A Necessary and Sufficient Condition for Existence of a Rational Point on an Elliptic Curve

TL;DR

This paper establishes a new necessary and sufficient condition for the existence of rational points on elliptic curves, transforming the problem into finding integer solutions to a related Diophantine equation, distinct from the Birch and Swinnerton-Dyer conjecture.

Contribution

It introduces a novel criterion based on a new formula for set intersections, providing an alternative to L-function-based conditions for rational points on elliptic curves.

Findings

New necessary and sufficient condition for rational points

A formula for intersection of finite sets

Distinct from Birch and Swinnerton-Dyer conjecture

Abstract

In this paper, the proof of the existence of a rational point on an elliptic curve is transformed into the proof of the existence of an integer solution for a Diophantine equation. By a new formula for calculating the number of elements in intersection of two finite sets, a necessary and sufficient condition for existence of a rational point on an elliptic curve is established. This condition is different from L-function in the Birch and Swinnerton-Dyer conjecture.