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arXiv · 0904.3709

Ranks of twists of elliptic curves and Hilbert's Tenth Problem

Barry Mazur·Karl Rubin
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Abstract

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 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.

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Barry Mazur, Karl Rubin. 2010-04-28. Ranks of twists of elliptic curves and Hilbert's Tenth Problem. https://doi.org/10.1007/s00222-010-0252-0

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