Ranks of twists of elliptic curves and Hilbert’s tenth problem
Ranks of twists of elliptic curves and Hilbert’s tenth problem
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椭圆曲线的扭曲等级和希尔伯特第十问题
DOI：
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发表时间：
2009
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通讯作者：
K. Rubin
中科院分区：
文献类型：
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作者：
B. Mazur;K. Rubin
本文研究了任意数域上二次扭椭圆曲线族的2-塞尔默秩。给出了椭圆曲线具有任意2-塞尔默秩的扭转的充分条件，并给出了具有给定2-塞尔默秩的扭转数的下界.因此，在适当的假设下，我们可以发现许多平凡的Mordell-Weil群的扭曲，以及（假设Shafarevich-Tate猜想）许多无限循环的Mordell-Weil群的其他扭曲。利用Poonen和Shlapentokh的工作，从我们的结果可以得出，如果Shafarevich-Tate猜想成立，那么Hilbert第十问题在每个数域的整数环上都有否定解.
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.