Let p ≡ 4, 7 mod 9 be a rational prime number such that 3 mod p is not a cube. In this paper we prove the 3-part of |X(E p )| · |X(E 3p 2 )| is as predicted by the Birch and Swinnerton-Dyer conjecture, where E p : x 3 + y 3 = p and E 3p 2 : x 3 + y 3 = 3p 2 are the elliptic curves related to the Sylvester conjecture and cube sum problems.
In this paper we discuss Waldspurger's local period integral for newforms in new cases. The main ingredient is the work [HN] on Waldspurger's period integral using the minimal vectors, and the explicit relation between the newforms and the minimal vectors. We use a representation theoretical trick to simplify computations for newforms. As an example, we compute the local integral coming from a special arithmetic setting which was used to study 3-part full BSD conjecture in [HSY].
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