2 Deleted incorrect paragraph about Tate uniformization

I personally would say that the root of the problem is the absence of a globally defined exponential map on $\mathbb{C}_p$. In the complex world $z \mapsto q(z) = \exp(2 \pi i z)$ is an isomorphism between $\mathbb{C} / \mathbb{Z}$ and $\mathbb{C}^\times$; but $\mathbb{C}_p / \mathbb{Z}$ is a horrible mess, and has nothing to do with $\mathbb{C}_p^\times$.

This makes it hard to bridge between the world of $q$-expansions and the world of analytic functions satisfying nice functional equations. The functional equation of a modular form looks nice if you write it in terms of $z$, but if you try and write it in terms of $q$ alone, it will make no sense.

That's why one has to take a different approach, building $p$-adic modular forms directly as $p$-adic analytic functions (more precisely: analytic sections of line bundles) on modular curves (more precisely: the rigid spaces attached to modular curves), rather than trying to build them as functions on some other space satisfying functional equations which have the post facto consequence that they descend to modular curves, as in the theory over $\mathbb{C}$.

(You might be interested to know that the power series in $q$ defining certain natural modular forms do converge $p$-adically for $q$ < 1, and this is very important in Tate's theory of uniformization of elliptic curves over $\mathbb{Q}_p$; but these series aren't $p$-adic modular forms as such.)

1

I personally would say that the root of the problem is the absence of a globally defined exponential map on $\mathbb{C}_p$. In the complex world $z \mapsto q(z) = \exp(2 \pi i z)$ is an isomorphism between $\mathbb{C} / \mathbb{Z}$ and $\mathbb{C}^\times$; but $\mathbb{C}_p / \mathbb{Z}$ is a horrible mess, and has nothing to do with $\mathbb{C}_p^\times$.

This makes it hard to bridge between the world of $q$-expansions and the world of analytic functions satisfying nice functional equations. The functional equation of a modular form looks nice if you write it in terms of $z$, but if you try and write it in terms of $q$ alone, it will make no sense.

That's why one has to take a different approach, building $p$-adic modular forms directly as $p$-adic analytic functions (more precisely: analytic sections of line bundles) on modular curves (more precisely: the rigid spaces attached to modular curves), rather than trying to build them as functions on some other space satisfying functional equations which have the post facto consequence that they descend to modular curves, as in the theory over $\mathbb{C}$.

(You might be interested to know that the power series in $q$ defining certain natural modular forms do converge $p$-adically for $q$ < 1, and this is very important in Tate's theory of uniformization of elliptic curves over $\mathbb{Q}_p$; but these series aren't $p$-adic modular forms as such.)