Number/Representation Theory

Let $E$ be a non-trivial elliptic curve over $\mathbb{Q}(t)$ of the form $E:y^2=x^3+f(t)$ for some polynomial $f(t)$. In this talk, I will first review some results on the rank and generators of the Mordell-Weil group $E(\mathbb{Q}(t))$ for certain families of polynomials $f(t)$, and then sketch the proof of my result for $f(t)=At^6+Bt^3+C$, which generalizes the result by Kloosterman (2026) and the result by Desjardins and Naskrecki (2024).