Galois Representations for non-self dual forms, Part III

Here are some complements to the previous remarks, considered in Part I and Part II.

First, in order to deal with non-zero weights, one has to replace the Shimura varieties \(Y\), \(X\), \(W\) by Kuga-Satake varieties over these spaces. This “only” adds technical difficulties.

Second, in order to work over the most general bases \(F\), one seems to require good minimal models and compactifications \(X_U\), \(W_U\) in characteristic \(p\), for a prime \(p\) which may be very ramified in \(F\). This is a genuine problem. The way to avoid this problem is amusing. It turns out that one only needs a good model of \(X^{\mathrm{ord}}\) and \(W^{\mathrm{ord}}\). In other words, one only has to understand integral models and toroidal compactifications at the ordinary cusps. However, the ordinariness is exactly what allows one to give appropriate models at these cusps, without having to deal with the more complicated cusps except in some fairly superficial way (say by taking normalizations over an integral model of a universal moduli space of abelian varieties). This seems quite clever.

Third, I was going to talk in more detail about \(n=2\), but having written down the argument it seems a little pointless now, since it is not going to simplify things very much. The only thing that is (perhaps) easier is to understand why the higher direct images of the pushforward of the subcanonical bundle to the minimal compactification vanishes; yet the example of \(\mathcal{A}_2\) in the previous post gives the idea, I think. I was also going to talk about the combinatorics of the boundary and their relationship to the cohomology of \(\mathrm{GL}(n)\), but on second thoughts I’m not.

Fourth, how close is \(H^*_{c,\partial}(\overline{X}^{\mathrm{ord}})\) to \(H^*_{c,\mathrm{Betti}}(X)\), the compactly supported Betti cohomology of the Shimura variety? It’s not so clear.

Fifth, the argument really only uses the ordinary locus in a fairly loose way, namely, it is (in the minimal compactification) affinoid, and it is compatible with Hecke correspondences. On the other hand, at finite level, this is pretty much the only possible such choice. However, perhaps at infinite level there may be other possible choices (in a perfect[-oid] world, as it were…).