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

Fix a prime number $\mathrm{\ell }\ge 5$. Let $K={\mathbb{F}}_q(t)$ be a global function field of characteristic $p$ coprime to $2,3$, and $q\equiv 1\text{ mod }\mathrm{\ell }$. Let $C:y^{\mathrm{\ell }}=F(x)$ be a non-isotrivial superelliptic curve over $K$ such that $F$ is a degree $3$ polynomial over ${\mathbb{F}}_q(t)$. Denote by $C_f:f{y}^{\mathrm{\ell }}=F(x)$ 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 $99\mathrm{\%}$ of such curves $C_f$ have at most $(3p{)}^{5\mathrm{\ell }}\cdot \mathrm{\ell }!$ 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 $1-{\zeta }_{\mathrm{\ell }}$ 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 $50\mathrm{\%}$, a disparity phenomena observed in previous works by Klagsbrun–Mazur–Rubin, Yu, and Morgan for quadratic twist families of principally polarized abelian varieties.