In the 70s, Mumford discovered p-adic analogues of classical uniformizations of curves and abelian varieties, which generalized Tate's p-adic uniformization of elliptic curves. Besides its significance for moduli, Mumford's construction can be also viewed as a highly nontrivial example of rigid analytic geometry. We shall start by reviewing the classical Schottky uniformization of compact Riemann surfaces and then introduce the dictionary between Mumford curves and p-adic Schottky groups. With the aid of the Bruhat-Tits tree of $PGL_2(\mathbb{Q}_p)$, we can illustrate examples of Mumford curves whose geometry and arithmetic are rich, and explain why the answer to life, the universe and everything should be changed.

This is a note I prepared for my second Trivial Notions talk at Harvard, Fall 2012. Our main sources are [1], [2] and [3]. Some pictures are taken from [4], [1], and [5].

TopTate curves

A well-known example of complex uniformization is the uniformization of any elliptic curve $E/\mathbb{C}$ by the complex plane $\mathbb{C}$. Namely, we have a complex-analytic isomorphism $$\mathbb{C}/\Lambda_\tau\cong E(\mathbb{C}),$$ for some lattice $\Lambda=\mathbb{Z}\oplus \mathbb{Z}\tau\subseteq \mathbb{C}$ and $\Im(\tau)>0$. The general scheme of uniformization is to find a certain universal (usually analytic) object and realize algebraic curves and varieties as the quotient of this universal object by a group action. This can yield results immediately: in the example of elliptic curves, we easily know that the $n$-torsion group $E[n]\cong (\mathbb{Z}/n \mathbb{Z})^2$, which is not entirely obvious in the purely algebraic setting.

The idea of finding a $p$-adic analogue of the uniformization of elliptic curves goes back to Tate. Replacing $\mathbb{R}$ and $\mathbb{C}$ by $\mathbb{Q}_p$ and $\mathbb{C}_p$, we can ask the following naive question: for an elliptic curve $E/\mathbb{C}_p$, does there exist a $\mathbb{Z}$-lattice $\Lambda\subseteq \mathbb{C}_p$ such that $$\mathbb{C}_p/\Lambda\overset{?}{=}E(\mathbb{C}_p).$$ This question does not quite make sense: the $\mathbb{Z}$-span of any element $v\in \mathbb{C}_p$ is not discrete since $p^k\rightarrow 0$ when $k\rightarrow+\infty$ under the $p$-adic absolute value. However, the multiplicative group $\mathbb{C}_p^\times$ has lots of $\mathbb{Z}$-lattices: $q^\mathbb{Z}$ for any $q\in \mathbb{C}_p^\times$. So we may seek a $p$-adic analogue of $$\xymatrix{\mathbb{C}^\times/q^\mathbb{Z}\ar[rr]^\cong & &E(\mathbb{C}),\\ & \mathbb{C}/\Lambda_\tau \ar[lu]^{\exp(2\pi i\cdot)} \ar[ru]^\cong& }$$ where $q=\exp(2\pi i \tau)$ ($|q|<1$ since $\Im(\tau)>0$).

Recall that the isomorphism $\mathbb{C}/\Lambda_\tau\cong E(\mathbb{C})$ is given by $z\mapsto (\wp_\tau(z),\wp_\tau'(z))$ and $E(\mathbb{C})$ is defined by the equation $$y^2=4x^3-g_2 (\tau)x-g_3(\tau).$$ Since $\wp_\tau(z)$ and $\wp_\tau'(z)$ are translation invariant, we can write them as a Fourier series $(X_q(u), Y_q(u))$ in terms of $u=\exp(2\pi i z)$. After an explicit change of coordinates to get rid of factors of $2\pi i$ and denominators, we obtain the equation $$y^2+xy=x^3+a_4(q)x+a_6(q),$$ where $a_4(q), a_6(q)\in q \mathbb{Z}[{[}q]]$, together with the universal power series \begin{align*} X_q(u)&=\sum_{n\in \mathbb{Z}}\frac{q^nu}{(1-q^nu)^2}-\sum_{n\ge1}\frac{2q^n}{(1-q^n)^2}, \\ Y_q(u)&=\sum_{n\in \mathbb{Z}}\frac{(q^nu)^2}{(1-q^nu)^3}+\sum_{n\ge1}\frac{q^n}{(1-q^n)^2}, \end{align*} which converge as long as $|q|<1$. The miracle is that these power series make perfect sense over any field; in particular, they converge for $q\in \mathbb{C}_p^\times$, $|q|<1$. In this way, Tate proved the following theorem.

