The Gross--Kohnen--Zagier theorem via $p$-adic uniformization

This article gives a new proof of the Gross–Kohnen–Zagier theorem for Shimura curves which exploits the $p$-adic uniformization of Cerednik–Drinfeld. The explicit description of CM points via this uniformization leads to an expression relating the Gross–Kohnen–Zagier generating series to the ordinary projection of the first derivative, with respect to a weight variable, of a $p$-adic family of positive definite ternary theta series.