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Counting pairs of conics over finite fields that satisfy the Poncelet $n$-gon condition

Tianhao Wang

TL;DR

This work studies densities of ordered pairs of smooth conics with transversal intersection over finite fields that satisfy Poncelet $n$-gon conditions. By tying each conic pair to an associated elliptic curve $E$ via the Griffiths–Harris framework and translating the $n$-gon condition into an $n$-torsion criterion on $E$, the authors reduce counts to torsion-point statistics on a family of elliptic curves over $K=\mathbb{F}_q(\lambda)$ and apply Chebotarev density for degree-1 reductions. They obtain an exact triangle-density $\frac{q-1}{q^2-q+1}$, and asymptotic densities $\frac{1}{q}+O(q^{-3/2})$ for tetragons and $\frac{d(n)-1}{q}+O(q^{-3/2})$ for odd $n$ coprime to $q$, plus a conjectural general density $\frac{d'(n)}{q}$; the analysis hinges on the $n$-torsion structure of specific elliptic curves (notably the Legendre family) and on the Galois action on torsion points. The paper also details how pencils of conics classify into five intersection-types, and provides explicit determinant-based criteria (via Cayley’s and Hankel determinants) that govern Poncelet conditions across pencils. Collectively, these results advance understanding of finite-field instances of Poncelet’s theorem and illuminate how Frobenius conjugacy drives conic-pair counts through elliptic-torsion phenomena.

Abstract

An ordered pair of smooth conics satisfies the Poncelet triangle condition if there is a triangle inscribed in the first conic and circumscribed in the second conic. Over a finite field $\mathbb{F}_q$ with characteristic greater than $3$, Chipalkatti showed that the density of pairs of smooth conics satisfying the Poncelet triangle condition is $\frac{1}{q}+O(q^{-2})$. We improve this result, showing that the density is exactly $\frac{q-1}{q^2-q+1}$. We consider the problem of determining the density of pairs of conics satisfying the Poncelet $n$-gon condition for larger $n$. We prove a corrected version of a conjecture of Chipalkatti, showing that the proportion of pairs of smooth conics satisfying the Poncelet tetragon condition is $\frac{1}{q} + O(q^{-{3/2}})$. We show that when $n$ is an odd integer coprime to $q$, the density of pairs of smooth conics satisfying this condition is $\frac{d(n)-1}{q}+O(q^{-3/2})$, where $d(n)$ is the number of divisors of $n$. More generally, we conjecture that the density of pairs of conics satisfying the Poncelet $n$-gon condition is $d'(n)/q$ in general, where $d'(n)$ is the number of divisors of $n$ not equal to $1$ or $2$. Our argument involves analyzing the $n$-torsion points on a certain elliptic curve over the function field $K = \mathbb{F}_q(λ)$.

Counting pairs of conics over finite fields that satisfy the Poncelet $n$-gon condition

TL;DR

This work studies densities of ordered pairs of smooth conics with transversal intersection over finite fields that satisfy Poncelet -gon conditions. By tying each conic pair to an associated elliptic curve via the Griffiths–Harris framework and translating the -gon condition into an -torsion criterion on , the authors reduce counts to torsion-point statistics on a family of elliptic curves over and apply Chebotarev density for degree-1 reductions. They obtain an exact triangle-density , and asymptotic densities for tetragons and for odd coprime to , plus a conjectural general density ; the analysis hinges on the -torsion structure of specific elliptic curves (notably the Legendre family) and on the Galois action on torsion points. The paper also details how pencils of conics classify into five intersection-types, and provides explicit determinant-based criteria (via Cayley’s and Hankel determinants) that govern Poncelet conditions across pencils. Collectively, these results advance understanding of finite-field instances of Poncelet’s theorem and illuminate how Frobenius conjugacy drives conic-pair counts through elliptic-torsion phenomena.

Abstract

An ordered pair of smooth conics satisfies the Poncelet triangle condition if there is a triangle inscribed in the first conic and circumscribed in the second conic. Over a finite field with characteristic greater than , Chipalkatti showed that the density of pairs of smooth conics satisfying the Poncelet triangle condition is . We improve this result, showing that the density is exactly . We consider the problem of determining the density of pairs of conics satisfying the Poncelet -gon condition for larger . We prove a corrected version of a conjecture of Chipalkatti, showing that the proportion of pairs of smooth conics satisfying the Poncelet tetragon condition is . We show that when is an odd integer coprime to , the density of pairs of smooth conics satisfying this condition is , where is the number of divisors of . More generally, we conjecture that the density of pairs of conics satisfying the Poncelet -gon condition is in general, where is the number of divisors of not equal to or . Our argument involves analyzing the -torsion points on a certain elliptic curve over the function field .
Paper Structure (29 sections, 22 theorems, 105 equations, 2 figures)

This paper contains 29 sections, 22 theorems, 105 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Poncelet triangles and Poncelet $n$-gons
  3. Main results
  4. Poncelet triangles and tetragons
  5. Pencils of conics and Cayley's criterion
  6. Number of pairs of smooth conics satisfying the Poncelet triangle condition
  7. Intersection type (1,1,1,1)
  8. Intersection type (2,1,1)
  9. Intersection type (2,2)
  10. Intersection type (4)
  11. Intersection type (3,1)
  12. Asymptotic density of pairs of smooth conics satisfying the Poncelet tetragon condition
  13. Poncelet $n$-gons and $n$-torsion points of elliptic curves over function fields
  14. Connection between Poncelet $n$-gon and $n$-torsion points of a family of elliptic curves
  15. Proof of Theorem \ref{['thm:n-gon']}
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $(\mathcal{A}, \mathcal{B})$ be an ordered pair of smooth plane conics with transversal intersection, and $(A,B)$ be their matrix representations. Then the pair $(\mathcal{A}, \mathcal{B})$ satisfies the Poncelet $n$-gon condition if and only if the point $(0, \sqrt{\det(B)})$ is an $n$-torsion

Figures (2)

  • Figure 1: An example of a Poncelet triangle. This image is taken from Chi.
  • Figure 2: The Poncelet construction. This image is taken from GH.

Theorems & Definitions (36)

  • Theorem 1.1: Griffiths-Harris
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • Theorem 3.2
  • ...and 26 more