On the non-vanishing of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-adic heights on CM abelian varieties, and the arithmetic of Katz <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-adic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>L</mml:mi></mml:math>-functions

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