On central L-values and the growth of the 3-part of the Tate–Shafarevich group

Abstract

Given any cube-free integer [Formula: see text], we study the [Formula: see text]-adic valuation of the algebraic part of the central [Formula: see text]-value of the elliptic curve [Formula: see text] We give a lower bound in terms of the number of distinct prime factors of [Formula: see text], which, in the case [Formula: see text] divides [Formula: see text], also depends on the power of [Formula: see text] in [Formula: see text]. This extends an earlier result of the author in which it was assumed that [Formula: see text] is coprime to [Formula: see text]. We also study the [Formula: see text]-part of the Tate–Shafarevich group for these curves and show that the lower bound is as expected from the conjecture of Birch and Swinnerton-Dyer, taking into account also the growth of the Tate–Shafarevich group.