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

## TL;DR

This paper proves the generic non-vanishing of cyclotomic p-adic heights on certain CM abelian varieties, providing evidence for Schneider's conjecture, by linking Katz p-adic L-functions with rational points via a Gross--Zagier formula.

## Contribution

It establishes the non-vanishing of p-adic heights on CM abelian varieties in new cases, extending previous results from CM elliptic curves to higher-dimensional varieties.

## Key findings

- Non-vanishing of p-adic heights under specific conditions
- Connection between Katz p-adic L-functions and rational points
- Support for Schneider's conjecture on p-adic heights

## 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 $\Z_{p}$-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 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 $B$.