Abstract

Abstract:

For any nonzero $h\in\Bbb{Z}$, 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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Additional Information

ISSN
1080-6377
Print ISSN
0002-9327
Pages
pp. 283-307
Launched on MUSE
2019-03-08
Open Access
No
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