Report from Luminy

For how long has Luminy been infested with bloodthirsty mosquitoes? The combination of mosquitoes in my room with the fact that my bed was 6′ long with a completely unnecessary headboard (which meant that I had to sleep on an angle with my ankles exposed) did not end well.

As for the math, there were plenty of interesting talks, most of which I will not discuss here. Jan Nekovar gave a nice talk explaining how one could prove that the cohomology of compact Shimura varieties of \(\mathrm{GL}(2)\)-type were semi-simple. For concreteness, imagine that \(X\) is a Hilbert modular surface associated to a real quadratic field \(F\). Suppose that \(\rho\) is the Galois representation associated to a cuspidal Hilbert modular form of parallel weight two. Then the Langlands-Kottwitz method shows that the semi-simplification of \(\rho^{\otimes 2}\) should occur inside \(H^2\). On the other hand, this argument only ever deals with the trace of Hecke operators and so cannot say anything about semi-simplifications. Nekovar’s argument is to use the Eichler-Shimura relation applied to partial Hecke operators for primes which split completely in both \(F\) and the corresponding reflex field. The point is that these operators satisfy a quadratic relation (with distinct eigenvalues for generic elements of the Galois group), and so act semi-simply on \(H^2\) (imagine everything is compact here). Then, by pure group theory, if the image of \(\rho\) is large enough, the sheer number of such elements is enough to force semi-simplicity. It is perhaps useful to note that if \(V\) is a representation such that \(V^{ss} = (\rho^{\otimes 2})^{ss}\) and \(V\) is geometric and pure, then \(V\) should automatically be semi-simple. This follows from any number of combinations of bits of the standard conjectures, but one way to see it is that if \(W\) is geometric and of weight zero, then (by Bloch-Kato) one should have \(H^1_f(F,W) = 0\). The relevant \(W\) in the example above is \(\mathrm{ad}^0(\rho)\). So in fact one can give an alternate proof of the theorem using the full power of modularity lifting theorems, providing one is willing to omit finitely many primes \(p\). This is really an explanation of why Jan’s result is nice! For example, as soon as one replaces \(\rho^{\otimes 2}\) by \(\rho^{\otimes n}\), one has to start dealing with \(H^1_f(F,\mathrm{Sym}^{2n}(\rho)(-n))\), which gets a little tricky.

These fellows turned up for the Bouillabaisse.

Ana Caraiani talked about a very nice result concerning the sign of Galois representations associated to torsion classes for \(\mathrm{GL}(n)/F^{+}\) for totally real fields \(F^{+}\) (this was joint work with Bao Le Hung). Namely, the trace of any complex conjugation lies in \(\{-1,0,1\}\) (in fact, the result identified the exact characteristic polynomial, which is more general in small characteristic). The basic strategy is to follow Scholze’s construction and “reduce” the problem to the case of essentially self-dual forms, where one has previous results by Taylor, Bellaïche-Chenevier, and Taïbi. However, there is a problem, which is that the regular self-dual automorphic forms one finds congruences with need not be globally irreducible, and perhaps not even cuspidal. Suppose one can show that they decompose into an isobaric sum \(\pi = \boxplus \pi_i\) where the \(\pi_i\) are self-dual. One runs into problems if too many of the \(\pi_i\) are of dimension \(n_i\) with \(n_i\) odd. However, by considering the weights, only one of the \(\pi_i\) can be odd, because otherwise the Hodge-Tate weight zero would occur with multiplicity which would violate the fact that \(\pi\) is regular. There is still something to check for the even \(n_i\) also, because previous results required some sign assumption on the character \(\eta\) such that \(\pi = \pi^{\vee} \eta\). I believe that even getting to this point required a further assumption on the torsion class not coming from the boundary. In the boundary case, there was also a reduction/induction case which also required careful handling of “odd” dimensional pieces, and some computation of a restriction of Hecke operators from the relevant Parabolic/Levi which required a sign to come out correctly. One clever technical step was working with the cohomology of adelic quotients \(G(F)\backslash G(\mathbf{A})/UK\) where \(K\) is a maximal compact of \(G(\mathbf{R})\) rather than the connected component \(K^0\). The advantage of this is that, in the odd dimensional case, this pins down the trace of complex conjugation to be \(+1\) rather than \(\pm 1\). This is clear when \(n = 1\), and that one should expect it to be true follows for \(n\) odd by taking determinants.

Peter Scholze gave a talk on his new functor. The basic elements in the construction of this functor are as follows. The Gross-Hopkins period map allows one to view the (infinite level) Lubin-Tate tower as a \(\mathrm{GL}_n(F)\)-torsor over the (\(D^{\times}\) Severi-Brauer variety) \(\mathbf{P}^{n-1}_{\mathbf{C}_p}\). So, given an admissible representation \(\pi\), one can form the “local system” \(\mathscr{F}_{\pi}\) on the base, and then take its cohomology. The key technical point of this construction is to show that the result is admissible for \(D^{\times}\), which amounts to proving finiteness of \(K\)-invariants for suitable compact open \(K\) of \(D^{\times}\). The first step is to pull back to the (lowest level) part of the Lubin-Tate tower, which one can do because the GH map splits. Now the map from infinite level to the base of the Lubin-Tate tower is really a \(\mathrm{GL}_n(\mathcal{O}_F)\)-torsor, so one only has to consider the restriction of \(\pi\) to \(\mathrm{GL}_n(\mathcal{O}_F)\). But then using the admissibility of \(\pi\), one can look instead at the regular representation of \(\mathrm{GL}_n(\mathcal{O}_F)\). Now, by some sort of Shapiro’s Lemma, one can pull everything back up to infinite level. At infinite level, however, one can replace the Lubin-Tate space by the corresponding Drinfeld tower. Now taking \(K\)-invariants is something that is “easy” to do, because there is an action of \(K\) on the space, and the quotient by \(K\) is some sufficiently nice object for which one has (again by Peter) some nice finiteness theorems for cohomology. I should probably have mentioned that at some point we are working with coefficients in \(\mathcal{O}^+/p\), i.e. in the almost world. The main application in the talk was to show that when one patches completed cohomology (a la CEGGPS), then one can recover the Galois representation from the result. This essentially amounts to showing that when one patches together suitable admissible \(\pi_i\), one can also patch the functor. This requires more than admissibility of the functor, but some sort of “uniform” admissibility (which is always required for patching). I think the key point here is that if \(\pi_i\) is something patched with a group of diamond operators \(\Delta\), then \(\pi_i\) has a filtration by \(|\Delta|\) copies of the original \(\pi\), and so \(\mathscr{F}_{\pi_i}\) has a corresponding uniformly bounded filtration by \(\mathscr{F}_{\pi}\), and so \(H^{n-1}(\mathbf{P}^{n-1}_{\mathbf{C}_p},\mathscr{F}_{\pi})^K\) has length at most \(|\Delta|\) times the corresponding length for the (fixed for all time) version for \(\pi\). On the other hand, Peter instead pulled out a new piece of kit by patching using ultra-filters. My own feeling about logic is that it is never really necessary to prove anything, and I think PS agreed that it wasn’t strictly required for this particular application. Now I understand that my prejudice may not be justified (for example, it is probably hard to prove various identities concerning orbital integrals in small characteristic directly), but I think it applies in this case. Plus, as a purely expositional remark, if you are going to whip out ultrafilters during a number theory talk then everyone is just going to talk about ultrafilters rather than the beautiful construction!