptic Curves,” Astérisque, Vol, 137, Société Mathématique de France, 1986. · Zbl 0602.14024 |

[4] | Lang, S.; Tate, J., Principal homogeneous space over abelian varieties, Amer. J. math., 80, 659-684, (1958) · Zbl 0097.36203 |

[5] | Lichtenbaum, S., The period-index problem for elliptic curves, Amer. J. math., 90, 1209-1223, (1968) · Zbl 0187.18602 |

[6] | Mumford, D., “abelian varieties,”, (1985), Oxford Univ. Press Oxford |

[7] | O’Neil, C., Jacobians of genus one curves, Mathematical research letters, 8, 25-140, (2001) · Zbl 1024.14011 |

[8] | C. O’Neil, Explicit descent over X(3) and X(5), preprint 328 on the Algebraic Number Theory Preprint Archives. |

[9] | Serre, J.-P., “A course in arithmetic,”, (1979), Springer-Verlag New York |

[10] | Serre, J.-P., “local fields,”, (1979), Springer-Verlag New York |

[11] | Silverman, J.H., “the arithmetic of elliptic curves,”, (1986), Springer-Verlag New York · Zbl 0585.14026 |

[12] | Stein, W.A., There are genus one curves over Q of every odd index, J. reine angew. math., 547, 139-147, (2002) · Zbl 1002.11050 |

[13] | Tate, J., Relations between K_{2} and Galois cohomology, Invent. math., 36, 257-274, (1976) · Zbl 0359.12011 |

[14] | Zarkhin, Yu.G., Noncommutative cohomologies and Mumford groups, Mathematical notes, 15, 241-244, (1974) · Zbl 0291.14015 |

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