Concrete Examples of Shimura Surfaces

Question

First a disclaimer: I am at best a part-time arithmetic geometer, so please accept my apologies when I am too naive or get something wrong.

From time to time I have tried to learn something about Shimura varieties. The friendliest example I came across so far are Shimura curves arising from quaternion algebras. By this, I will mean the following:

Choose an indefinite quaternion algebra $B$ over $\mathbb{Q}$ with discriminant $D\neq 1$ and a maximal order $\Lambda \subset B$. Then $\mathcal{X}^B$ is the stack over $\mathbb{Z}[\frac1D]$ classifying, for $R$ an $\mathbb{Z}[\frac1D]$-algebra, projective abelian surfaces $A$ over $R$ with an action $\Lambda \to End(A)$ satisfying the following condition: If $R\to S$ is a ring map such that $\Lambda \otimes S \cong M_2(S)$, then the two summands of the $M_2(S)$-module $T_eA \otimes_R S$ are locally free of rank $1$.

These kinds of Shimura curves have (among others) the following nice properties:

1. $\mathcal{X}^B$ is a smooth and proper Deligne-Mumford stack.
2. The deformation theory is controlled by the associated $p$-divisible group. More precisely: We have an étale map $(\mathcal{X}^B)_p \to \mathcal{M}_{p-div}$ to the moduli stack of 1-dimensional $p$-divisible groups.
3. One has a good understanding of the coarse moduli space over $\mathbb{C}$ (quotient of upper half-plane).
4. In many cases one can write down equations of coarse moduli spaces over $\mathbb{Q}$ or even $\mathbb{Z}$ of the curve or its quotient by the Atkin-Lehner involutions.
5. Points (especially of maximal height of the $1$-dimensional formal group) can have interesting automorphism groups; but one can choose level structures so that points have only trivial automorphisms.
6. Its definition does not require adèles. (This probably only an advantage due to my inexperience.)

Is there anything like this for Shimura varieties of dimension $2$? I do not need huge families, but only a few examples I can touch with my hands, with a reasonably simple definition and a good moduli interpretation (deformation theory controlled by $1$-dimensional $p$-divisible group).

Answer 1

As Dylan says, the examples you are looking for are called Picard modular surfaces.

So far as I know, the simplest example of a Shimura surface is from a paper "Sur des fonctions de deux variables indépendantes analogues aux modulaires" of Picard (yes, that Picard) from 1883. From the formal group perspective, it's attached to the abelian integral $$ \int \frac{dz}{\sqrt[3]{z^4 + a_1 z^3 + a_2 z^2 + a_3 z + a_4}}. $$

Roughly, the development goes as follows:

• Given a monic degree-four polynomial $f(z)$ without double roots, the equation $w^3 = f(z)$ defines a plane curve whose closure is a smooth curve of genus 3 with an action of $\Bbb Z/3$ (on $w$). The only isomorphisms between them which are $\Bbb Z/3$-equivariant are induced by the action itself and the maps $z \mapsto z+t$.

• The Jacobian of said curve is 3-dimensional abelian variety with an action of $\Bbb Z/3$. It's also principally polarized -- this roughly corresponds to the cohomology pairing on $H^1$ of the curve.

• There is a basis $$ \frac{dz}{w^2}, \frac{z dz}{w^2}, \text{ and }\frac{dz}{w} $$ of the 1-forms. This splits the formal completion of the Jacobian into two $\Bbb Z/3$-invariant summands, one 2-dimensional, one 1-dimensional. If $p \equiv 1 \mod 3$, then this splitting extends to the $p$-divisible group in the desired manner. (The formal group is actually additive concentrated in even heights for primes congruent to 2.)

• This defines for you a map of moduli from the moduli of these curves to a corresponding Shimura variety of $3$-dimensional polarized abelian surfaces with an action of the ring $\Bbb Z[\zeta_3]$. The action $[-1]$ on the abelian varieties doesn't lift to any of these curves (none of them are hyperelliptic), and so the map of moduli factors through a $\Bbb Z/2$-quotient.

• However, I think (?) that the Torelli theorem implies that after you take this quotient, the result is an open substack. It's not the whole Shimura surface. Roughly, there are situations where the polynomial $f(z)$ degenerates, but the abelian variety does not. In terms of the Deligne-Mumford compactification, if you have only double roots then the smooth curve can be made to degenerate down to a stable curve of arithmetic genus $3$, whose "Jacobian" $Pic^0$ is still an abelian variety (but which now decomposes as the product of some 1- and 2-dimensional abelian varieties with $\Bbb Z/3$-action).

• If you close up under this procedure, you get the whole Shimura surface. The surface is not compact in this case; there is one "point at infinity." There are two compactifications: the Satake-Baily-Borel compactification adds points to the moduli parametrizing so-called semi-abelian schemes (this compactification is singular), and the smooth compactification expands it so that these points are actually (quotients of) elliptic curves.

• In this case, the moduli is pretty simple because it's described entirely in terms of the $a_i$. Away from the primes $2$ and $3$, you can eliminate the coefficient $a_1$ explicitly by a translation, and the Satake-Baily-Borel compactification is the weighted projective stack $$ Proj(\Bbb Z[1/6, a_2, a_4, a_3^2]) $$ with $|a_i| = 3i$. (The element $a_3$ gets a square because of this $\pm 1$ issue.) The moduli of curves hits the portion where the quartic discriminant of $f(z)$ doesn't vanish, and the uncompactified part is the portion where $f(z)$ has no triple zeros. This is the complement of the part defined by the ideal $(a_2^2 + 12 a_4,27 a_3^2 + 8 a_2^3)$.

• The largest automorphism group occuring here is for the curves $w^3 = x^4 + a_3 x$. The resulting abelian scheme has $18$ (equivariant) automorphisms. At $p=7$ this is a height-$3$ point (but there's another one with only 6 automorphisms).

• Unless I'm mistaken, the height-$3$ locus has mass $(p-1)(p^2-1)/1296$.

For other results, many have studied this modular surface and other Picard modular surfaces in the intervening years, but most of them can't be analyzed so concisely. The book "The Zeta Functions of Picard Modular Surfaces" by Langlands et al. has a wealth of information about their general structure (in particular, about calculations with the trace formula), and can lead to a number of other useful places in the literature.

Feel free to contact me if you have further interest.