  We develop an analytic theory of cusps for Scholze's $p$-adic modular curves
at infinite level in terms of perfectoid parameter spaces for Tate curves. As
an application, we describe a canonical tilting isomorphism between an
anticanonical overconvergent neighbourhood of the ordinary locus of the modular
curve at level $\Gamma_1(p^\infty)$ and the analogous locus of an infinite
level perfected Igusa variety. We also prove various $q$-expansion principles
for functions on modular curves at infinite level, namely that the properties
of extending to the cusps, vanishing, coming from finite level, and being
bounded, can all be detected on $q$-expansions.
