## Galois Representations for non self-dual forms, Part I

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:

• $\pi_{\infty}$ has trivial infinitesimal character.
• $\pi_p$ is unramified.
• $F$ is an imaginary quadratic field in which $p$ splits.

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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### 4 Responses to Galois Representations for non self-dual forms, Part I

1. Dear GR,

Is the double negative “not to avoid thinking about” deliberate? These have been
a helpful series of post, by the way.

Cheers,

Matt

2. Pingback: Scholze on Torsion 0 | Persiflage