Mazur, BarryRubin, Karl2013-02-282010Mazur, Barry C., and Karl Rubin. 2010. Ranks of twists of elliptic curves and Hilbert’s tenth problem. Inventiones Mathematicae 181(3): 541-575.0020-9910http://nrs.harvard.edu/urn-3:HUL.InstRepos:10356588In 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 with infinite cyclic Mordell-Weil group. Using work of Poonen and Shlapentokh, it follows from our results that if the Shafarevich-Tate conjecture holds, then Hilbert’s Tenth Problem has a negative answer over the ring of integers of every number field.en-USJournal Article2013-02-2810.1007/s00222-010-0252-0