A question about points on an elliptic curve with prime denominator

2023/07/13 by Myerson, Simon L Rydin
#11G05 #11N05 (Primary) 11N36 #11N32 (Secondary) #14G05 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2307.09406

Abstract

Let E be an elliptic curve defined by a Weierstrass equation with integer coefficients. Any rational point on E other than the identity is of the form ( x(P) / z(P)² , y(P) / z(P)³ ) where x(P), y(P) ∈ \mathbb Z and z(P) ∈ \mathbb N and in addition both gcd( x(P) , z(P) ) = 1 and gcd( y(P) , z(P) ) = 1 hold. The question is: how often is z(P) a prime number?

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