  Following a suggestion of Peter Scholze, we construct an action of
$\hat{\mathbb{G}}_m$ on the Katz moduli problem, a profinite-\'{e}tale cover of
the ordinary locus of the $p$-adic modular curve whose ring of functions is
Serre's space of $p$-adic modular functions. This action is a local, $p$-adic
analog of a global, archimedean action of the circle group $S^1$ on the
lattice-unstable locus of the modular curve over $\mathbb{C}$. To construct the
$\hat{\mathbb{G}}_m$-action, we descend a moduli-theoretic action of a larger
group on the (big) ordinary Igusa variety of Caraiani-Scholze. We compute the
action explicitly on local expansions and find it is given by a simple
multiplication of the cuspidal and Serre-Tate coordinates $q$; along the way we
also prove a natural generalization of Dwork's equation $\tau=\log q$ for
extensions of $\mathbb{Q}_p/\mathbb{Z}_p$ by $\mu_{p^\infty}$ valid over a
non-Artinian base. Finally, we give a direct argument (without appealing to
local expansions) to show that the action of $\hat{\mathbb{G}}_m$ integrates
the differential operator $\theta$ coming from the Gauss-Manin connection and
unit root splitting, and explain an application to Eisenstein measures and
$p$-adic $L$-functions.
