DUALITY THEOREMS IN GALOIS COHOMOLOGY OVER NUMBER FIELDS

By J. Tate. 2010

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On the Birch-Swinnerton-Dyer quotients modulo squares (2006, background): “(If A is an elliptic curve or has a principal polarisation arising from a K-rational divisor, then the order of X is a square, see [5, 34, 30].”
Ranks of twists of elliptic curves and Hilbert’s tenth problem (2009, background): “By Poitou-Tate global duality (see for example [17, Theorem I.4.10], [ 28, Theorem 3.1 ], or [22, Theorem 1.7.3]), the images of the right-hand maps are orthogonal complements under the (nondegenerate) sum of the local Tate pairings, so their F2-dimensions sum to v∈T dimF2 H 1 f (Kv ,E [2]) .T he le”