The Period-Index Obstruction for Elliptic Curves

By Catherine O’Neil. 2002

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Let K be a field and let E be an elliptic curve over K . Let GK be the absolute Galois group of K . The elements of the group HðGK ;EÞ are in one-to-one correspondence with isomorphism classes of principal homogeneous spaces C of E. The period of such a C is its order in HðGK ;EÞ, a torsion group. The index of such a C is the smallest positive integer d such that there is a K-rational line bundle of degree d on C. The period-index problem is to determine when these invariants are equal. We denote the period of C by n and the index by d; then Lang and Tate [4] showed that n divides d and that d and n have the same prime factors. If K is a local field, Lichtenbaum [5] showed that d 1⁄4 n. If E is defined over Q and its analytic rank is 0, Stein [12] showed that for infinitely many integers n that there exist homogeneous spaces C with nðCÞ 1⁄4 dðCÞ 1⁄4 n. If the elliptic curve E has a rational n torsion point, Lang and Tate [4] showed that there are infinitely many homogeneous spaces C with nðCÞ 1⁄4 dðCÞ 1⁄4 n. There are also examples where nad (see [1, 4, 5]). Finally, Cassels originally showed [1] that elements C of Sha have nðCÞ 1⁄4 dðCÞ. In fact, we give another easy proof of this below. In this paper we will first examine the special case when the elliptic curve E is defined over a large enough field K so that the points of E1⁄2n ð % KÞ are all defined over K . We will relate the period-index obstruction map on such elliptic curves to the Hilbert symbol. We will then derive Hilbert symbol-like properties of this obstruction map in the general case. In particular, we show that the obstruction map is quadratic.

Subfields: Applied Mathematics, Geometry and Topology, Information Systems

View Paper: https://www.semanticscholar.org/paper/9685e97ec494b920b82cd37e99b20d697160f0e3