  We give a new construction of overconvergent modular forms of arbitrary
weights, defining them in terms of functions on certain affinoid subsets of
Scholze's infinite-level modular curve. These affinoid subsets, and a certain
canonical coordinate on them, play a role in our construction which is strongly
analogous with the role of the upper half-plane and its coordinate `z' in the
classical analytic theory of modular forms. As one application of these ideas,
we define and study an overconvergent Eichler-Shimura map in the context of
compact Shimura curves over $\mathbb{Q}$, proving stronger analogues of results
of Andreatta-Iovita-Stevens.
