arXiv:1910.12409v2 [math.NT] 24 Apr 2020 MOST INTEGRAL ODD-DEGREE BINARY FORMS FAIL TO PROPERLY REPRESENT A SQUARE ASHVIN A. SWAMINATHAN Abstract. Let F ∈ Z[x, z ] be a binary form of degree 2n +1 ≥ 5. A result of Darmon and Granville known as “Faltings plus epsilon” states that when F is separable, the superelliptic equation y 2 = F (x, z ) has finitely many primitive integer solutions. In this paper, we prove a strong asymptotic version of “Faltings plus epsilon” which states that in families of superelliptic equations of sufficiently large degree and having a fixed non-square leading coefficient, a positive proportion of members have no primitive integer solutions (put another way, a positive proportion of the corresponding binary forms fail to properly represent a square). Moreover, we show that in these families, a positive proportion of everywhere locally soluble members have a Brauer-Manin obstruction to satisfying the Hasse principle. Our result can be viewed as an analogue for superelliptic equations of Bhargava’s result that most even-degree hyperelliptic curves over Q have no rational points. 1. Introduction Let F ∈ Z[x,z] be a binary form of degree N ≥ 5, and consider the equation y 2 = F (x,z). When N =2n is even, this equation cuts out a subscheme C F of the weighted projective plane P 2 Q (1,n, 1). We say that C F is a hyperelliptic curve if it is smooth and geometrically irreducible, and in this case, Faltings’ Theorem (see [Fal83]) states that the curve C F has finitely many rational points. In [Bha13, Theorem 1], Bhargava proved the following “strong asymptotic form” of Faltings’ Theorem: when the family of binary forms over Z of fixed even degree 2n is enumerated by height, the density of forms F such that the equation y 2 = F (x,z) has a rational solution is o(2 −n ). In simpler terms, most integral even-degree binary forms fail to represent a square. The objective of this paper is to prove a strong asymptotic form of Faltings’ Theorem for superelliptic equations y 2 = F (x,z), where F is of odd degree N =2n +1 ≥ 5. In this case, the problem of studying rational solutions is trivial: given any (x 0 ,z 0 ) ∈ Q 2 , the triple (1) (x,y,z)=(x 0 · F (x 0 ,z 0 ),F (x 0 ,z 0 ) n+1 ,z 0 · F (x 0 ,z 0 )) is readily seen to be a rational solution to y 2 = F (x,z). This triviality can be expressed in geometric terms as follows: the equation y 2 = F (x,z) cuts out a subscheme S F of the weighted projective plane P 2 Q (2, 2n+1, 2), and the rational map P 2 Q (2, 2n+1, 2)  P 1 Q sending [x : y : z]  [x : z] restricts to an isomorphism S F ∼ − P 1 Q . As is evident from (1), what makes the problem of studying rational solutions to the superelliptic equation y 2 = F (x,z) trivial is that the coordinates of a solution are allowed to have common factors. Indeed, the problem of studying primitive integer solutions — triples (x 0 ,c,z 0 ) ∈ Z 3 such that c 2 = F (x 0 ,z 0 ) and gcd(x 0 ,z 0 ) = 1 (i.e., proper representations of a square by the binary form F ) — is considerably more interesting. In [DG95, Theorem 1 ′ ], Darmon and Granville prove that y 2 = F (x,z) has finitely many primitive integer solutions when F is separable of degree 2n +1 ≥ 5 (see also [Dar97], where Darmon dubs this result “Faltings plus epsilon”). Given this analogue of Faltings’ Theorem, it is natural to hope Date : April 27, 2020. 2010 Mathematics Subject Classification. 11D45, 14G05 (primary), and 20G25 14H25 (secondary). 1