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 , 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].

A well-known example of complex uniformization is the uniformization of any elliptic curve by the complex plane . Namely, we have a complex-analytic isomorphism for some lattice and . 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 -torsion group , which is not entirely obvious in the purely algebraic setting.

The idea of finding a -adic analogue of the uniformization of elliptic curves goes back to Tate. Replacing and by and , we can ask the following naive question: for an elliptic curve , does there exist a -lattice such that This question does not quite make sense: the -span of any element is not discrete since when under the -adic absolute value. However, the multiplicative group has lots of -lattices: for any . So we may seek a -adic analogue of where ( since ).

Recall that the isomorphism is given by and is defined by the equation Since and are translation invariant, we can write them as a Fourier series in terms of . After an explicit change of coordinates to get rid of factors of and denominators, we obtain the equation where , together with the universal power series
which *converge* as long as . The miracle is that these power series make perfect sense over any field; in particular, they converge for , . In this way, Tate proved the following theorem.

Theorem 1 (Tate)
For with , there exists an elliptic curve such that there is a Galois-equivariant "-adic analytic" isomorphism

Observe that implies that , hence reducing mod we obtain the equation which defines a singular cubic curve with a node and tangent lines and at . In other words, has *split multiplicative reduction*. Conversely, Tate also proved that any elliptic curve with split multiplicative reduction over is isomorphic to a unique with , . These elliptic curves are called *Tate curves*, best viewed as elliptic curves over . As an immediate consequence of the -adic uniformization, one can easily compute the Galois action on the Tate modules of Tate curves.

Now the natural question becomes: can we construct the -adic uniformization for smooth projective curves of genus ? One may think of Koebe's uniformization of compact Riemann surfaces as the quotient of the upper half plane by Fuchsian groups . Unfortunately, since the -adic topology is totally disconnected, the notion of "simply-connected" in the -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 , and in the complex plane with disjoint interiors, the two Mobius transformations sending the exterior of to the interior of generate a discrete subgroup of (i.e., a Kleinian group). is a free group of rank 2 and its limit set consists of the dust left out by iterations of on the common exterior of the 4 circles. It is easy to see that a fundamental domain of acting on can be chosen as the common exterior of the 4 circles with two pairs of circle boundaries identified. In this way becomes a compact Riemann surface of genus 2.

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

Analogously, for a -adic Schottky group, we denote by the set of limit points and . We now hope that the quotient admits a structure of an algebraic curve. Let us show an example to illustrate that the expectation is not completely ridiculous.

Example 2
Let be the free group of rank 1 generated by . Then is discrete, thus is a -adic Schottky group. The limit set is exactly . So 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 is a -adic Schottky group of rank . Then there is a -adic analytic isomorphism , where is smooth projective curve of genus over . Such a curve is called a *Mumford curve*.

You may wonder whether an arbitrary smooth projective curve of genus admits such a -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 is a -adic Schottky group of rank . Then the Mumford curve 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 -rational with two -rational branches.

To see the reason why the theorem is plausible, we need to go a bit into Mumford's construction of as a rigid analytic space. Remarkably, the analytic reduction of is closely related to the Bruhat-Tits tree of .

Definition 2
A *lattice* in is a free -module of rank two . Two lattices and are said to be equivalent if for some . For any two lattices , , we can find a -basis of such that is a -basis of (), then the *distance* is well defined on lattice classes.

Definition 3
The *Bruhat-Tits tree* of is a tree consisting of

- vertices: lattice classes .
- edges: and are adjacent if and only if .

Example 3
The tree for is shown in the following picture.

Notice that edges coming out of a vertex correspond bijectively to lines in , i.e., points in , all vertices with given distance to correspond bijectively to and the infinite ends correspond bijectively to .

The fact that acts on the tree 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 -adic Schottky groups arithmetically (e.g., groups coming from quaternionic orders).

Theorem 4
is a -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 acting on a vertex of is conjugate to , a compact subgroup. Since is discrete, we know this stabilizer must be a finite group. But is torsion-free, so we know that the action on is actually free. It follows that is the universal covering of the quotient and is the fundamental group of . There is a finite subgraph such that retracts to it, hence the fundamental group is a free group generated by the loops of .
¡õ

More importantly, the tree helps us to understand the analytic reduction of . Since the topology on 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 . It corresponds to the affinoid algebra . The analytic reduction is simply , an affine line. Now consider a covering of by two affinoids and . They correspond to the affinoid algebras and . So the analytic reduction of with respect to this covering becomes a projective line (in ) and an affine line (in ) 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 using smaller and smaller affinoids around the rational points . The analytic reduction of then becomes a huge tree of s crossing at nodes, with dual graph being exactly .

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

Example 5
Suppose and has action on the tree shown on the left ( and can be computed explicitly, cf., [4]). Then resulting quotient graph and reduction are shown on the right.

From Mumford's dictionary, we know that is a hyperelliptic curve of genus with no -rational points and whose reduction is two rational curves crossing at three -rational points (indeed, its equation can be written down explicitly using -function). You can understand its "rich" geometry through staring at the dollar sign.

To summarize, compared to the complex uniformization, Mumford's -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 (among others). Here is our final example due to Herrlich, which enormously improves the classical Hurwitz bound for the number of automorphisms of curves of genus over any field of characteristic 0.

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

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

[2]Schottky Groups and Mumford Curves (Lecture Notes in Mathematics), Springer, 1980.

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

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

[5]Indra's Pearls: The Vision of Felix Klein, Cambridge University Press, 2002.

[6]Rigid Analytic Geometry and Its Applications (Progress in Mathematics), Birkhauser Boston, 2003.