A question about points on an elliptic curve with prime denominator

Simon L Rydin Myerson

A question about points on an elliptic curve with prime denominator

Number Theory 2023-07-19 v1

Authors: Simon L Rydin Myerson

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 $x(P), y(P) \in \mathbb Z$ and $z(P) \in \mathbb N$ 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?

Keywords

elliptic curve rational points weierstrass equation prime denominator integer coefficients

Cite

@article{arxiv.2307.09406,
  title  = {A question about points on an elliptic curve with prime denominator},
  author = {Simon L Rydin Myerson},
  journal= {arXiv preprint arXiv:2307.09406},
  year   = {2023}
}

Comments

3 pages. This is a slightly edited version of an open question I submitted to the problem session of the JA2023 conference, where it was communicated by Michel Waldschmidt

A question about points on an elliptic curve with prime denominator

Abstract

Keywords

Cite

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