Most odd-degree binary forms fail to primitively represent a square

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 FN (f₀) of degree- N superelliptic equations with fixed leading coefficient f₀ Z Z², ordered by height. For every sufficiently large N, we prove that among equations in the family FN (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.