Theorem 1 (Tate) For $q\in \mathbb{C}_p^\times$ with $|q|<1$, there exists an elliptic curve $E_q/\mathbb{C}_p$ such that there is a Galois-equivariant "$p$-adic analytic" isomorphism $$\mathbb{C}_p^\times/q^\mathbb{Z}\cong E_q(\mathbb{C}_p).$$

Observe that $|q|<1$ implies that $|a_4(q)|, |a_6(q)|<1$, hence reducing mod $p$ we obtain the equation $$y^2+xy=x^3,$$ which defines a singular cubic curve with a node and tangent lines $y=0$ and $x+y=0$ at $(0,0)$. In other words, $E_q$ has split multiplicative reduction. Conversely, Tate also proved that any elliptic curve with split multiplicative reduction over $\mathbb{Q}_p$ is isomorphic to a unique $E_q$ with $|q|<1$, $q \in \mathbb{Q}_p^\times$. These elliptic curves are called Tate curves, best viewed as elliptic curves over $\Spec \mathbb{Z}[{[}q]]$. As an immediate consequence of the $p$-adic uniformization, one can easily compute the Galois action on the Tate modules of Tate curves.

TopSchottky uniformization

Now the natural question becomes: can we construct the $p$-adic uniformization for smooth projective curves of genus $g\ge2$? One may think of Koebe's uniformization of compact Riemann surfaces as the quotient of the upper half plane $\mathbb{H}$ by Fuchsian groups $\Gamma\subseteq PGL_2(\mathbb{R})$. Unfortunately, since the $p$-adic topology is totally disconnected, the notion of "simply-connected" in the $p$-adic world is more subtle than in the complex world (e.g., all curves with good reduction are "simply-connected", if defined properly). It turns out that the right analogue Mumford discovered is the Schottky uniformization.

Example 1 Given two pairs of circles $A_1$, $B_1$ and $A_2,B_2$ in the complex plane with disjoint interiors, the two Mobius transformations $\gamma_i$ sending the exterior of $A_i$ to the interior of $B_i$ generate a discrete subgroup $\Gamma$ of $PGL_2(\mathbb{C})$ (i.e., a Kleinian group). $\Gamma$ is a free group of rank 2 and its limit set $\mathcal{L}_\Gamma$ consists of the dust left out by iterations of $\gamma_i$ on the common exterior of the 4 circles. It is easy to see that a fundamental domain of $\Gamma$ acting on $\Omega_\Gamma=\mathbb{P}^1(\mathbb{C})-\mathcal{L}_\Gamma$ can be chosen as the common exterior of the 4 circles with two pairs of circle boundaries identified. In this way $\Omega_\Gamma/\Gamma$ becomes a compact Riemann surface of genus 2.

In general, a Schottky group of rank $g$ is a free group constructed as above using $g$ pairs of Jordan curves. For a Schottky group $\Gamma$ of rank $g$, $\Omega_\Gamma/\Gamma$ is a compact Riemann surface of genus $g$. Conversely, any compact Riemann surface can be obtained from some Schottky group. Motivated by this, we define

Definition 1 A $p$-adic Schottky group is a discrete, finitely generated, free group $\Gamma\subseteq PGL_2(\mathbb{Q}_p)$.

