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 Q, but over 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

CRM-CICMA Québec Vermont Seminar Series

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