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

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.

Minor changes. To appear in Inventiones mathematicae

References in corpus (2)

• Elliptic curves with all quadratic twists of positive rank
• On the arithmetic of twists of superelliptic curves

Cited by in corpus (15)

• Random maximal isotropic subspaces and Selmer groups
• Root numbers and parity of ranks of elliptic curves
• Disparity in Selmer ranks of quadratic twists of elliptic curves
• On the Ranks of the 2-Selmer Groups of Twists of a Given Elliptic Curve
• Ranks of elliptic curves with prescribed torsion over number fields
• Hasse principle for generalised Kummer varieties
• On 2-Selmer groups of twists after quadratic extension
• Rank growth of elliptic curves over $N$-th root extensions
• Second descent and rational points on Kummer varieties
• Rank stability of elliptic curves in certain non-abelian extensions
• Rank growth of elliptic curves in $S_4$ and $A_4$ quartic extensions of the rationals
• Rational homotopy type and computability
• On Conjectural Rank Parities of Quartic and Sextic Twists of Elliptic Curves
• A positive density of elliptic curves are diophantine stable in certain Galois extensions
• Selmer stability for elliptic curves in Galois $\ell$-extensions