On a conjecture of Agashe

2021/02/25 by Melistas, Mentzelos
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2102.12618

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.

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