2019/10/28 by Ashvin Swaminathan, Swaminathan, Ashvin
Computer Science · Mathematics · #11D45 #14G05 #14H25 #20G25 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1910.12409
openalex publication_date 2019/10/28 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
Let F be a separable integral binary form of odd degree N ≥ 5. A result of Darmon and Granville known as ``Faltings plus epsilon'' implies that the degree-N superelliptic equation y2 = F(x,z) has finitely many primitive integer solutions. In this paper, we consider the family \mathscrFN(f0) of degree-N superelliptic equations with fixed leading coefficient f0 ∈ ℤ \smallsetminus ±ℤ2, ordered by height. For every sufficiently large N, we prove that among equations in the family \mathscrFN(f0), more than 74.9% are insoluble, and more than 71.8% are everywhere locally soluble but fail the Hasse principle due to the Brauer--Manin obstruction. We further show that these proportions rise to at least 99.9% and 96.7%, respectively, when f0 has sufficiently many prime divisors of odd multiplicity. Our result can be viewed as a strong asymptotic form of ``Faltings plus epsilon'' for superelliptic equations and constitutes an analogue of Bhargava's result that most hyperelliptic curves over ℚ have no rational points.