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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 $$ q\prod_{n=1}^\infty (1-q^n)^{24}. $$ 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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Because of convergence issues, I don't think one can make sense of the notion that a $p$-adic modular form is a function on $\mathbf{C}_p$. But here's something one can do: a $p$-adic modular form still has a $q$-expansion which converges for $|q|<1$, because $|q|<1$ is a (rather small) region on the modular curve where these things are defined, and so you could think of a $p$-adic modular form as a function on this disc defined by a power series. But the hard part is saying what "transformation law" it has to satisfy, because "every elliptic curve over the complexes has a $q$, but..." –  Kevin Buzzard May 24 '11 at 19:31
    
..."over the p-adics most elliptic curves don't; only the ones with good reduction do". So $q$ is not the full story (although of course by analytic continuation properties, the power series determines the form). –  Kevin Buzzard May 24 '11 at 19:32
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Here's a more sensible and I hope comprehensible way of thinking about it. A modular form isn't really a function on the upper half plane! It's a function on elliptic curves, or pairs consisting of an elliptic curve and a differential. This definition works well $p$-adically and over the complexes. Now over the complexes every elliptic curve is isomorphic to $\mathbf{C}/\langle 1,\tau\rangle$ for some $\tau$ in the upper half plane. But over the $p$-adics this is not true; the curves that can be uniformised are those with bad reduction. So for a classical modular form... –  Kevin Buzzard May 24 '11 at 20:04
    
...you get a function on $\tau$'s obeying the usual relations -- but in the $p$-adic world you need more than this so the theory is perhaps a bit deeper. –  Kevin Buzzard May 24 '11 at 20:05
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$E_4$ converges nowhere ,so it is completely boring. Proof: for any fixed $\tau$, choose $n$ a big enough power of $p$ so that $n$ times anything near $\tau$ is small. You get infinitely many summands with $m$ congruent to $1$ mod $p$. $\Delta$ converges in the $p$-adic unit disc, since it is the $q$-expansion of a modular form. –  S. Carnahan May 24 '11 at 20:58

3 Answers 3

up vote 8 down vote accepted

There is such a theory, but the analytic object that the forms live on is an analytified modular curve, not simply $\mathbb{C}_p$ (though there is a "$p$-adic upper-half plane" that can be used to uniformize some similar moduli spaces, but as far as I know not the usual modular curves).

Basically, if $f$ is a classical modular form of some weight $k$, $f$ can be realized as a section of a sheaf $\omega^{\otimes k}$ on a complex-analytic modular curve such as $X_1(N)$ obtained via quotient from the complex upper-half plane. These curves and sheaves have algebraic models defined over $\mathbb{Q}$, and (under mild hypotheses) the form $f$ actually arises from a section of the associated sheaf $\omega^{\otimes k}$ on the modular curve defined over $\mathbb{Q}$ (or some finite extension).

Now you can go in another direction and consider the rigid-analytic space over $\mathbb{Q}_p$ associated to the smooth algebraic curve $X_1(N)$ and your form $f$ gives rise to a $p$-adic analytic object on this curve. Now one can play games like considering subspaces obtained by removing disks around supersingular points to obtain general $p$-adic modular forms (such as limits of classical forms of varying weight) and overconvergent modular forms.

For what it's worth, these curves are $p$-adic analytic moduli spaces, and this point of view on $p$-adic modular forms essentially differs from Katz's by thinking about rigid spaces as opposed to formal schemes (the Raynaud point of view on $p$-adic analytic geometry).

Look at the papers of Coleman (such as his $p$-adic Banach spaces paper or the eigencurve paper with Mazur) if you would like to read more on this point of view.

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The $p$-adic upper half plane (over $\mathbb{C}_p$, say) does analytically uniformize some usual modular curves, e.g. $X_0(p)$. But the point here is that it only uniformizes curves which have totally degenerate multiplicative bad reduction, so you are missing out on a lot compared to the classical case. (Probably you knew this already: we did talk about some math in between all those beers...) –  Pete L. Clark May 24 '11 at 20:08
    
(Unfortunately I never asked you about eigencurve stuff -- I should have.) –  Pete L. Clark May 24 '11 at 20:08

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 $\mathbb{C}$.

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You're right. That paragraph is wrong and I will delete it. –  David Loeffler May 24 '11 at 19:37

Since I'm not an expert, this will be just a minor note in addition to the excellent answers already given.

The example modular forms $E_4$ and $\Delta$ you've written down are defined over the integers (after suitable normalization). That is, they eat elliptic curves with nowhere-vanishing differentials over any base, and produce numbers (i.e., functions on the base) in a suitably canonical way. In particular, they are already "classical" $p$-adic modular forms by straightforward base change, and they are analytic forms, because they can be canonically extended to rigid-analytic input data.

The problem with the definition of $E_4$ that you wrote is that it uses the additive uniformization of elliptic curves ($\mathbb{C}/\Lambda \overset{\simeq}{\longrightarrow} E$) in an essential way, and elliptic curves only have an additive uniformization over the complex numbers. However, the $q$-expansion $E_4(q) = 1 + 240 \sum_{n \geq 1} \sigma_3(n) q^n$ makes sense $p$-adic analytically, since the multiplicative uniformization $\mathbb{G}_m/q^{\mathbb{Z}} \overset{\simeq}{\longrightarrow} E$ works in both analytic worlds.

Random points:

  1. Katz's definition of modular form (see his paper in Springer Lecture Notes 350, scanned on his web page) can be straightforwardly modified to take analytic inputs (real, complex, or $p$-adic).
  2. When people say "$p$-adic modular form", they might mean only non-classical forms, i.e., those that aren't defined in the "too supersingular" locus. The overconvergent forms, namely those that extend from the ordinary locus to the "not too supersingular" locus, are necessarily analytic in nature.
  3. These objects are interesting in part because they can be used to construct interesting Galois representations through some magic I don't understand. I should say something clever using phrases like "$p$-adic Langlands" and "eigenvariety" here.
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