Persiflage

I recently wrote a paper (with Toby Gee and George Boxer, see also here) on constructing regular algebraic automorphic representations \(\pi\) of (cohomological) weight zero and level one, and therefore also cuspidal cohomology classes in the cohomology of \(\mathrm{GL}_n(\mathbf{Z})\) for some values of \(n\).

There was one slightly subtle point which we had to address concerning the relation between the cohomology of \(\mathrm{SL}_n(\mathbf{Z})\) and \(\mathrm{GL}_n(\mathbf{Z})\), or at least the relationship between the parts of cohomology which come from cuspidal modular forms. I have observed this issue turn up in some different contexts, and that is what I wanted to talk about today. The main message is that from the perspective of the Langlands program, the cohomology of \(\mathrm{GL}_n(\mathcal{O}_F)\) is more fundamental than tbe cohomology of \(\mathrm{SL}_n(\mathcal{O}_F)\). When \(F = \mathbf{Q}\) these groups are “more or less” the same (more on that below), but the differences are more pronounced and significant when \(F \ne \mathbf{Q}\). But let’s start by talking about the case of classical modular forms, where there is already something a little bit interesting to say. A regular algebraic automorphic representation \(\pi\) for \(\mathrm{GL}(2)/\mathbf{Q}\) of level one corresponds to a cuspidal modular eigenform of weight \(k \ge 2\) and level one. We know that cuspidal modular forms of weight \(k \ge 2\) and level one contribute via Eichler-Shimura to the Betti cohomology groups of the modular curve. As an orbifold, the modular curve can be realized as \(\mathbf{H}/\Gamma\) where now \(\Gamma = \mathrm{SL}_2(\mathbf{Z})\) rather than \(\mathrm{GL}_2(\mathbf{Z})\). In this situation at least, we understand quite well what is happening. These eigenforms give rise to a two-dimensional space inside \(H^1\) of the modular curve, and thus inside \(H^1(\Gamma)\), and we understand what the “extra” action of the element
\[ \displaystyle{ \left( \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right) } \]
is; namely under the Eichler-Shimura isomorphism, it corresponds to the action of complex conjugation (so from the perspective of the Hodge filtration, it takes the holomorphic forms to the antiholomorphic forms and vice-versa). It acts on the relevant piece of cohomology with trace zero. Note that this no longer holds on non-cuspidal cohomology, for example \(H^0\) is one dimensional in both cases. Of course in cohomological weight zero (which corresponds to weight \(k = 2\)), there turn out to be no such forms, but the point is that the vanishing of the cuspidal cohomology for \(\mathrm{GL}_2(\mathbf{Z})\) is equivalent to the same statement for \(\mathrm{SL}_2(\mathbf{Z})\). (Something similar is also true in higher weight as well when there really do exist such forms.)

For larger \(n\) there is a similar equivalence; but now the behavior depends on the parity of \(n\). For \(n\) odd, the cohomology of \(\mathrm{GL}_n(\mathbf{Z})\) and \(\mathrm{SL}_n(\mathbf{Z})\) is (rationally) the same because \(\mathrm{GL}_n(\mathbf{Z}) \simeq \mathrm{SL}_n(\mathbf{Z}) \times \mathbf{Z}/2 \mathbf{Z}\) (then by the Künneth formula). But for \(n\) even, a level one weight zero \(\pi\) gives rise to two copies of the exterior algebra

\[ \bigwedge^* \mathbf{C}^{\ell_0} \]

in degrees \([q_0,\ldots,q_0 + \ell_0]\), with \(\ell_0 = (n-2)/2\), and the action of the “extra” element acts freely on these two copies. All this comes down to the differences in the real representation theory of \(\mathrm{GL}_n(\mathbf{R})\) and \(\mathrm{GL}_n(\mathbf{R})^{+}\) which is discussed briefly in the paper but which I won’t talk about here.

