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The most elementary way to define $p$-adic modular forms is via limits of classical modular forms. More precisely $f \in \mathbb{Z}_p[[q]]$ is called a $p$-adic modular form if there are modular forms $f_n$ with integral coefficients such that $f \equiv f_n \mod p^n$ (as $q$-expansions). Note it does not really make sense to attribute 'a weight' to $f$ since the $f_n$ are allowed to have different (increasing weights). This is the older definition by Serre.

I know (but I do not understand) a newer definition by Katz, which has a more geometric flavor. See here math.arizona.edu/~swc/notes/files/01BuzzardL2.pdf.

So we have an approach using the $q$-expansion and we have an approach to $p$-adic modular forms using geometric ideas.

My question now is, whether there is also a developed theory on analytic p-adic modular forms?

Some ideas what this might mean. For example we could consider the Eisenstein series $$E_4(\tau)=\sum_{n,m \in \mathbb{Z}} \frac{1}{(n\tau+m)^4}$$ as function of $\tau$ not being an element of the upper half-plane but of some subset of $\mathbb{C}_p$.

Does this sum even converge somewhere in $\mathbb{C}_p$. And is it (up to a constant) a classical or Katz $p$-adic modular form? Does it even equal (mod p) the Eisenstein series $E_4$?

A similar question could be posed for the infinite product usually defining the $\Delta$-function $$q\prod_{n=1}^\infty (1-q^n). 1-q^n)^{24}.$$ Does this converge somewhere when $q$ is in some subset of $\mathbb{C}_p$. Is it a $p$-adic modular form?

If there is no such theory? Why not? Is it not interesting?

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# Are there 'analytic' $p$-adic modular forms.

The most elementary way to define $p$-adic modular forms is via limits of classical modular forms. More precisely $f \in \mathbb{Z}_p[[q]]$ is called a $p$-adic modular form if there are modular forms $f_n$ with integral coefficients such that $f \equiv f_n \mod p^n$ (as $q$-expansions). Note it does not really make sense to attribute 'a weight' to $f$ since the $f_n$ are allowed to have different (increasing weights). This is the older definition by Serre.

I know (but I do not understand) a newer definition by Katz, which has a more geometric flavor. See here math.arizona.edu/~swc/notes/files/01BuzzardL2.pdf.

So we have an approach using the $q$-expansion and we have an approach to $p$-adic modular forms using geometric ideas.

My question now is, whether there is also a developed theory on analytic p-adic modular forms?

Some ideas what this might mean. For example we could consider the Eisenstein series $$E_4(\tau)=\sum_{n,m \in \mathbb{Z}} \frac{1}{(n\tau+m)^4}$$ as function of $\tau$ not being an element of the upper half-plane but of some subset of $\mathbb{C}_p$.

Does this sum even converge somewhere in $\mathbb{C}_p$. And is it (up to a constant) a classical or Katz $p$-adic modular form? Does it even equal (mod p) the Eisenstein series $E_4$?

A similar question could be posed for the infinite product usually defining the $\Delta$-function $$q\prod_{n=1}^\infty (1-q^n).$$ Does this converge somewhere when $q$ is in some subset of $\mathbb{C}_p$. Is it a $p$-adic modular form?

If there is no such theory? Why not? Is it not interesting?