Infinitely many elliptic curves of rank exactly two, II

TL;DR

This paper constructs an infinite family of elliptic curves with rank exactly two and specific torsion subgroups, advancing understanding of elliptic curve ranks under the parity conjecture.

Contribution

It introduces a method to generate infinitely many elliptic curves with rank exactly two and prescribed torsion subgroups, assuming the parity conjecture.

Findings

Infinite family of elliptic curves with rank exactly two

Elliptic curves with torsion subgroup cyclic of order two or three

Results depend on the parity conjecture

Abstract

In this paper, we construct an infinite family of elliptic curves whose rank is exactly two and the torsion subgroup is a cyclic group of order two or three, under the parity conjecture.