On the Local Divisibility of Heegner Points

Heegner points on the modular curve X0(N), and their images on elliptic curve factors E of the Jacobian, enjoy many remarkable properties. These points are the moduli of level structures with endomorphisms by the ring of integers of an imaginary quadratic field K. Their traces to E(K) have height given by the first derivative at s = 1 of the L-function of E over K (cf. [GZ86]), and their l-divisibility in the Mordell-Weil group controls the first l-descent on E over K (cf. [Gro]). In this note, we show how their l-divisibility in the local group E(Kp), where p is a prime that is inert in K, often determines a first descent over K on a related abelian variety A over Q. The abelian variety A is associated to a modular form of weight 2 and level Np that is obtained by Ribet’s level-raising theorem from the modular form of level N associated to E. This descent result is Theorem 2 below. To prove the descent theorem, we compare the local conditions defining a certain Selmer group for A with those defining the l-Selmer group for E. The conditions agree at places of K prime to p, and at p the condition changes from the unramified local condition to a transverse condition. The parity lemma proved in §5.3 then compares the ranks of the corresponding Selmer groups in terms of the l-divisibility of P in E(Kp) and allows us to understand a first descent on A/K based on Kolyvagin’s determination of the first l-descent on E/K. Some related work on the Selmer group can be found in [BD99, Prop 1.5] and [BD05]; a comparison with the value of the L-function at s = 1 is given in [BD99, Thm 1.3].