A positive proportion of thue equations fail the integral hasse principle

Shabnam Akhtari, Manjul Bhargava

Research output: Contribution to journalArticle

Abstract

For any nonzero h ∈ 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 ≥ 3, provided that these integral binary n-ic forms G are ordered by the maximum of the absolute values of their coefficients.

Original languageEnglish (US)
Pages (from-to)283-307
Number of pages25
JournalAmerican Journal of Mathematics
Volume141
Issue number2
DOIs
StatePublished - Apr 2019

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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