On the non-vanishing of p-adic heights on CM abelian varieties, and the arithmetic of Katz p-adic L-functions

Abstract

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 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.

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