Let me first run through the setting of my question in an example I understand well; that of modular curves. If $Y_1(N)$ denotes the usual modular curve over the complexes, the quotient of the upper half plane by the congruence subgroup $\Gamma=\Gamma_1(N)$, then there are two kinds of sheaves that one often sees showing up in the theory of automorphic forms in this setting:

1) Locally constant sheaves. The ones showing up typically come from representations of $\Gamma$ of the form $Symm^{k-2}(\mathbf{C}^2)$, with $\Gamma$ acting in the obvious way on $\mathbf{C}^2$. These sheaves---call them $V_k$---are related to classical modular forms of weight $k$ via the Eichler-Shimura correspondence. They only exist for $k\geq2$ (weight 1 forms are not cohomological) and representation-theoreticically the sheaves are associated to representations of the algebraic group $SL(2)$ (the reason one starts in weight 2 rather than weight 0 is that there is a correction factor of "half the sum of the positive roots").

2) Coherent sheaves. The ones showing up here are powers $\omega^k$ of a canonical line bundle $\omega$ coming from the universal elliptic curve. The global sections of $\omega^k$ (which are bounded at the cusps) are classical modular forms of weight $k$. Although there are no classical modular forms of negative weight, the sheaf $\omega^k$ still makes sense for $k<0$ (in contrast to case 1 above). I am much vaguer about what is conceptually going on here. I have it in my mind that here $k$ is somehow a representation of the group $SO(2,\mathbf{R})$.

Now my question: what is the generalisation of this to arbitrary, say, PEL Shimura varieties? Part (1) I understand: I can consider algebraic representations of the reductive group I'm working with and for each such gadget I can make a locally constant sheaf. But Part (2) I understand less. I am guessing I can construct a big vector bundle on my moduli space coming from the abelian variety. Now, given some representation of some group or other, I can build coherent sheaves somehow, possibly by "changing the structure group" somehow. For which representations of which group does this give me a coherent sheaf on the moduli space?

Basically---what is the general yoga for supplying natural coherent sheaves on Shimura varieties, which specialises to the construction of $\omega^k$ in the modular curve case, and which explains why $\omega^k$ exists even for $k<0$?

  • $\begingroup$ I always imagined this was answered in Blasius-Harris-Ramakrishnan, MR1262930. Did you look there? $\endgroup$ – TSG May 20 '10 at 12:31
  • $\begingroup$ Kevin, is there a reason one shouldn't just attribute $\omega^{\otimes k}$ for all integers $k$ (allowing $k < 0$) to the fact that $\omega$ is a vector bundle? There are papers of Milne on "automorphic vector bundles"... $\endgroup$ – BCnrd May 20 '10 at 12:31
  • $\begingroup$ @BCnrd: as you know, the point is not that it's a vector bundle, it's (as I suspect you meant to say) that it's a line bundle. Consider the case of Siegel modular forms on Sp_4: I have two weights k1,k2. I think that if k1>=k2 there's a coherent sheaf and if k1>=k2>=3 there's an etale sheaf. But I can't invert most of the coherent sheaves because I think they have rank > 1. $\endgroup$ – Kevin Buzzard May 20 '10 at 18:13

If the Shimura variety is attached to the Shimura data $h:\mathbb S \to G_{/\mathbb R}$, and if as usual $K$ denotes the centralizer in $G_{/\mathbb R}$ of $h$, then the automorphic vector bundles are $G(\mathbb R)$-equivariant bundles on $X:= G(\mathbb R)/K(\mathbb R)$ attached to representations of the algebraic group $K$.

Specifically, if $V$ is the representation (over $\mathbb C$, say), then the associated vector bundle $\mathcal V$ is given by $(G(\mathbb R)\times V)/K(\mathbb R).$

Note that if $G = GL_2$ and we are in the modular curve case, then $K$ is equal to $\mathbb C^{\times}$ (thought of as an algebraic group over $\mathbb R$), its complexification is $\mathbb G_m\times \mathbb G_m$, and so there are two integral parameters describing its irreps (which are just one-dimensional in this case). One of them is the usual $k$ of $\omega^k$; the other can be chosen so as to correspond to twisting by a power of the determinant, which does not change the underlying line bundle, but changes the equivariant structure (which we don't think about so explicitly in the classical language).