Analogously, for a $p$-adic Schottky group, we denote by $\mathcal{L}_\Gamma$ the set of limit points and $\Omega_\Gamma=\mathbb{P}^1(\mathbb{C}_p)-\mathcal{L}_\Gamma$. We now hope that the quotient $\Omega_\Gamma/\Gamma$ admits a structure of an algebraic curve. Let us show an example to illustrate that the expectation is not completely ridiculous.

Example 2 Let $\Gamma$ be the free group of rank 1 generated by $\begin{bmatrix} q & 0 \\ 0 & 1 \end{bmatrix}$. Then $\Gamma$ is discrete, thus is a $p$-adic Schottky group. The limit set $\mathcal{L}_\Gamma$ is exactly $\{0,\infty\}$. So $\Omega_\Gamma/\Gamma=\mathbb{C}_p^\times/q^\mathbb{Z}$ is a Tate curve, which has genus 1 and split multiplicative reduction as we have already seen.

In general, Mumford proved the following influential theorem.

Theorem 2 (Mumford) Suppose $\Gamma$ is a $p$-adic Schottky group of rank $g$. Then there is a $p$-adic analytic isomorphism $\Omega_\Gamma/\Gamma\cong X_\Gamma(\mathbb{C}_p)$, where $X_\Gamma$ is smooth projective curve of genus $g$ over $\mathbb{Q}_p$. Such a curve $X_\Gamma$ is called a Mumford curve.

TopMumford curves and Trees

You may wonder whether an arbitrary smooth projective curve of genus $g\ge2$ admits such a $p$-adic uniformization. But you are smart enough to figure out the answer at once: no, otherwise it would be meaningless to invent the terminology "Mumford curve". At least, as we already know, the elliptic curves which are Mumford curves should have a specific reduction type. This actually generalizes.

Theorem 3 (Mumford) Suppose $\Gamma$ is a $p$-adic Schottky group of rank $g\ge2$. Then the Mumford curve $X_\Gamma$ has split degenerate stable reduction. Conversely, any smooth projective curve with split degenerate stable reduction is a Mumford curve.

Here stable means (as usual) that the reduction has at most ordinary double points (a.k.a. nodes) and any rational component (if any) meets other components at least 3 points; split degenerate means that the normalization of all components are rational and all nodes are $\mathbb{Q}_p$-rational with two $\mathbb{Q}_p$-rational branches.

To see the reason why the theorem is plausible, we need to go a bit into Mumford's construction of $\Omega_\Gamma$ as a rigid analytic space. Remarkably, the analytic reduction of $\Omega_\Gamma$ is closely related to the Bruhat-Tits tree of $PGL_2(\mathbb{Q}_p)$.

Definition 2 A lattice in $\mathbb{Q}_p^2$ is a free $\mathbb{Z}_p$-module of rank two $\Lambda\subseteq \mathbb{Q}_p^2$. Two lattices $\Lambda$ and $\Lambda'$ are said to be equivalent if $\Lambda=\Lambda'x$ for some $x\in \mathbb{Q}_p^\times$. For any two lattices $\Lambda$, $\Lambda'$, we can find a $\mathbb{Z}_p$-basis $(e_1, e_2)$ of $\Lambda$ such that $(p^ae_1, p^b e_2)$ is a $\mathbb{Z}_p$-basis of $\Lambda'$ ($a,b\in \mathbb{Z}$), then the distance $d([\Lambda], [\Lambda'])=|a-b|$ is well defined on lattice classes.
Definition 3 The Bruhat-Tits tree $\Delta$ of $PGL_2(\mathbb{Q}_p)$ is a tree consisting of
  • vertices: lattice classes $[\Lambda]$.
  • edges: $[\Lambda]$ and $[\Lambda']$ are adjacent if and only if $d([\Lambda],[\Lambda'])=1$.
Example 3 The tree $\Delta$ for $p=2$ is shown in the following picture.

