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Root numbers and parity phenomena
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The parity conjecture has a long and distinguished history. It gives a way of predicting the existence of points of infinite order on elliptic curves without having to construct them, and is responsible for a wide range of unexplained arithmetic phenomena. It is one of the main consequences of the Birch and Swinnerton-Dyer conjecture and lets one calculate the parity of the rank of an elliptic curve using root numbers. In this handbook, we explain how to use local root numbers of elliptic curves to realise some of these phenomena, with an emphasis on explicit calculations. The text is aimed at a `user' and, as such, we will not be concerned with the proofs of known cases of the parity conjecture, but instead we will demonstrate the use of the theory by means of examples.
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Elliptic curves of rank one over number fields
For every number field K and every 3-generic elliptic curve E/K with full rational 2-torsion, there are infinitely many quadratic twists of E with rank exactly 1.
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