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

2009/04/30 by Barry Mazur, B. Mazur, Karl Rubin +1 · 3 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #Cryptography and Residue Arithmetic #math.NT #msc:11G05 #msc:11R11 #msc:11U05

paper · pdf · doi:10.1007/s00222-010-0252-0

Minor changes. To appear in Inventiones mathematicae

arxiv created 2010/04/28 · arxiv updated 2010/04/29 · openalex publication_date 2010/05/18 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

Citations

• Finding large Selmer rank via an arithmetic theory of local constants
• Kolyvagin systems

Cited by

• Quadratic spaces and Selmer groups of abelian varieties with multiplication
• Selmer stability for elliptic curves in Galois ℓ-extensions
• Second descent and rational points on Kummer varieties

Related

• On 2-Selmer ranks of quadratic twists of elliptic curves
• Selmer Ranks of Quadratic Twists of Elliptic Curves with Partial Rational Two-Torsion
• Increasing the p-Selmer rank by twisting
• Iwasawa theory and ranks of elliptic curves in quadratic twist families
• Quadratic Twists of Elliptic Curves with Small Selmer Rank