Definition 4.2.2 of https://www.math.arizona.edu/~swc/aws/2018/2018SharifiNotes.pdf defines a $\Lambda$-adic form (where $\Lambda$ is an Iwasawa algebra) to be a cusp form if all but finitely many of its specializations are cusp forms.
However, since cuspidality is given by a vanishing condition, wouldn't such a thing be a cusp form at every specialization? Because something that vanishes at all but finitely many specializations should automatically vanish identically, right?