A p-adic modular form is a limit of modular forms under a topology that encodes congruences between its Fourier coefficients.

Definition

The original definition is due to Serre. Consider the space of modular forms over $\mathbb{Q}$, and recall that they are fully determined by their q-expansions, therefore we may write them as power series$f=\sum_{n=0}^{\infty}{a_{n}}q^{n}\in\mathbb{Q}[[q]]$. Since $\mathbb{Q}$ embeds into the p-adic numbers$\mathbb{Q}_{p}$, we may consider these modular forms as elements of $\mathbb{Q}_{p}[[q]]$ as well. We define a valuation$v_{p}$ on this space by

$v_{p}(f):=\inf_{n}( v_{p}(a_{n}))$

where the $v_{p}$ on the right-hand side is the p-adic valuation on $\mathbb{Q}_{p}$ with $v_{p}(p)=1$.

A p-adic modular form is a power series $f=\sum_{n=0}^{\infty}{a_{n}}q^{n}\in\mathbb{Q}_{p}[[q]]$ such that there exists a sequence of modular forms $f_{1},f_{2},\ldots$ with $f_{i}\to f$.

Another different definition was later formulated by Katz.

Let $X$ be a modular curve and let $X^{\rig}$ be its associated rigid analytic space. The p-adic modular forms of weight $k$ are the global sections $H^{0}(X^{\rig}[0],\omega^{k})$, where $X^{\rig}[0]$ denotes the ordinary locus of $X^{\rig}$.