On a conjecture of Agashe

Abstract

Let E/Q be an optimal elliptic curve, −D be a negative fundamental discriminant coprime to the conductor N of E/Q and let E −D /Q be the twist of E/Q by −D. A conjecture of Agashe predicts that if E −D /Q has analytic rank 0, then the square of the order of the torsion subgroup of E −D /Q divides the product of the order of the Shafarevich-Tate group of E −D /Q and the orders of the arithmetic component groups of E −D /Q, up to a power of 2. This conjecture can be viewed as evidence for the second part of the Birch and Swinnerton-Dyer conjecture for elliptic curves of analytic rank zero. We provide a proof of a slightly more general statement without using the optimality hypothesis. c p (E −D ), up to a power of 2. Theorem 1.2 below, which is the main theorem of this paper, implies Conjecture 1.1 (see Corollary 3.19 for the proof of this implication). We note that Conjecture 1.1 and, hence, our