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

Abstract

Fix a prime number $\ell \ge 5$. Let $K={\mathbb{F}}_q(t)$ be a global function field of characteristic $p$ coprime to $2,3$, and $q\equiv 1\text{ mod }\ell$. 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 $C_f:f{y}^{\ell }=F(x)$ 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 $(3p{)}^{4\ell }\cdot \ell !$ 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 Swinnerton-Dyer, Klagsbrun--Mazur--Rubin, Yu, and the author.

Version 3: 42 pages, 1 figure. Substantial revisions made, pertaining to the fact that the Weil pairing over $1-{\zeta }_{\ell }$ torsion module of the Jacobian of $C_f$ is symmetric. We devise a new Markov model which governs the $1-{\zeta }_{\ell }$ Selmer groups of Jacobians of $C_f$. Comments Welcome