The problem of constructing Galois representations associated to Maass forms with eigenvalue 1/4 is, by now, a fairly notorious problem. The only known strategy, first explained by Carayol, is to first transfer the representation to a unitary group over an imaginary quadrtic field, where one can realize the corresponding transfer in the coherent cohomology of a related “Griffiths-Schmid” variety X. Then one hopes to study the action of Hecke operators on this space and relate it to some (hopefully existing) rational structure on the cohomology. The wrinkle is that X is not algebraic but merely a complex manifold, so it’s not so easy to see how to impose any rational structure on the (higher) coherent cohomology. I have nothing intelligent to say about whether this approach will work. However, suppose one is as optimistic as possible, and thinks about what one might *hope* to be true — not only to construct Galois representations but also prove the converse (Artin). Then, following [CG], one might hope to find an *integral* structure on this cohomology (with interesting torsion) on which to study congruences and then glue together torsion classes using Taylor-Wiles to produce a patched complex of the right length. What is the invariant \(l_0\) in this case? One might (generally) hope in this context to study conjugate self-dual representations
\(\rho: G_E \rightarrow \mathrm{GL}_3(A)\)
for an imaginary quadratic field E (in which p splits) for local Artinian rings (A,m) with A/m of characteristic p which are unramified at p. The difference in dimension between the ordinary local deformation ring and the unramified deformation ring appears to be 3, and thus we expect \(l_0 = 3.\) Correspondingly, we expect cohomology to occur in a range of cohomological degrees \([q_0,q_0 + 3]\) for some \(q_0.\) Moreover, in the presence of cohomology in characteristic zero, we expect to see cohomologies in all such degrees. Yet this doesn’t happen for X; in fact, the cohomology only occurs (in characteristic zero) in degrees 1 and 2 (according to RLT). This suggests not only that we *won’t* be able to prove modularity using integral cohomology of X, but even that — in the most naive sense — we should not expect an integral structure at least with the usual properties. Namely, if we patch a complex of integral cohomology of length 1, then the corresponding patched modules in cohomology will be too big for any unramified deformation ring to act. So it appears that the best possible scenario is too good to be true.
On a different matter, there is another pressing issue I would like to bring to my readers. In the papers I have written with Matt and David (and some by myself), we have used the notation \(l_0\) — which has its origins in the book of Borel and Wallach. There is, however, a competing notation in some of Akshay’s papers, namely \(\delta.\) One argument for the latter is that \(l_0\) specifically comes from a particular calculation in \((\mathfrak{g},K)\)-cohomology, and is not compatible with other situations in which one might want to consider the problem of cohomology in various degrees. (For example, for weight one modular forms, the Galois \(l_0 = 1\) whereas GL(2)/Q has discrete series.) My argument is that there will never be any confusion when using \(l_0,\) and that it has the property of being unlikely to every conflict with any other notation. Moreover, the phenomenology in both coherent and Betti cohomology both depend on \(l_0\) in exactly the same way. Dear reader: what is your opinion?