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

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

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Title: Ranks of Twists of Elliptic Curves and Hilbert’s Tenth Problem
Author: Mazur, Barry C.; Rubin, Karl

Note: Order does not necessarily reflect citation order of authors.

Citation: Mazur, Barry C., and Karl Rubin. 2010. Ranks of twists of elliptic curves and Hilbert’s tenth problem. Inventiones Mathematicae 181(3): 541-575.
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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.
Published Version: doi:10.1007/s00222-010-0252-0
Terms of Use: This article is made available under the terms and conditions applicable to Open Access Policy Articles, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAP
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:10356588
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