Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

My answer to the question:

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?

share|improve this question
    
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
1  
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

1 Answer 1

Perhaps I'm not understanding what you're asking, but there's a paper of Scholl's (Modular Forms and de Rham Cohomology) where he proves a version of the ASD congruences using some relatively sophisticated machinery.

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.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.