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On the prime Selmer ranks of cyclic prime twist families of elliptic curves over global function fields

Sun Woo Park

TL;DR

The paper proves that the $\pi$-Selmer ranks in cyclic order-$p$ twist families of a fixed non-isotrivial elliptic curve over the global function field $K=\mathbb{F}_q(t)$ (with $q\equiv1\pmod p$) follow the Poonen–Rains distribution, up to explicit error terms that decay as $n$ grows. The approach blends effective function-field Chebotarev and Erdős–Kac theorems with a governing Markov operator that models the evolution of local Selmer ranks under twists; by coupling local and global data and exploiting equidistribution of local characters, the authors derive a sharp convergence rate to the PR distribution. Under mild hypotheses (including that the Galois action on $E[p]$ is large, e.g., contains $\mathrm{SL}_2(\mathbb{F}_p)$), they also obtain corollaries about the growth of ranks in the twisted fields, showing that rank increases are rare and tightly controlled. This establishes a function-field analogue of the Bai–Mazur–Rains framework for Selmer distributions, with explicit, computable error bounds and a clear mechanism via Markov dynamics for the convergence to the universal PR law.

Abstract

Fix a prime number $p$. Let $\mathbb{F}_q$ be a finite field of characteristic coprime to 2, 3, and $p$, which also contains the primitive $p$-th root of unity $μ_p$. Based on the works by Swinnerton-Dyer and Klagsbrun, Mazur, and Rubin, we prove that the probability distribution of the sizes of prime Selmer groups over a family of cyclic prime twists of non-isotrivial elliptic curves over $\mathbb{F}_q(t)$ satisfying a number of mild constraints conforms to the distribution conjectured by Bhargava, Kane, Lenstra, Poonen, and Rains with explicit error bounds. The key tools used in proving these results are the Riemann hypothesis over global function fields, the Erdös-Kac theorem, and the geometric ergodicity of Markov chains.

On the prime Selmer ranks of cyclic prime twist families of elliptic curves over global function fields

TL;DR

The paper proves that the -Selmer ranks in cyclic order- twist families of a fixed non-isotrivial elliptic curve over the global function field (with ) follow the Poonen–Rains distribution, up to explicit error terms that decay as grows. The approach blends effective function-field Chebotarev and Erdős–Kac theorems with a governing Markov operator that models the evolution of local Selmer ranks under twists; by coupling local and global data and exploiting equidistribution of local characters, the authors derive a sharp convergence rate to the PR distribution. Under mild hypotheses (including that the Galois action on is large, e.g., contains ), they also obtain corollaries about the growth of ranks in the twisted fields, showing that rank increases are rare and tightly controlled. This establishes a function-field analogue of the Bai–Mazur–Rains framework for Selmer distributions, with explicit, computable error bounds and a clear mechanism via Markov dynamics for the convergence to the universal PR law.

Abstract

Fix a prime number . Let be a finite field of characteristic coprime to 2, 3, and , which also contains the primitive -th root of unity . Based on the works by Swinnerton-Dyer and Klagsbrun, Mazur, and Rubin, we prove that the probability distribution of the sizes of prime Selmer groups over a family of cyclic prime twists of non-isotrivial elliptic curves over satisfying a number of mild constraints conforms to the distribution conjectured by Bhargava, Kane, Lenstra, Poonen, and Rains with explicit error bounds. The key tools used in proving these results are the Riemann hypothesis over global function fields, the Erdös-Kac theorem, and the geometric ergodicity of Markov chains.
Paper Structure (18 sections, 21 theorems, 185 equations)

This paper contains 18 sections, 21 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. Remarks and Outlines
  3. Key Ingredients
  4. Outline of the proof
  5. Relevant works
  6. Effective theorems from the Riemann hypothesis
  7. Effective Chebotarev density theorem
  8. Erdös-Kac Theorem
  9. Splitting partitions of polynomials
  10. Some sets of places
  11. Splitting partition of polynomials over finite fields
  12. Equidistribution of local characters
  13. Local Selmer groups
  14. Local twists
  15. Auxiliary places
  16. ...and 3 more sections

Key Result

Theorem 1.2

Main Theorem. Fix a prime number $p$. Let $K = \mathbb{F}_q(t)$ be a global function field whose characteristic is coprime to $2$,$3$, and $q \equiv 1 \text{ mod } p$. Let $E: y^2 = F(x) = x^3 + Ax + B$ be an elliptic curve over $K$ which satisfies the following conditions. Let $\alpha(p)$ be a constant defined as where $0 < \gamma_p < 1$ is a constant depending on $p$ as defined in Corollary co

Theorems & Definitions (70)

  • Conjecture
  • Theorem 1.2
  • Remark 1.3
  • Corollary 1.4
  • proof
  • Remark 1.5
  • Remark 2.1
  • Theorem 3.1: Effective Chebotarev density theorem
  • Corollary 3.2
  • Remark 3.3
  • ...and 60 more