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# Ranks of Twists of Elliptic Curves and Hilbert ’ s Tenth Problem

@inproceedings{Mazur2010RanksOT, title={Ranks of Twists of Elliptic Curves and Hilbert ’ s Tenth Problem}, author={Barry Mazur and Karl Rubin}, year={2010} }

- Published 2010

In this paper we investigate the 2-Selmer rank in families of quadratic twists of elliptic curves over arbitrary number fields. We give sufficient conditions on an elliptic curve so that it has twists of arbitrary 2-Selmer rank, and we give lower bounds for the number of twists (with bounded conductor) that have a given 2-Selmer rank. As a consequence, under appropriate hypotheses we can find many twists with trivial Mordell-Weil group, and (assuming the Shafarevich-Tate conjecture) many others… CONTINUE READING