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

2025/04/29 by Park, Sun Woo
#11G10 #11G30 #11M50 #60J10 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.20594

Abstract

Fix a prime number ℓ ≥ 5. Let K = \mathbbF_q(t) be a global function field of characteristic p coprime to 2,3, and q ≡ 1 mod ℓ. Let C:y^ℓ = F(x) be a non-isotrivial superelliptic curve over K such that F is a degree 3 polynomial over \mathbbF_q(t). Denote by C_f: fy^ℓ = F(x) the twist of C by a polynomial f over \mathbbF_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)^5ℓ ⋅ ℓ! many K-rational points, as f ranges over the set of polynomials of sufficiently large degree over \mathbbF_q. To achieve this, we compute the distribution of dimensions of 1-ζ_ℓ 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.

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