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arXiv · 1603.08623

A positive proportion of Thue equations fail the integral Hasse principle

Shabnam Akhtari·Manjul Bhargava
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Abstract

For any nonzero $h\in\mathbb{Z}$, 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 $n \geq 3$, provided that these integral binary $n$-ic forms $G$ are ordered by the maximum of the absolute values of their coefficients.

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Shabnam Akhtari, Manjul Bhargava. 2016-03-29. A positive proportion of Thue equations fail the integral Hasse principle. https://doi.org/10.1353/ajm.2019.0006

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