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

• 成果类型: Article
• 署名作者: Mazur, B.; Rubin, K.
• 署名单位: Harvard University; University of California System; University of California Irvine
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-010-0252-0
• 发表日期: 2010
• 页码: 541-575
• 关键词: quadratic twists; diophantine sets; rational-points

摘要

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.