# Are there $p$-adic modular forms for non-congruence subgroups?

moduli interpretations for modular curves

led me to wonder about the question in the present title.

It seems that modular forms for non-congruence subgroups may not "arise" from algebraic geometry in the same way modular forms for congruence subgroups do (as is evidenced by the lack of a good moduli problem and this whole unbounded denominators thing). Nonetheless, they are clearly complex-analytic objects. I wonder if there's a $p$-adic analytic analog to be had.

Here are two more precise questions:

1. (less ambitious) Are there interesting congruences to be had between such modular forms? Interesting $p$-adic limits? Interesting $p$-adic families?

2. (more ambitious) Is there some (perhaps inherently analytic) description of non-congruence forms in terms of moduli of elliptic curves that can be mimicked in the $p$-adic analytic setting?

My rather vague feeling is that the Atkin and Swinnerton-Dyer congruences suggest that there's something to be said here, but I haven't been able to dig up much on these questions in particular. Does anyone know of such work?

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The theory of division algebras is parallel to that of congruence subgroups. The division algebra stuff gives you compact Riemann surfaces, coming from non-congruence, cocompact subgroups. Although I do not know the definition of a $p$-adic modular forms, these should have counterparts. – Marc Palm Aug 13 '12 at 11:17
Do you know about rigid analytic modular forms and are you dissatisfied with them? – Rob Harron Aug 13 '12 at 17:30
I do not know about rigid-analytic modular forms for non-congruence subgroups. That's precisely the sort of thing I'm asking for. All such forms that I'm aware of live on rigid-analytic modular curves that parameterize elliptic curves with some extra structure. All this is unique to congruence subgroups as far as I know. – Ramsey Aug 13 '12 at 18:06
I don't really know much about these things, but there are non-congruence discrete subgroups of $\mathrm{SL}_2(\mathbf{Q}_p)$ (right?) and hence there are rigid analytic modular forms with respect to these subgroups. Am I missing something? – Rob Harron Aug 13 '12 at 21:53
Here's one approach: noncongruence modular curves have models over number fields, which map to $X(1)$, so I guess you can take $\mathbb{C}_p$ points and look at the preimage of the ordinary / $r$-overconvergent locus of $X(1)$, and define p-adic modular forms a la Katz / Coleman as the sections of sheaves over the ordinary or overconvergent loci. Then you certainly get a space of p-adic objects that includes the classical ones in a natural way. – David Loeffler Aug 28 '12 at 9:21

Specifically, if $\Gamma$ is any finite index subgroup of $\text{SL}_2(\mathbb{Z})$, and suppose $k > 2$ is even, then if $d = \text{dim }S_k(\Gamma)$, then there exists integers $A_0,\ldots,A_{2d}$ such that $A_0 = 1$, $A_{2d} = p^{(k-1)d}$, for which we have the congruence:
$$a_{np^d}(f) + A_1a_{np^{d-1}}(f) + \cdots + A_da_n(f) + A_{d+1}a_{n/p}(f) + \cdots + A_{2d}a_{n/p^d}(f)\equiv 0\mod p^{(k-1)(1+\text{ord}_p n)}$$
If $d = 1$, then this is exactly the ASD congruence. The $A_i$'s are actually the coefficients of the characteristic polynomial of Frobenius acting on a certain $2d$-dimensional $\ell$-adic sheaf.