Let B be a simple CM abelian variety over a CM field E, p a rational
prime. Suppose that B has potentially ordinary reduction above p and is
self-dual with root number −1. Under some further conditions, we prove the
generic non-vanishing of (cyclotomic) p-adic heights on B along
anticyclotomic Zp-extensions of E. This provides evidence towards
Schneider's conjecture on the non-vanishing of p-adic heights. For CM
elliptic curves over \Q, the result was previously known as a consequence of
work of Bertrand, Gross--Zagier and Rohrlich in the 1980s. Our proof is based
on non-vanishing results for Katz p-adic L-functions and a Gross--Zagier
formula relating the latter to families of rational points on B.Comment: Ann. Inst. Fourier, to appea