Ranks of twists of elliptic curves and Hilbert’s tenth problem

By Barry Mazur, Karl Rubin. 2009

163 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Information Systems

View Paper: https://www.semanticscholar.org/paper/57bb066623a3a5ebc24e76b268565f3650321502

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Citations in the corpus

Root numbers and parity of ranks of elliptic curves (2009, unknown): “Also, a slightly weaker form of (a)⇔(d) is proved in [18] §9.”
Random maximal isotropic subspaces and Selmer groups (2010, background): “size of Sel2 is 3 [BS10a] and the average size of Sel3 is 4 [BS10b] (cf. Proposition 2.20). See [dJ02] for some earlier related investigations in the function ﬁeld case. Other evidence is provided in [MR10], which, for an elliptic curve E over a number ﬁeld k, gives lower bounds on the number of qua”