ON THE NON-VANISHING OF p-ADIC HEIGHTS ON CM ABELIAN VARIETIES, AND THE ARITHMETIC OF KATZ p-ADIC L-FUNCTIONS

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 Z(p)-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 works 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.