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. Pingback: Galois Representations for non self-dual forms, Part II | Persiflage

    2. Pingback: Galois Representations for non-self dual forms, Part III | Persiflage

    3. 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

    4. Pingback: Scholze on Torsion 0 | Persiflage

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