TY - JOUR

T1 - A positive proportion of thue equations fail the integral hasse principle

AU - Akhtari, Shabnam

AU - Bhargava, Manjul

N1 - Funding Information:
Manuscript received March 2, 2016; revised September 16, 2017. Research of the first author supported in part by NSF grant DMS-1601837; research of the second author supported in part by a Simons Investigator grant and NSF grant DMS-1001828. American Journal of Mathematics 141 (2019), 283–307. ©c 2019 by Johns Hopkins University Press.
Publisher Copyright:
© 2019 by Johns Hopkins University Press.
Copyright:
Copyright 2019 Elsevier B.V., All rights reserved.

PY - 2019/4

Y1 - 2019/4

N2 - 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.

AB - 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.

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U2 - 10.1353/ajm.2019.0006

DO - 10.1353/ajm.2019.0006

M3 - Article

AN - SCOPUS:85063217199

SN - 0002-9327

VL - 141

SP - 283

EP - 307

JO - American Journal of Mathematics

JF - American Journal of Mathematics

IS - 2

ER -