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

Sun Woo Park

TL;DR

The paper addresses how Mordell–Lang-type bounds behave for twist families of a class of superelliptic curves over global function fields. It develops a Markov-operator framework to compute the distribution of $1-\zeta_\ell$ Selmer groups of Jacobians of twisted curves, extending prior cyclic-twist results to admissible rank-2 Galois modules and providing explicit convergence rates. Under suitable hypotheses, it proves uniform upper bounds on the average and high-probability bounds on the number of $K$-rational points across twists, and demonstrates a genuine disparity in the parity distribution of Selmer ranks that deviates from the $1/2$ PR-prediction. By applying these results to twists of $C: y^\ell = F(x)$ with $\deg F=3$, it yields explicit constants bounding the moments of $\#C_f(K)$, shows tight concentration for most twists, and clarifies how these function-field phenomena relate to, yet differ from, quadratic-twist heuristics in the number field setting. Overall, the work advances the statistical understanding of Mordell–Lang-type behavior for higher-genus curves over function fields and highlights nuanced departures from Poonen–Rains predictions in this context.

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 $1-ζ_\ell$ 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.

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

TL;DR

The paper addresses how Mordell–Lang-type bounds behave for twist families of a class of superelliptic curves over global function fields. It develops a Markov-operator framework to compute the distribution of Selmer groups of Jacobians of twisted curves, extending prior cyclic-twist results to admissible rank-2 Galois modules and providing explicit convergence rates. Under suitable hypotheses, it proves uniform upper bounds on the average and high-probability bounds on the number of -rational points across twists, and demonstrates a genuine disparity in the parity distribution of Selmer ranks that deviates from the PR-prediction. By applying these results to twists of with , it yields explicit constants bounding the moments of , shows tight concentration for most twists, and clarifies how these function-field phenomena relate to, yet differ from, quadratic-twist heuristics in the number field setting. Overall, the work advances the statistical understanding of Mordell–Lang-type behavior for higher-genus curves over function fields and highlights nuanced departures from Poonen–Rains predictions in this context.

Abstract

Fix a prime number . Let be a global function field of characteristic coprime to , and . Let be a non-isotrivial superelliptic curve over such that is a degree polynomial over . Denote by the twist of by a polynomial over . Assuming some conditions on , we show that the expected number of -rational points of is bounded, and at least of such curves have at most many -rational points, as ranges over the set of polynomials of sufficiently large degree over . To achieve this, we compute the distribution of dimensions of 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 , a disparity phenomena observed in previous works by Klagsbrun--Mazur--Rubin, Yu, and Morgan for quadratic twist families of principally polarized abelian varieties.
Paper Structure (11 sections, 17 theorems, 60 equations, 1 table)

This paper contains 11 sections, 17 theorems, 60 equations, 1 table.

Table of Contents

  1. Introduction
  2. Main result
  3. Selmer groups of Jacobians
  4. Organization
  5. Distribution of twisted Selmer structures of rank 2 Galois modules
  6. Effective Chebotarev density theorem
  7. Irreducible factors
  8. Markov chains
  9. Local Selmer groups
  10. Proof of Theorem \ref{['thm:Galois_module']}
  11. Applications to twists of some superelliptic curves

Key Result

Theorem 1.1

Let $g \geq 2$ be an integer, and let $K$ be a global field. Given any smooth projective curve $C$ of genus $g$ over $K$, there exists an absolute constant $c := c(g,K) > 0$ such that where $\mathrm{Jac}(C)$ is the Jacobian variety of the curve $C$ over $K$.

Theorems & Definitions (39)

  • Theorem 1.1: Uniform Mordell--Lang for curves
  • Theorem 1.2: A simplification of Theorem \ref{['thm:main']}
  • Theorem 1.3: A simplification of Theorem \ref{['thm:superelliptic']}
  • Definition 2.2: Definition 5.8, Lemma 5.9 of KMR14, Definition 4.2 of park2022prime
  • Definition 2.3
  • Definition 2.4: Definition 5.4 of KMR14
  • Lemma 2.5: Lemma 5.5 of KMR14
  • Definition 2.6: Definition 5.10 of KMR14
  • Definition 2.7: c.f. Definition 5.6, Definition 5.12 of KMR14
  • Definition 2.8
  • ...and 29 more