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

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 𝔽 q with characteristic greater than 3 , Chipalkatti showed that the density of pairs of smooth conics satisfying the Poncelet triangle condition is 1 q + O ( q - 2 ) . We improve this result, showing that the density is exactly 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 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 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 = 𝔽 q ( λ ) .