Now $K$ is the Levi in a parabolic $P$ (defined over $\mathbb R$), and there is an open embedding (of complex analytic spaces) $X := G(\mathbb R)/K(\mathbb R) \hookrightarrow G(\mathbb C)/P(\mathbb C) =: D$. (If you look at Deligne's Corvalis article and unravel things, you'll see that this is how the complex structure on $X$ is defined: in the Hodge-theoretic formulation given there, $P$ is the parabolic preserving the Hodge filtration on the Hodge-structure corresponding to the base point of $X$.) The automorphic vector bundles are naturally defined on the flag variety $D$ (as $\mathcal V := (G(\mathbb C)\times V)/P(\mathbb C)$), and then restricted to $X$. (So in the modular curve case, $D$ is the projective line, and $\omega^k$ comes from $\mathcal O(k)$.)

Now the automorphic bundles, being $G$-equivariant, descend to bundles on each Shimura variety quotient $Sh(X,K_f)$ of $X \times G(\mathbb A_f)$, and have canonical models defined over the reflex field. In the PEL case, one will be able to construct these canonical models using data from the abelian varieties (think about how the abelian varieties give rise to Hodge structures with Mumford--Tate group equal to $G$ in the first place). In general, this result is due to Milne (see his answer).

Added: A colleague pointed out to me, regarding Kevin's initial question, that it can be understood much more generally. Namely, all automorphic vector bundles are locally free coherent sheaves, equivariant under the action of the finite adeles, but not all coherent sheaves necessarily have these properties! Practically nothing is known about the $K$-theory of Shimura varieties, although the question is obviously of fundamental interest. The very deep current work on cycles on Shimura varieties, from various points of view, must be the beginning of a substantial theory whose ultimate shape no one is in a position to imagine.  So Kevin's question should be seen as a program for future collaboration between number theorists and algebraic geometers (at least).

Also, let me remark that I'm told that the term "automorphic vector bundle" was invented in a conversation between Michael Harris and Jim Milne in the second half of the 80s.

  • $\begingroup$ This answer also answers my "next question"---how to get everything over a number field. $\endgroup$ – Kevin Buzzard May 20 '10 at 18:20

As others have pointed out, the word you are looking for is automorphic vector bundle. The holomorphic automorphic forms are exactly the sections of these bundles. To define them, you start with a homogeneous bundle on the (compact) dual hermitian symmetric space. Initially, the construction of the automorphic vector bundle attached to the homogeneous bundle is analytic, but it can be made algebraic by the introduction of the standard principal bundle. It is known that the standard principle bundle has a canonical model over the reflex field, so this all works arithmetically. [The definition and proof of the existence of a canonical model of the standard principal bundle is in my 1988 Inventiones paper, after earlier work of Shimura, Harris, and others; there is a discussion of such things in Chapter III of my article in the Proceedings of the 1988 Ann Arbor conference, which are available on my website.]

  • $\begingroup$ Dear Jim, I edited my answer to give the correct attribution for the construction of canonical models. $\endgroup$ – Emerton May 20 '10 at 15:51
  • $\begingroup$ This answer not only answers the question I asked, but also the question after that! $\endgroup$ – Kevin Buzzard May 20 '10 at 18:16

I think that a good way to think about the second setting is in terms of automorphic representations (and Harris is "the" authority, although I don't remember BHR reference specifically). Let $G=GL_{+}(2,\mathbb{R}), \Gamma=SL(2,\mathbb{Z})$. A modular form $f$ of weight $k$ can be lifted in a standard way to a function on $G/\Gamma$, real group $G$ acts on functions by translation and $f$ generates a representation of $G.$ This function transforms as the $k$th power of the standard 1-dim representation of $K=SO(2,\mathbb{R})$ and the holomorphy of $f$ translates into the condition that it's the lowest $K$-type (which is scalar).

In the Shimura case, $G$ is of Hermitian type, so it contains a $U(1)$ factor and hence admits scalar representations labeled by an integer $k$. Holomorphic representations (=lowest weight modules) with scalar minimal (here: lowest) $K$-type form a direct analogue of the classical case, i.e. scalar automorphic forms. More generally, as Brian and Emerton said, you can start with any $K$-module and form the corresponding automorphic vector bundle, coherent sheaf, etc. In the group theoretic language, you'd be considering a lowest weight module with a general LKT $\sigma$ (minimal $K$-type always has multiplicity 1) realized in functions $G\to V_\sigma$ equivariant under $K$.

  • $\begingroup$ This started as a short comment that would hopefully give a condensed representation-theoretic version of Emerton's answer. I guess I failed... $\endgroup$ – Victor Protsak May 20 '10 at 14:26

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