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

2018/09/30 by Peng Gao, Gao, P.
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1810.00320

openalex publication_date 2018/09/30 · openalex created_date 2018/10/05 · openalex updated_date 2026/07/28

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.

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