Uniform distribution of Heegner points

Abstract

Let E be a (modular!) elliptic curve over Q, of conductor N . Let K denote an imaginary quadratic field of discriminant D, with (N ,D) = 1. If p is a prime, then there exists a unique Zp-extension K∞/K such that Gal(K/Q) acts nontrivially on Gal(K∞/K ). The field K∞ is called the anticyclotomic Zp-extension of K . Let E(K∞ ) denote the Mordell-Weil group of E over K∞. Then a fundamental conjecture of Mazur [Maz] predicts that the size of E(K∞ ) is controlled by the prime factorization of N in K . Equivalently, Mazur’s conjecture relates the size of the Mordell-Weil group to the sign in the functional equation of certain L-series. The conjecture was verified by Greenberg, Rohrlich, and Rubin, in what Mazur calls the exceptional case, when E has complex multiplication by K . More generally, they settled the conjecture for certain abelian varieties with complex multiplication. For a discussion of this CM case, we refer the reader to [Gre], [Roh], and [Rub]. Our goal in this paper is to treat the generic case, which occurs either when E has no CM, or when the field of complexmultiplications is distinct from K . Under certain conditions on E and K , Mazur’s conjecture predicts that the group E(K∞ ) is finitely generated; our main result asserts that this is in fact the case, at least when p is an ordinary prime for E, or when the class number of K is prime to p. The main new ingredient we introduce is that of equidistribution, following ideas used by Ferrero andWashington to study the cyclotomicμ-invariant. More precisely, we show that theHeegner points associated to definite quaternion algebras are uniformly distributed on the components of a certain curve X , and that the elements of a certain Galois group act independently, in a suitable sense. This, combined with a special value formula due to Gross, allows us to conclude that the special values of anticyclotomic L-functions are almost always nonzero, so that the statement about the Mordell-Weil groups follows from the machinery of Euler systems as developed by Bertolini and Darmon [BD].