But what happens for general number fields \(F\)? There’s a confusion which I have seen in various places even for \(n=2\) about whether one should be considering the cohomology of \(\mathrm{SL}_n(\mathcal{O}_F)\) or \(\mathrm{GL}_n(\mathcal{O}_F)\). Of course it depends on what exactly one wants to do. But at least if one is interested in computing automorphic representations conjecturally associated to motives which have level one, one should really be considering the cohomology of \(\mathrm{GL}_2(\mathcal{O}_F)\). This confusion comes with good pedigree — It turns up in the Serre-Tate correspondence! Tate mentions (October 1969, page 382) a colloquium by Swan who “disappointed everybody” by computing that \(H_1(\mathrm{SL}_2(\mathbf{Z}[\sqrt{-14}]),\mathbf{Z})\) has rank three, compared to the lower bound (coming from the boundary tori) of two. (Side remark: Tate notes in a later letter [Nov 15] it should be \(\sqrt{-10}\), not \(\sqrt{-14}\).) Serre responds (October 15, page 384) that he doesn’t find this at all surprising, and in fact:

(via la théorie de Weil cela signifait qu’il existe de courbes elliptiques sur le corps en question qui n’ont pas de multiplication complexe — on n’en doute pas). En fait, vu Weil, il s’impose d’essayer de construire une courbe elliptique sur \(\mathbf{Q}(\sqrt{-56})\) ayant bonne réduction partout;

Now I confess that when I first read this quote I interpreted it as a misapprehension on Serre’s part, because (since this is \(\mathrm{SL}_2\) not \(\mathrm{GL}_2\)) there need not exist any such elliptic curve. But looking it up again now, I started to have my doubts, and Serre was perhaps more circumspect than I had assumed. Indeed chatgpt tells me:

The phrase “il s’impose d’essayer” in French does not have the same strict sense of necessity as “it is necessary” in English. A more nuanced translation could be “it is imperative to try” or “it is important to try.” It suggests a strong recommendation or importance, rather than an absolute necessity.

(Possibly Colmez can confirm this; AI has rendered his go playing superfluous but not yet his skills interpreting for anglophones the nuances of Serre’s words.) That’s also consistent with how Serre continues:

je connais trop mal la théorie de Weil pour être sûr que ça doit exister; mais il vaut la peine d’essayer

Later (note the remark on \(d=-56\) versus \(d=-40\) abpve), Serre says:

C’est bien \(\mathbf{Q}(\sqrt{-40})\) le corps où Mennicke a trouve que le rang de \(\mathrm{SL}_2\) rendu abélien est nombre de classes. Mais il a un corps encore plus beau: \(\mathbf{Q}(\sqrt{-109})\) où le \(\mathrm{GL}_2\) rendu abélien est infini (c’est une propriété plus forte S1 que la précédente). Ici aussi, on a envie de chercher des courbes elliptiques à bonne réduction.

Perhaps worth adding the modern footnote as well:

«via la théorie de Weil cela signifiait que…» je m’avançais beaucoup en disant ça (I was talking through my hat).

Of course, 45 years later things have been clarified, at least conjecturally. (We still have no general way to produce motives from cohomology, even for Hilbert modular forms of parallel weight \(2\).) One perspective which I think is helpful (at least to those who care more about Galois representations) is thinking about the differences between the Galois representations associated to automorphic forms on \(\mathrm{SL}_n\) versus \(\mathrm{GL}_n\). Given a \(\pi\) for the former (say cuspidal algebraic of weight zero and level one), you should think about this as giving a compatible family of projective representations:
\[\rho(\pi): G_F \rightarrow \mathrm{PGL}_n(\overline{\mathbf{Q}}_p)\]
which are absolutely irreducible and crystalline of the expected weights and unramified outside \(v|p\). Now in this situation,one knows (following for example Patrikis) that there exists for any such \(\rho\) a lift to a genuine representation of \(G_F\) which is crystalline at \(v|p\) of the right weight for all \(v|p\) — this generally requires some parity condition on the weight but we are assuming that here. What is not automatic, however, is that this lift has level \(N=1\) any more; that is, the image of inertia at other primes \(v\) may be non-trivial (though of course the image lies in the center). Here there is something special which happens only for \(F = \mathbf{Q}\); as observed by Tate, you can globalize these local characters and then twist to eliminate all the auxiliary ramification. (This argument is explained by Serre in his 1975 Durham paper which is always impossible to find online; it is used to show that a complex
projective representation can be lifted to an Artin representation ramified at the same set of primes.) For other fields, even if the class number is trivial, you get global obstructions coming (via class field theory) from the unit group. (Even for imaginary quadratic fields, where the unit group is not very big, this is still an issue, and the general problem can only be avoided for fields for which the unit group has order \(2\) and which have a real place, which is quite a restrictive condition when you think about it.) The direct automorphic argument is ultimately quite similar, but there are some traps waiting for the unwary (related to Grunwald-Wang); see the discussion in this paper.

