A positive proportion of Thue equations fail the integral Hasse principle
A positive proportion of Thue equations fail the integral Hasse principle
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Thue 方程的正比例不符合积分哈斯原理
DOI：
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发表时间：
2016
影响因子：
1.7
通讯作者：
M. Bhargava
中科院分区：
文献类型：
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作者：
S. Akhtari;M. Bhargava
摘要：对于任意非零的$hinBbb{Z}$，证明了整数二元三次形式$F$的正比例在局部处处表示$h$，但不全局表示$h$；即，正比例的三次Thue方程F(x,y)=h$不符合积分Hasse原理。在这里，我们对这类积分二元三次形式$F$的所有类按它们的绝对判别式排序。对于任意定次方程$G(x,y)= $ h$，我们证明了同样的结果，只要这些积分二进制$n$-ic形式$G$按其系数绝对值的最大值排序。
Abstract:For any nonzero $hinBbb{Z}$, we prove that a positive proportion of integral binary cubic forms $F$ do locally everywhere represent $h$ but do not globally represent $h$; that is, a positive proportion of cubic Thue equations $F(x,y)=h$ fail the integral Hasse principle. Here, we order all classes of such integral binary cubic forms $F$ by their absolute discriminants. We prove the same result for Thue equations $G(x,y)=h$ of any fixed degree $ngeq 3$, provided that these integral binary $n$-ic forms $G$ are ordered by the maximum of the absolute values of their coefficients.