Mordell--Lang and disparate Selmer ranks of odd twists of some superelliptic curves over global function fields

Abstract

Fix a prime number $\ell \geq 5$. Let $K = \mathbb{F}_q(t)$ be a global function field of characteristic $p$ coprime to $2,3$, and $q \equiv 1 \text{ mod } \ell$. Let $C:y^\ell = F(x)$ be a non-isotrivial superelliptic curve over $K$ such that $F$ is a degree $3$ polynomial over $\mathbb{F}_q(t)$. Denote by $C_f: fy^\ell = F(x)$ the twist of $C$ by a polynomial $f$ over $\mathbb{F}_q$. Assuming some conditions on $C$, we show that the expected number of $K$-rational points of $C_f$ is bounded, and at least $99\%$ of such curves $C_f$ have at most $(3p)^{5\ell} \cdot \ell!$ many $K$-rational points, as $f$ ranges over the set of polynomials of sufficiently large degree over $\mathbb{F}_q$. To achieve this, we compute the distribution of dimensions of $1-\zeta_\ell$ Selmer groups of Jacobians of such superelliptic curves. This is done by generalizing the technique of constructing a governing Markov operator, as developed from previous studies by Klagsbrun--Mazur--Rubin, Yu, and the author. As a byproduct, we prove that the density of odd twist families of such superelliptic curves with even Selmer ranks cannot be equal to $50\%$, a disparity phenomena observed in previous works by Klagsbrun--Mazur--Rubin, Yu, and Morgan for quadratic twist families of principally polarized abelian varieties.