Bhargavaology and Chabauty implies weak Stewart’s conjecture ‘almost surely’

This is a continuation of a previous post. Recall that Stewart’s conjecture asserts that there exists a constant $c_0$ such that for all binary forms $F(x,y) \in \mathbb{Z}[x,y]$ with non-zero discriminant and degree $r$ at least 3, there exists a number $C_F$ which depends on the coefficients of $F$ such that for all integers $|h| \geq C_F$, the equation $F(x,y) = h$ has at most $c_0$ solutions. The so-called weak Stewart’s conjecture is the same statement with the absolute constant $c_0$ replaced by a constant $c(r)$ which is only allowed to depend on the degree of $F$, but otherwise independent of (the coefficients of) $F$. The weaker conjecture is ‘almost surely’ a consequence of a theorem of Bhargava and Gross, which is the main content of this paper and recent work of Michael Stoll using Chabauty’s method; seen here. Indeed, by the Bhargava-Gross theorem, we know that `almost all’ (in the sense that all but $o(2^{-g})$ when ordered by naive height) hyperellitpic curves of genus $g$ with $g$ large has Jacobian rank less than $g-3$, since the average size is only $3/2$, and from there we obtain the bound that $c(2g+2) \leq 8(g+1)(g-1) + (4g-12)g = 12g^2 - 12g - 8$.

The formulation is as follows. The equation $F(x,y) = hz^2$ defines the equation of a quadratic twist of the hyperelliptic curve $F(x,y) = z^2$. Assuming that the average over the family of quadratic twists does not deviate too much from the average over all curves, we should see that every curve in the family of quadratic twists have small rank ‘most of the time’, and hence obtain the weak version of Stewart’s conjecture.

However, we are far from being able to establish such a result. The most basic obstruction is that there could exist a set of exceptional curves with very large Jacobian rank in a set of density $o(2^{-g})$, which may still be positive. Secondly, it is very difficult to control the average size, much less an upper bound, for the Jacobian rank of twists of a curve, as there are infinitely many congruence conditions at play.

Nevertheless, with a sharper tool to deal with potential large rank cases or even an argument to dispense the large genus case altogether, Chabauty’s method and Bhargavaology would imply the weak version of Stewart’s conjecture. Exciting times!