In "Canonical models of Shimura curves" by J.S. Milne (avaliable at https://www.jmilne.org/math/articles/2003a.pdf), he explains the definition of quaternion Shimura curve, and explains the modern definition of Shimura varieties is the best (at Section 4 Page 33):

"More significantly, many Shimura varieties are not moduli varieties, not even conjecturally, and so Definition A doesn’t apply to such Shimura varieties."

And he writes in Section 6, Page 37:

"Conjecturally, the Shimura variety is a moduli variety (in general for motives) when $w_X$ is defined over $\mathbb Q$, and it is not a moduli variety when $w_X$ is not defined over $\mathbb Q$."

Here $w_X : \mathbb G_m \rightarrow G_{\mathbb R}$ is the weight homomorphism. The moduli variety is for general motives, and Milne proved in 1994 that all Shimura varieties of abelian type with rational weight are moduli varieties for abelian motives.

Why do we believe such conjecture? I can't find a reference and don't know whether it's proved. If the weight is not rational, what is the obstruction of our Shimura variety to have a moduli interpretation ? On the other hand, for non-abelian type Shimura variety with rational weight, what objects will it parametrize? Thanks for any enlightening example.

  • $\begingroup$ Associated to a representation of $G$, you get a representation of the fundamental group of your Shimura variety. If you can make this representation into a variation of rational Hodge structures, you can make the Shimura variety into a moduli space of these Hodge structures (and associated motives). So maybe one can see that this fails if the weight homomorphism is not rational. Maybe this is obvious because the weight of a rational variation of pure Hodge structures defines a rational homomorphism $\mathbb G_m \to GL_n$. $\endgroup$
    – Will Sawin
    Nov 27 '19 at 2:39

When the weight $w_X$ is rational, but the Shimura variety is not of abelian type, it is very much an open question whether it should be a moduli variety for motives. I would call it a "hope" rather than conjecture that it is (Deligne in his Corvallis article, p. 248, calls it a dream).

If a Shimura variety is moduli variety for motives, then it will carry a family of motives (at least if it is a fine moduli variety), which gives rise to a variation of $\mathbb{Q}$-Hodge structures on the Shimura variety. From this you get faithful representation of the reductive group $G$ defining the Shimura variety on a vector space underlying a rational Hodge structure. The composite of $w_X$ with the representation gives the weight map for the Hodge structure, which is defined over $\mathbb{Q}$.

  • $\begingroup$ Thank you for the reference! $\endgroup$
    – sawdada
    Nov 27 '19 at 19:03

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