# Are Shimura Varieties $K(\pi, 1)$'s?

Let $\mathscr{A}_g$ be the moduli space of principally polarized Abelian varieties of dimension $g$.  The complex-analytic space (stack) associated to $\mathscr{A}_g$ is a $K(\pi,1)$; that is, its only non-vanishing homotopy group is $\pi_1$, which is $Sp_{2g}(\mathbb{Z})$.  In particular, the cohomology of $\mathscr{A}_g$ is the same as the cohomlogy of $Sp_{2g}(\mathbb{Z})$.

Jesse Silliman, a graduate student at Stanford, has told me an argument that shows that this is in some sense maximally untrue when one considers the "etale homotopy type" of $\mathscr{A}_g$.  (Jesse proposed it as an argument for why (compactifications of) $\mathscr{A}_g$ and its level covers have trivial $H^1$ for $g>1$, but a slight enhancement actually shows much more).  Moreover, the argument is anabelian, which always gets me excited.  A similar argument should work for many Shimura varieties of rank $>1$, but I'll just sketch it in the case of $\mathscr{A}_g$ with $g>1$.

Let $G$ be the etale fundamental group of $\mathscr{A}_g$, namely $Sp_{2g}(\hat{\mathbb{Z}})$.  This is a pretty pathological group (for example, its center is infinitely generated), but whatever.  Given any continuous representation $$\rho: G\to GL_n(\mathbb{Z}_\ell),$$ there is a natural map $$\iota: H^*(G, \rho)\to H^*(\mathscr{A}_g, \mathscr{F}_\rho),$$ where $\mathscr{F}_\rho$ is the lisse sheaf on $\mathscr{A}_g$ associated to $\rho$, and the left hand side is continuous cohomology.  Now suppose that $\rho$ actually comes from some representation of $Sp_{2g}(\mathbb{Z})$ into $GL_n(\mathbb{Z})$.

I claim that in this case $\mathscr{F}_\rho$ shows up in the cohomology of some power of the universal Abelian variety, and is cut out by absolute Hodge cycles, so in particular $\rho$ canonically extends to a representation of the arithmetic fundamental group of $\mathscr{A}_g$ (possibly after some finite field extension).  Moreover, at any $K$-point $x$ of $\mathscr{A}_g$, the action of the Galois group of $K$ on $\rho$ factors through its tautological action on the Tate module of the fiber of the universal Abelian variety at $x$.

Let's choose a geometric point at some CM point $x\in \mathscr{A}_g$, with associated CM field $E$.  Then the action of the Galois group of $E$ on $G$ factors through the Artin reciprocity map by the main theorem on CM; hence the action on $H^*(G, \rho)$ is Abelian.  But the action on the etale cohomology of $\mathscr{A}_g$ is independent of basepoint, so running over all CM basepoints, we can see that anything in the image of $\iota$ has to be of Tate type!  In particular, none of the interesting cohomology of $\mathscr{A}_g$ comes from its etale fundamental group.  The same argument works for any level cover of $\mathscr{A}_g$.  Thus in some sense these things are as far from $K(\pi, 1)$'s as possible.

Of course, $H^1(\mathscr{A}_g)=H^1(G)$ more or less by definition, so this shows that $H^1(\mathscr{A}_g)$ (and any level cover thereof) is pure of Tate type --- in particular, any smooth compactification of (a level cover of) $\mathscr{A}_g$ must have $H^1=0$.

Exercise -- how did I use that $g>1$?  (Obviously this last sentence is untrue when $g=1$).  Hint:  It's when I said that the action of Galois on $G$ factors through the Artin reciprocity map; in this step, I used the solution to the congruence subgroup problem, which is of course false for $g=1$.