In this post I want to briefly summarize the following interesting paper of Milne and Suh http://www.jmilne.org/math/articles/2009c.pdf in which they give a general method for constructing connected Shimura varieties with the property that one can apply an automorphism of $\mathbb{C}$ to their defining equations and obtain a variety with a different fundamental group.

On the one hand, since (most) automorphisms of the complex numbers are extremely weird and certainly not continuous, perhaps this should not be surprising. On the other hand, the theory of etale cohomology (or, if you prefer, GAGA) implies that the cohomology of these spaces must always be the same, and the theory of the etale fundamental group implies that the profinite completion of the fundamental groups are the same. In light of these results, I find that one can obtain different honest fundamental groups rather surprising. I should probably mention that I think the first such example is credited to Serre, but I haven’t yet tracked down the relevant article, and Milne-Suh’s construction is also interesting for other reasons.

Working with connected Shimura varieties has, at the outset, two obvious advantages for tackling this problem. Firstly, a connected Shimura variety is something of the form $\Gamma \backslash X$, where $X$ is a symmetric hermitian domain and $\Gamma$ a torsion free subgroup of some algebraic group. The geometry of this situation is very closely related to the theory of the algebraic group in which $\Gamma$ resides, and $X$ is simply connected, so the fundamental group is simply $\Gamma$ itself. Secondly, Milne (following important work of Shimura, Deligne, Kazhdan, …) has published some wonderful detailed results on how connected Shimura varieties behave under automorphisms of $\mathbb{C}$, so by quoting some of these we can get to our result more quickly (but should note that in the case where our variety is of abelian type the corresponding results follow from Deligne’s theory, and in the PEL case from the classical theory of complex multiplication and moduli spaces).

There are two key ingredients.

Input 1: Margulis super-rigidity

Let $F$ be a totally real field, and $H,H'$ absolutely simple simply connected algebraic groups over $F$, and $H_*,H'_*$ their Weil restrictions down to $\mathbb{Q}$. Suppose we have lattices of $H_*(\mathbb{Q})$ and $latex H’_*(Q)$ that are isomorphic as groups. Then the principle of Margulis super-rigidity says that $H'$ is isomorphic to some twist of $H$ by a field automorphism of $F$.

In particular, suppose we were in the situation where the base changes of $H,H'$ to the finite adeles of $F$ were isomorphic, and fix $K$ an open compact subgroup (of both groups under this isomorphism). If the lattices cut out in the rational points of each group by $K$ were isomorphic, it would force $H' \cong \sigma H$ for $\sigma$ an automorphism of $F$. This is significant for us, because these lattices will be the fundamental groups of our pairs of conjugate Shimura varieties, and the result we have just stated says that we can guarantee they fail to be isomorphic by simply arranging for $H'$ to fail to be a conjugate of $H$ by an automorphism of $F$.

Input 2: Information about conjugates of connected Shimura varieties

Here is where I’d say the meat of the argument takes place. Let us say that a variety is of “type $(H,X)$” if it’s of the form $\Gamma \backslash X$ for $X$ a symmetric hermitian domain and $\Gamma$ a congruence subgroup of $H(F)$.

Main Theorem: Let $H$ be a simply connected group, and fix $V$ a variety of type $(H,X)$, and $\tau \in Aut(\mathbb{C})$. Then $\tau V$ is a variety of type $(H',X')$ and we can relate $H$ and $H'$ locally by the relations:

• At finite places, $H'_v \cong H_v$.
• At infinite places, $H'_v \cong H_{\tau v}$.

The proof of this statement goes roughly as follows. Firstly, Milne’s papers on conjugates of Shimura varieties give a description of $H'$ as a twist of $H$ by a torsor of a proalgebraic group $S^0_F$ which is Tannaka dual to the category of CM-motives over $\mathbb{C}$ with $F$-endomorphisms. In our situation (for a given Shimura datum) the map involved factors through an algebraic quotient, which implies it factors through the Mumford-Tate group of an actual CM abelian variety $A$ with endomorphisms by $F$. With this description, one can reduce the study of the torsor to that of a certain set of isomorphisms between Betti cohomology groups of $A$ and $\tau A$. At each place of $F$, one can then compute it explicitly via the canonical comparison isomorphisms to l-adic and de Rham cohomology, finding in particular that it is trivial at the finite places and deducing the required relation at the infinite places.

Payoff: Suppose we can find a group $H/F$  absolutely simple simply connected  associated with a symmetric hermitian domain $X$, and $\tau \in Aut(\mathbb{C})$ such that whenever $H'/F$ satisfies the relations of the main theorem it fails to be isomorphic to an $Aut(F)$-conjugate of $H$. Note that the isomorphisms (from the main theorem) at all the finite places give an isomorphism over the finite adeles, and it isn’t too hard to show that given $V=\Gamma \backslash X$, the corresponding open compact of $H(\mathbb{A}_{F,f})$ is identified with that of $H'$ corresponding to $\Gamma'$ with $\tau V = \Gamma'\backslash X'$. We therefore deduce from Margulis super-rigidity that $V$ and $\tau V$ have different fundamental groups.

This gives us a rather general lovely recipe for constructing such varieties, and I would advise reading the original paper to get an idea of the different kinds of things one can do. For the sake of completeness I will just sketch an example. Fix your favourite totally real field $F \not= \mathbb{Q}$ with no automorphisms (e.g. any cubic non-Galois field, but there exist lots of examples of all degrees at least 3). Now take the group of norm 1 elements of your favourite quaternion algebra over $F$ which split (say) at exactly one infinite place. Now take some automorphism of $\mathbb{C}$ which moves this place somewhere else. These visibly give data required by the ‘payoff’ and thus the connected Shimura varieties obtained will have the desired property.