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

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 ${\text{GL}}_2(\mathbb{Z})$-invariants, classically denoted by $I$ and $J$.

17 pages, to appear in Mathematical Proceedings of the Cambridge Philosophical Society

References in corpus (1)

• The average size of the 2-Selmer group of Jacobians of hyperelliptic curves having a rational Weierstrass point

Cited by in corpus (2)

• Thue equations that simultaneously fail the Hasse principle
• The average number of integral points on the congruent number curves