It occurs more and more that I ask a question on math stackexchange and then realize that it is more appropriate to mathoverflow. Hopefully this reflects well on myself... In any case, I copy here word for word my question from stackexchange, which received no answers, and I will erase it from there.

In a previous question of mine from a long time ago Matt E gave an excellent exposition to Langlands which has benefited me immensely. Now that I have matured a little bit, I wish to know certain things in more detail.

In the answer to my previous question, Matt E detailed how newforms are related to motives. Newforms are classifies as either classical Atkin Lehner newforms of some weight $k$; and Maass forms of weight $\lambda=\frac14$. Classical Atkin Lehner newforms of weight $k$ should correspond to motives with degree of purity $k-1$, and Maass forms of weight $\lambda=\frac14$ should correspond to degree of purity $0$ motives. (So degree of purity $0$ motives are those that come from classical newforms of weight $1$ and Maass forms.)

A classical Atkin Lehner newform is a cuspform in some $\Gamma_0(N)$ for some $N$, which is a simultaneous eigenvalue of all Hecke operators, such that $N$ is minimal with respect to the associated system of eigenvalues.

My question is: what, if any, is the role played by other congruence and non-congruence subgroups other than $\Gamma_0(N)$? Do they fit into the Langlands philosophy in some way?

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    $\begingroup$ Congruence subgroups other than $\Gamma_0(N)$ arise naturally in the theory of automorphic forms; in fact any congruence subgroup arises naturally. However in the "classical" Langlands picture, non-congruence subgroups do not arise, because the automorphic forms giving rise to an automorphic representation are all locally constant and hence constant on some open adelic subgroup, which is by definition defined using congruence conditions. There is a story involving non-congruence subgroups, but it is somehow another layer of complexity on top of the "usual" Langlands programme. $\endgroup$ Nov 25 '11 at 0:41
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    $\begingroup$ For representations attached to non-congruence subgroups, see the papers of Tony Scholl that you can find here: dpmms.cam.ac.uk/~ajs1005/#publications $\endgroup$ Nov 25 '11 at 1:13
  • $\begingroup$ Kevin, is it possible for me to convince you to put some reference for that story? $\endgroup$
    – Nicole
    Nov 25 '11 at 2:29
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    $\begingroup$ I was thinking of the work of Scholl mentioned above. I think that the basic idea is that a non-congruence subgroup still gives you a compact Riemann surface which I think is still defined over a number field, and inside the etale cohomology you can still see Galois representations. However they are e.g. no longer 2-dimensional and things are much murkier. $\endgroup$ Nov 25 '11 at 7:02

An automorphic representation $\pi$ in an adelic pircture has an Euler factrozation into $\otimes_v \pi_v$ for all places $v$.

Now at a finite place $v$, the restriction of $\pi_v$ to $GL(2, o_v)$ contains a subrepresentation of $Ind^{GL_2(o)}_{\Gamma_0(p^N)} 1$ for some $N$.

(Actually, more is true, i.e. it differs from $Ind_{B(o)}^{GL_2(o)} 1$ by a finite dimension.)

New forms are in this language the vectors, which occur for some for $p^N$. They will occur for all $p^M$ as well with $M>N$.

This is why it it sufficient to study $\Gamma_0(N)$, which is just the classical analog of the Atkin-Lehner theory.

To study other congruence subgroup seems reasonable and is actually finer. There is a theory of types, which was invented to describe the supercuspidals representations at the finite places. This is one ingredient, if not the ingredient, in the local Langlands correspondance.

For compact non congruence subgroups, Jacquet Langlands is one example.

For a general non compact non congruence subgroups in $SL(2,\mathbb{R})$, one does not even know the existence of cuspforms. But $SL(2,\mathbb{R})$ has a special lattice structure anyway, in the sense that Margulis arithmeticity theorem does not hold.

For higher rank groups, all lattices are commensurable with congruence lattices, but I guess, there are not congruence themself(?)

There is also Beyli's theorem, where the lattices are in general non congruence, but finite index in $SL(2, \mathbb{Z})$. But there does not exist a connection to the Langlands program, as far as I know.


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