I recently returned home from a trip to Paris for Clozel’s 70th birthday conference. Naturally I stayed in an airbnb downtown, and the RER B gods smiled on me with a hassle free commute for the entire week. Tekés was an interesting find, a fun (and surprisingly cheap) Israeli vegetarian restaurant right near where I was staying. But surely the food highlight of the week was the lunch spreads during the conference at Orsay — certainly the best conference food I’ve ever had! Great vegetarian food with amazing eggplant dishes, feta, figs, all the good stuff. (Rumor was it was chosen by Valentin Hernandez and paid for by Vincent Pilloni; I’m not sure if that’s true but a great job all round.)
There were quite a number of interesting talks, although as mentioned before I don’t like singling out because that sometimes seems like an implicit criticism of the other talks. But a few thoughts spurred by some of the talks (which you can more or less guess if you wanted to), some of which were already raised by others in conversation during the conference:
(Global) modules for Galois deformation rings. As a result of Taylor-Wiles patching, one usually constructs a CM-module \(M_{\infty}\) defined over some local deformation ring \(R\). Often quite a bit of mileage can be gained by exploiting the ring theoretic structure of \(R\) to conclude something about the module \(M_{\infty}\) and vice versa. Perhaps the ur-version of this argument is Diamond’s argument showing (in some circumstances) that the formal smoothness of \(R\) implies that \(M_{\infty}\) is free. A more recent example is in the work of Jeff Manning where he exploits the geometry of some particular \(R\) (by relating to a more geometric situation where one can perhaps understand the Picard group) to restrict the possible \(M_{\infty}\) to a very small number of possibilities from which one can then get some mileage. But one question raised is the extent to which this one can always do this. As one considers more and more complicated \(R\), is there some constraint on possible \(R\) which means that there are only going to ever exist a small number of faithful maximal rank one CM-modules \(M\), or are there going to be situations where \(R\) is very complicated and one can’t hope classify all such \(M\), but only (for some mysterious situation) a very small number of them turn up in global situations. Note that globally there is often a few possibilities of the type of cohomology one patches, and even for \(\mathrm{GL}(2)\) you can be in situations where you can force \(M\) to be free or self-dual by working in coherent cohomology or etale cohomology respectively and these modules are not always the same.
The work of Arthur: (Some) experts are at the point where they no longer expect Arthur to publish proofs of results he has claimed, leaving a huge gap in the literature. The summary seems to be that many very smart people are putting lots of effort into filling in some of these details, and that this seems to require new arguments. For example, it seems to be the case that one of Arthur’s proposed inductive arguments will not (at least naively) work. The mathematical community should be immensely grateful to people working on this!
Shimura Varieties: I once joked that today’s generation is more likely to learn about Galois representations associated to elliptic curves and modular curves before learning any class field theory. Well that generation has passed! We may be approaching a moment where people learn about Shimura varieties without ever thinking about modular curves, let alone getting close and personal with \(X_0(11)\). (Note: I don’t think that RLT knows the genus of \(X_0(11)\) and that never stopped him, of course.) Some people can look at the abstract definition of a Shimura variety and then start proving things; I am certainly not one of those people. Fortunately there are still many interesting open questions even about classical modular forms.
A result of Garland (at least) two talks reminded me of a vanishing result of Garland. Suppose that \(\Gamma\) is (for example) a lattice in \(\mathrm{SL}_n(\mathbf{Q}_p)\). Then the cohomology (with coefficients in \(\mathbf{Q}\)) vanish in positive degrees below \((n-1)\). But I think that much more should be true, namely, that the cohomology should all have a “trivial” Hecke action in the expected ways, i.e. the completed cohomology groups should all be finite in this range, as they are for \(\mathrm{SL}_n(\mathbf{Z})\) (more or less, let’s not be precise about ranges and what eactly is known). It feels like conjectures of this sort are not completely out of reach. Is this too optimistic? This is already an interesting problem in the case of \(H^2(\Gamma,\mathbf{F}_p)\).