Infinite family of elliptic curves of rank at least 4

TL;DR

This paper proves that there are infinitely many rational parameters for which the elliptic curve $E_t$ has a rank of at least 4, expanding understanding of the distribution of ranks in families of elliptic curves.

Contribution

It establishes the existence of infinitely many elliptic curves with rank at least 4 within a specific family parameterized by rational numbers.

Findings

Infinitely many $t$ yield rank $ ext{at least } 4$

Construction of explicit examples of high-rank elliptic curves

Advances knowledge on rank distribution in elliptic curve families

Abstract

We investigate $\mathbb{Q}$-ranks of the elliptic curve $E_t$: $y^2+txy=x^3+tx^2-x+1$ where $t$ is a rational parameter. We prove that for infinitely many values of $t$ the rank of $E_t(\mathbb{Q})$ is at least 4.