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

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

2009-04-23

arxiv.org/abs/0904.3709v3

Mathematics

Number Theory

Minor changes. To appear in Inventiones mathematicae

Scientific paper

10.1007/s00222-010-0252-0

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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