Twists of elliptic curves with CM

Abstract: We consider certain families of sextic twists of the elliptic curve $y^2=x^3+1$ that are not defined over $\mathbb{Q}$, but over $\mathbb{Q}[\sqrt{-3}]$. We compute a formula that relates the central value of their L-functions $L(E, 1)$ to the square of a trace of a modular function evaluated at a CM point. Assuming the Birch and Swinnerton-Dyer conjecture, when the value above is non-zero, we should recover the order of the Tate-Shafarevich group, and we show that the value is indeed an integer square.

number theory

Audience: researchers in the topic

Warsaw Number Theory Seminar