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.
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.
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.
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.
In general, Mumford proved the following influential theorem.
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.
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 .
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).
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., ), but the following example may be instructive.
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.
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 .
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