Notice that edges coming out of a vertex $[\Lambda]$ correspond bijectively to lines in $\Lambda/p\Lambda\cong\mathbb{F}_p^2$, i.e., points in $\mathbb{P}^1(\mathbb{F}_p)$, all vertices with given distance $m$ to $[\Lambda]$ correspond bijectively to $\mathbb{P}^1(\mathbb{Z}_p/p^m \mathbb{Z}_p)=\mathbb{P}^1(\mathbb{Z}/p^m \mathbb{Z})$ and the infinite ends correspond bijectively to $\mathbb{P}^1(\mathbb{Z}_p)=\mathbb{P}^1(\mathbb{Q}_p)$.

The fact that $PGL_2(\mathbb{Q}_p)$ acts on the tree $\Delta$ already helps us to retrieve the following theorem which is not obvious using purely group-theoretic methods. This replaces "free" by the weaker requirement "torsion-free", and consequently we can construct many $p$-adic Schottky groups arithmetically (e.g., groups coming from quaternionic orders).

Theorem 4 $\Gamma\subseteq PGL_2(\mathbb{Q}_p)$ is a $p$-adic Schottky group if and only if it is discrete, finitely generated and torsion-free.
Proof We need only show the "if" part. Notice that the stabilizer of $PGL_2(\mathbb{Q}_p)$ acting on a vertex of $\Delta$ is conjugate to $PGL_2(\mathbb{Z}_p)$, a compact subgroup. Since $\Gamma$ is discrete, we know this stabilizer must be a finite group. But $\Gamma$ is torsion-free, so we know that the action on $\Delta$ is actually free. It follows that $\Omega_\Gamma$ is the universal covering of the quotient $\Delta/\Gamma$ and $\Gamma$ is the fundamental group of $\Delta/\Gamma$. There is a finite subgraph $(\Delta/\Gamma)_0$ such that $\Delta/\Gamma$ retracts to it, hence the fundamental group is a free group generated by the loops of $(\Delta/\Gamma)_0$. ¡õ

More importantly, the tree $\Delta$ helps us to understand the analytic reduction of $\Omega=\mathbb{P}^1(\mathbb{C}_p)-\mathbb{P}^1(\mathbb{Q}_p)$. Since the topology on $\mathbb{Q}_p$ is totally disconnected, the idea of rigid analytic geometry is to "rigidify" the topology using affinoids, i.e., complements of open disks. We will not discuss the notion of analytic reduction in detail (cf., [6]), but the following example may be instructive.

Example 4 Consider the closed unit disk $D=\{z\in \mathbb{C}_p: |z|\le1\}$. It corresponds to the affinoid algebra $\mathbb{C}_p\langle z\rangle= \{\sum a_n z^n\in \mathbb{C}_p[{[}z]], a_n\rightarrow 0\}$. The analytic reduction is simply $\Spec \overline{\mathbb{F}}_p[z]$, an affine line. Now consider a covering of $D$ by two affinoids $X_1=\{|z|\le |p|\}$ and $X_2=\{|p|\le |z|\le1\}$. They correspond to the affinoid algebras $\mathbb{C}_p\langle z/p\rangle=\mathbb{C}_p\langle s^{-1}\rangle$ and $\mathbb{C}_p\langle p/z, z\rangle= \mathbb{C}_p\langle s,t\rangle/(s t-p)$. So the analytic reduction of $D$ with respect to this covering becomes a projective line (in $s$) and an affine line (in $z$) crossing at a node. Geometrically, this can be viewed as a "blow-up" operation at a closed point in the special fiber. In general, there is a bijection between projective integral model of algebraic curves and analytic reductions associated to its pure affinoid coverings.

Now one can cover $\Omega$ using smaller and smaller affinoids around the rational points $\mathbb{P}^1(\mathbb{Q}_p)$. The analytic reduction of $\Omega$ then becomes a huge tree of $\mathbb{P}^1$s crossing at nodes, with dual graph being exactly $\Delta$.

