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Let $B$ be a simple CM abelian variety over a CM field $E$, $p$ a rationalprime. Suppose that $B$ has potentially ordinary reduction above $p$ and isself-dual with root number $-1$. Under some further conditions, we prove thegeneric non-vanishing of (cyclotomic) $p$-adic heights on $B$ alonganticyclotomic $\Z_{p}$-extensions of $E$. This provides evidence towardsSchneider's conjecture on the non-vanishing of $p$-adic heights. For CMelliptic curves over $\Q$, the result was previously known as a consequence ofwork of Bertrand, Gross--Zagier and Rohrlich in the 1980s. Our proof is basedon non-vanishing results for Katz $p$-adic $L$-functions and a Gross--Zagierformula relating the latter to families of rational points on $B$.

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