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While reading "Automorphic Forms and L-functions for the Group $GL(n,R)$" by D. Goldfeld, I've got a feeling that linear groups over $\mathbb{R}$ and $\mathbb{Z}$ are considered only as technical tools (beside their presence in the definitions). The author often uses various decompositions such as Iwasawa and Bruhat, Levi and Langlands decomposition of parabolic subgroups, Hermite and Smith normal forms and other straightforward matrix calculations. But I don't see if this works the other way around. The question is:

What can the theory of automorphic forms for $SL(n,\mathbb{Z})$ say about the structure of $SL(n,\mathbb{Z})$?

The question is motivated by the fact that from the viewpoint of structure theory of algebraic groups $SL_n,~n\geqslant3$ behaves much better than $SL_2$, while modular forms for $SL_2$ seem to be studied more that those for groups of higher ranks. So let me formulate another version of the above question (not sure if it is less vague):

In what aspects, if any, automorphic forms for $SL(n,\mathbb{Z}),~n\geqslant3$ behave better than those for $SL(2,\mathbb{Z})$?

I apologize, if all of the above is trivial (or nonsense), but I am as far from being an expert in modular forms as possible.

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  • $\begingroup$ For one example, Higher rank groups are Kazhdan groups, hence immediately you have the so-called $\tau$ property for congruence towers... $\endgroup$
    – Asaf
    Commented Mar 16, 2014 at 12:34
  • $\begingroup$ And you should discriminate between modular forms and automorphic forms, as $SLn$ for $n\geq 3$ does not have discrete series, by Harish-Chandra's theorem. That is also reflected in the alg. group structure, not every maximal torus is Cartan in $SL3$. $\endgroup$
    – Asaf
    Commented Mar 16, 2014 at 12:42
  • $\begingroup$ @Asaf: the $\tau$ property for congruence towers also works for $SL_2$ by work of Brooks, so unlike property T, does not distinguish $SL_2$ from higher $SL_n$. Also it's not obtained, as far as I know, from automorphic forms $\endgroup$
    – YCor
    Commented Mar 16, 2014 at 12:42
  • $\begingroup$ @Yves, OK, I agree, the thing is that the proof of the $\tau$ property for $SL2$ would require a-lot of arithmetic machinery which is of-course in the nature of $SL2(Z)$, where in the higher-rank case, it follows directly from representation theory in general. I agree that if one insists here on $SL2(Z)$ the comment is probably not clarifying the point. I vaguely recall that Lubotzky once mentioned to me about arithmetic lattice in $SL2$ which does not have $\tau$... Moreover, $\tau$ is reflected by automorphic forms as mixing rates in the standard representation... $\endgroup$
    – Asaf
    Commented Mar 16, 2014 at 12:50
  • $\begingroup$ @Asaf: you mean, an arithmetic lattice in $SL_2(R)$ not $(\tau)$ w.r.t. congruence subgroups? $\endgroup$
    – YCor
    Commented Mar 16, 2014 at 12:58

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I don't exactly see how the second question is another version of the first, so let me answer them separately (and each very partially):

In what aspects, if any, automorphic forms for $SL(n,\mathbb Z)$, $n⩾3$ behave better than those for $SL(2,\mathbb Z)$?

I would say "in none" if you take the question strictly, but if, as is natural, you are interesting in modular/automorphic forms for finite index subgroups of $SL(n,\mathbb Z)$ as well, then an aspect of the theory that becomes much simpler when $n \geq 3$ is that in this case, all those finite index subgroups are congruence subgroups by a theorem of Bass, Milnor and Serre. In the case $n=2$, this is not true, and modular forms for non-congruences subgroups are much less well-understood that modular forms for congruences subgroups. They don't have a large set of Hecke operators acting on them for instance.

What can the theory of automorphic forms for $SL(n,\mathbb Z)$ say about the structure of $SL(n,\mathbb Z)$?

If you interpret "structure" largely, as "algebraic properties, then the main thing that the theory of automorphic forms on $SL(n,\mathbb Z)$ can teach you is about the size and structure of cohomology groups $H^*(SL(n,\mathbb Z),A)$. Here the coefficients $A$ can be constant (A=$\mathbb Q$, say, or more generally an algebraic representation of $SL_n(\mathbb Z)$). In fact standard theorems provides a decomposition of those cohomology groups, which are very subtle invariants, in terms of automorphic forms. You can translate information obtained on automorphic forms by different methods (e.g. trace formula, or their relation with Galois representations and arithmetic) to deduce things that can not be obtained otherwise on these cohomology groups.

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    $\begingroup$ There is no Eichler-Selberg type trace formula for SL(3) because of the abscence of discrete series for SL(3,R). $\endgroup$
    – Marc Palm
    Commented Mar 16, 2014 at 22:41

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