At this stage Mumford's result may be a bit more transparent. For a $p$-adic Schottky group $\Gamma$, we can construct the quotient of $\Omega_\Gamma$ by gluing the affinoids under the action of $\Gamma$ to form a rigid analytic quotient curve and then apply a GAGA-type theorem to algebraize it. In particular, the reduction $\bar X$ should coincide with the quotient of the reduction of $\Omega_\Gamma$ by $\Gamma$, in other words, $X$ has split degenerate reduction with dual graph $(\Delta/\Gamma)_0$! We list the beautiful dictionary as by-products of Mumford's construction.

\begin{center} \begin{tabular}[h]{p{2.6cm}|p{5.7cm}} dual graph of $\bar X$ & $(\Delta/\Gamma)_0$\\ \hline $X(\mathbb{Q}_p)$ & ends of $\Delta/\Gamma$\\ \hline $\bar X(\mathbb{F}_p)$ & edges coming from vertices of $(\Delta/\Gamma)_0$\\ \hline Reduction map $X(\mathbb{Q}_p)\rightarrow \bar X(\mathbb{F}_p)$ & \centering $\{\text{ends of } \Delta/\Gamma\}\rightarrow \{\text{edges coming from vertices of } (\Delta/\Gamma)_0\}$ \end{tabular} \end{center}

Example 5 Suppose $p=2$ and $\Gamma=\langle\alpha,\beta\rangle$ has action on the tree $\Delta$ shown on the left ($\alpha$ and $\beta$ can be computed explicitly, cf., [4]). Then resulting quotient graph and reduction are shown on the right.

From Mumford's dictionary, we know that $X_\Gamma$ is a hyperelliptic curve of genus $g=2$ with no $\mathbb{Q}_2$-rational points and whose reduction is two rational curves crossing at three $\mathbb{Q}_2$-rational points (indeed, its equation can be written down explicitly using $\theta$-function). You can understand its "rich" geometry through staring at the dollar sign.

To summarize, compared to the complex uniformization, Mumford's $p$-adic uniformization is weaker in the sense that not all curves can arise this way. But it may also be viewed as stronger in the sense that stronger results concerning its geometry and arithmetic may be achieved with the aid of the tree $\Delta$ (among others). Here is our final example due to Herrlich, which enormously improves the classical Hurwitz bound $42\cdot(2g-2)$ for the number of automorphisms of curves of genus $g\ge2$ over any field of characteristic 0.

Theorem 5 (Herrlich) Suppose $X$ is a Mumford curve over $\mathbb{Q}_p$ of genus $g\ge2$, then $$\#\Aut(X)\le \begin{cases} 24\cdot(2g-2) & p=2, \\ 12\cdot(2g-2) & p=3, \\ 15\cdot(2g-2) & p=5, \\ 6\cdot(2g-2) & p\ge7. \end{cases}$$

Therefore you may want to change the answer to life, the universe and everything according to your favorite prime $p$.

References

[1]Mumford, D., An analytic construction of degenerating curves over complete local rings, Compositio Math 24 (1972), no.2, 129--174.

[2]L. Gerritzen and M. van der Put, Schottky Groups and Mumford Curves (Lecture Notes in Mathematics), Springer, 1980.

[3]Mihran Papikian, Non-archimedean uniformization and monodromy pairing, http://www.math.psu.edu/papikian/Research/RAU.pdf.

[4]Cornelissen, G. and Kato, F., The p-adic icosahedron, Notices of the AMS 52 (2005), no.7, 720--727.

[5]David Mumford and Caroline Series and David Wright, Indra's Pearls: The Vision of Felix Klein, Cambridge University Press, 2002.

[6]Jean Fresnel and Marius van der Put, Rigid Analytic Geometry and Its Applications (Progress in Mathematics), Birkhauser Boston, 2003.

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