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 $d^{\prime }(n)/q$ in general, where $d^{\prime }(n)$ 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 $K = \mathbb{F}_q(\lambda )$.
Un couple ordonné de coniques lisses satisfait à la condition du triangle de Poncelet s’il existe un triangle inscrit dans la première conique et circonscrit à la seconde. Sur un corps fini $\mathbb{F}_q$ de caractéristique strictement supérieure à $3$, Chipalkatti a montré que la densité des couples de coniques lisses satisfaisant à la condition du triangle de Poncelet est $\frac{1}{q}+O(q^{-2})$. Nous améliorons ce résultat en montrant que la densité est exactement $\frac{q-1}{q^2-q+1}$.
Nous considérons le problème de la détermination de la densité des couples de coniques satisfaisant à la condition du $n$-gone de Poncelet pour des valeurs de $n$ plus grandes. Nous prouvons une version corrigée d’une conjecture de Chipalkatti, en montrant que la proportion de couples de coniques lisses satisfaisant à la condition du quadrilatère de Poncelet est $\frac{1}{q} + O(q^{-3/2})$. Nous montrons que lorsque $n$ est un entier impair premier avec $q$, la densité des couples de coniques lisses satisfaisant à cette condition est $\frac{d(n)-1}{q}+O(q^{-3/2})$, où $d(n)$ est le nombre de diviseurs de $n$. Plus généralement, nous conjecturons que la densité des couples de coniques satisfaisant à la condition du $n$-gone de Poncelet est $d^{\prime }(n)/q$ en général, où $d^{\prime }(n)$ est le nombre de diviseurs de $n$ différents de $1$ et $2$. Notre argument repose sur l’analyse des points de $n$-torsion sur une certaine courbe elliptique sur le corps de fonctions $K = \mathbb{F}_q(\lambda )$.
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Keywords: Poncelet n-gons, elliptic curves, torsion points, Galois representations
Tianhao Wang 1
CC-BY-ND 4.0
Tianhao Wang. Counting pairs of conics over finite fields that satisfy the Poncelet $n$-gon condition. Journal de théorie des nombres de Bordeaux, Tome 38 (2026) no. 2, pp. 411-449. doi: 10.5802/jtnb.1366
@article{JTNB_2026__38_2_411_0,
author = {Tianhao Wang},
title = {Counting pairs of conics over finite fields that satisfy the {Poncelet} $n$-gon condition},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {411--449},
year = {2026},
publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
volume = {38},
number = {2},
doi = {10.5802/jtnb.1366},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1366/}
}
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%0 Journal Article %A Tianhao Wang %T Counting pairs of conics over finite fields that satisfy the Poncelet $n$-gon condition %J Journal de théorie des nombres de Bordeaux %D 2026 %P 411-449 %V 38 %N 2 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1366/ %R 10.5802/jtnb.1366 %G en %F JTNB_2026__38_2_411_0
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