This is the first of a series of posts discussing the recent work of Harris, Lan, Taylor, and Thorne on constructing Galois representations associated to regular algebraic automorphic forms for \(\mathrm{GL}(n)\) over a CM field \(F/F^{+}\). I will dispense with any niceties about why one should care, and try simply to decipher the scribbles I made during a talk RLT gave at the Drinfeld seminar. I should warn the reader of two difficulties: this paper does not exist as a public manuscript, and it also involves technical details which I generally prefer not to avoid thinking about. So caveat emptor.
First, some simplifying assumptions. Let’s assume that:
For examples, I will generally consider the case \(n = 1\) and \(n = 2\).
The goal will be to construct a Galois representation
\(R_p(\pi) = r_p(\pi) \oplus \epsilon^{1-2n} r_p(\pi^{c,\vee})\)
If one can do this for \(\pi\) and for \(\pi \otimes \chi\) for enough characters \(\chi\), then one can recover \(r_p(\pi)\). Naturally enough, \(R_p(\pi)\) will be associated to an automorphic form \(\Pi\) for a bigger group. Now \(\pi \boxplus \epsilon^{1-2n} \pi^{c,\vee}\) is automorphic for \(\mathrm{GL}(2n)/F\); it is, moreover, an essentially conjugate self-dual (RAESD) although no longer cuspidal. It does, however, come from a smaller group, namely, the unitary similitude group \(G\) which is ubiquitous in the papers of of Harris and Taylor. Over the complex numbers, \(G\) looks like \(\mathrm{GL}(2n) \times \mathrm{GL}(1)\), but over the real numbers I think it must look like \(\mathrm{GU}(n,n)\). Although it’s true that the natural — i.e. occurring in cohomology of \(X(G\)) — Galois representations associated to RAESDC forms \(\varpi\) for \(G\) will actually be nth exterior powers, I don’t think that matters so much, since once one has congruences between \(\varpi\) and \(\Pi\) one gets Galois representations of the right degree for \(\Pi\).
OK. Now associated to \(G\) and an open compact \(U\) of \(G(\mathbf{A}^f)\) one has three natural objects: a smooth quasi-projective Shimura variety \(Y = Y_U\), a (typically non-smooth) normal minimal compactification \(X = X_U\), and a (family of) smooth toroidal compactifications \(W = W_U\). The complement of \(Y\) in \(W\) is SNCD (smooth normal crossing divisor). I’m using somewhat non-standard terminology as far as the letters go because I don’t want too many subscripts. If \(n = 1\), then \(Y\) is an open modular curve, \(X = W\) is a smooth compactification, and the complement of \(Y\) in \(W\) is a finite number of points (cusps). If \(n = 2\), then \(Y\) has complex dimension \(4\). More on that example later.
As usual, one has the Hodge bundle \(\mathbb{E} = \pi_* \Omega^{1}_{A/Y}\), from which one may build automorphic bundles \(\xi_{\rho}\) in the usual way for suitable algebraic representations \(\rho\) of what I guess amounts to the levi of \(G(\mathbf{C})\). In my notes I have written:
\(\xi_{st} = \mathrm{st}_{\tau} \oplus \mathrm{st’}_{\tau’}\)
Here \(\mathrm{st}\) means the standard \(n\)-dimensional representation of \(\mathrm{GL}_n\), and \(\mathrm{st’}\) denotes the complex conjugate representation. One must have has \(\mathbb{E} = \xi_{st}\), where the decomposition into a direct sum of two rank \(n\)-modules comes from the action of the auxiliary ring on the tangent space to the universal abelian variety (built into the definition of \(G\) which I have omitted). I also have written:
\( \mathrm{KS} = \mathrm{st}_{\tau} \otimes \mathrm{st’}_{\tau’}\)
This presumably relates to the Kodaira–Spencer isomorphism. It’s certainly consistent with a surjection:
\(\bigwedge^2 \pi_* \Omega^{1}_{A/Y} \rightarrow \Omega^1_{Y/k}\)
Now it turns out that \(\xi_{\rho}\) extends to \( W\) in two natural ways, there is the canonical extension \(\xi^{\mathrm{can}}_{\rho}\) and the sub-canonical extension \(\xi^{\mathrm{sub}}_{\rho}\); they differ by the divisor corresponding to the boundary. Just as in the case \(n = 1\), the bundle \(\xi^{\mathrm{can}}\) should be though of as having log-poles at the boundary. Last but not least, for the one dimensional representation \(\wedge^{2n}(\mathrm{st}_{\tau} \oplus \mathrm{st’}_{\tau’})\), one has the line bundle \(\omega\) on \(Y\). Denote the canonical extension of \(\omega\) to \(W\) by \(\omega\). Then it turns out that \(\omega\) is the pull-back of an ample line bundle \(\omega\) on \(X\). Of course, if \(n =1\), then \(\omega\) is what you think it is — well, almost, since we are using \(GU(1,1)\) Shimura varieties rather than \(\mathrm{GL}(2)\). However, for general \(n\), things are a little trickier. For example, \(\omega\) is ample on \(X\), but not (in general) on \(W\).
If \(U\) is maximal at \(p\), then the previous constructions also work over a finite field \(k\) of characteristic \(p\) and the appropriate smoothness claims are still true. One has the Hasse invariant \(H\), which is a section of \(\omega^{p-1}\) over \(X/k\). Since \(\omega\) is ample on \(X\), the complement of the zero divisor of \(H\) is affine, it is of course the ordinary locus. In particular, one has Galois representations of the correct flavor associated to forms in the infinite dimensional space
\(H^0(X^{\mathrm{ord}}, \xi_{\rho})\)
This follows in the “usual” way; RLT sketched an argument, it goes as expected, although I think the Kocher principle must have slipped in at some point.
So far, I haven’t really said anything related to the actual argument, but I think I will stop here for now. The next step is to connect \(\Pi\) in any way to classes in the p-adic modular forms arising in the cohomology group above.
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