A positive proportion of locally soluble quartic Thue equations are globally insoluble

Shabnam Akhtari

Type: Article

Publication Date: 2021-08-05

Citations: 2

DOI: https://doi.org/10.1017/s0305004121000554

Abstract

For any fixed nonzero integer $h$, we show that a positive proportion of integral binary quartic forms $F$ do locally everywhere represent $h$, but do not globally represent $h$. We order classes of integral binary quartic forms by the two generators of their ring of $\textrm{GL}_{2}(\mathbb{Z})$-invariants, classically denoted by $I$ and $J$.

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Mathematical Proceedings of the Cambridge Philosophical Society - View
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Works Cited by This (10)

Action	Title	Year	Authors
+	Effective finiteness results for binary forms with given discriminant	1991	Jan‐Hendrik Evertse Kálmán Győry
+	Representation of Small Integers by Binary Forms	2015	Shabnam Akhtari
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+	The number of points on a singular curve over a finite field	1994	David B. Leep Charles C. Yeomans
+	On the density of discriminants of cubic fields. II	1971	H. Davenport H. Heilbronn
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+ PDF Chat	On the Number of Solutions of Polynomial Congruences and Thue Equations	1991	C. L. Stewart