So for example, it is true that as \(F\) ranges over all imaginary quadratic fields, one has
\[H^1_{\mathrm{cusp}}(\mathrm{SL}_2(\mathcal{O}_F),\mathbf{C}) \ne 0\]
for all but finitely many \(F\). But the analogue for \(\mathrm{GL}_2(\mathcal{O}_F)\) is not only unknown, but
we certainly have:

Conjecture: There are infinitely many imaginary quadratic fields \(F\) with
\[H^1_{\mathrm{cusp}}(\mathrm{GL}_2(\mathcal{O}_F),\mathbf{C}) = 0.\]

By the way, from the perspective of Galois representations, one can see why the group above should be non-zero in the case of \(\mathrm{SL}_2(\mathcal{O}_F)\). Let \(F = \mathbf{Q}(\sqrt{-D})\). All we need to find are modular forms \(\pi\) of weight two with the property that, locally at primes \(p|D\), the corresponding Weil-(Deligne) representation on restriction to inertia becomes trivial after restriction to \(\mathbf{Q}_p(\sqrt{-D})\) up to twist. One easy way to achieve this is to take ramified principal series \(\mathrm{PS}(1,\chi)\) for some (local) ramified quadratic character \(\chi\). The problem is this leads (globally) to a sign difficulty; if \(F\) has prime discriminant, then globally you would want the weight of \(\pi\) to be two and the Nebentypus character to be the quadratic character of conductor \(\Delta_F\) which is odd, which is a problem. (Sometimes it is not; if \(F = \mathbf{Q}(\sqrt{-p})\) and \(p \equiv 1 \bmod 4\) then you can take the real character of conductor \(p\), but if \(p \equiv -1 \bmod 4\) this doesn’t work.) But instead of principal series, one can take certain supercuspidal representations: Assume that \(F_p/\mathbf{Q}_p\) is a ramified quadratic extension. Then if \(\chi\) is a totally ramified character of \(F^{\times}_p\) of order \(2^m\) where \(2^{m} \| p-1\), then the base change of this supercuspidal representation will be unramified up to twist, but the original representation will not be unramified up to twist. It’s now easy to construct such forms (and even compute how many of them there are), and see there are plenty of them when the discriminant of \(\Delta_K\) gets large (one has to avoid CM forms over \(K\) which can become non-cuspidal but these are easy to bound.) It’s also easy to see that while these base changes are unramified at every place up to a local twist they are not in general unramified everywhere up to a global twist.

The forms one finds in this way by base change are invariant under complex conjugation (now acting on the group), and there is another “geometric” way to show they exist which was originally done by Rohlfs (see this paper), who I believe was the first person to prove the non-vanishing claim above. (In fact, this is one way to start proving base change in this situation.)

When it comes to general number fields, one certainly expects (by functoriality!) that \(H^*_{\mathrm{cusp}}(\mathrm{GL}_n(\mathcal{O}_F),\mathbf{C})\) should be non-zero for \(n=79\) say and every number field \(F\), but this is hopeless for almost all fields. Using our arguments (and Newton-Thorne for totally real fields!) One certainly can prove it for many totally real and CM fields (some ramification conditions are required for the arguments to work) using the exact same argument. Of course, when for such fields there exists a cuspidal Hilbert modular form of weight two and level one then you can just used Newton-Thorne directly! For general fields, as usual, the problem of understanding automorphic forms eludes us.

Curiously enough, while writing this post, there appeared a very recent preprint by Darshan and Raghuram here which constructs, for example, cuspidal cohomology classes for \(\mathrm{GL}_n/F\) of (for example) cohomological weight zero for any number field \(F\) which is Galois over a totally real field \(F^{+}\) of some deep enough level, but with no further assumptions on \(F\). Clozel (in this paper) did something similar when \(n\) is even by automorphic induction, but already for \(n=3\) this no longer works. Assuming all conjectures, the simplest way to construct such forms for \(F = \mathbf{Q}\) or any totally real field is to take symmetric squares of Hilbert modular forms (these more or less constitute all the self-dual forms). It seems to me that the forms found by Darshan and Raghuram must be some shadow of these forms over the largest totally real subfield \(F^{+}\) of \(F\) and so one is seeing a hint of non-cyclic base change here which is intriguing! I hope to return to this later when I understand it better.