Algorithms for Mumford Curves

Curves over non-Archimedean fields are of fundamental importance to algebraic geometry and number theory. Mumford curves are a family of such curves, and are interesting from both a theoretical and computational perspective. In non-Archimedean geometry, they are quotients of projective space by Schottky groups. In tropical geometry, which looks at the images in $\mathbb{R}^{n}$ of curves under coordinate-wise valuation, these are balanced graphs with the maximal number of cycles. For instance, the tropicalization of an elliptic Mumford curve can be realized as a plane cubic in honeycomb form [CS].

Let $K$ be an algebraically closed field complete with respect to a nontrivial non-Archimedean valuation. Unless otherwise stated, $|\cdot|$ will denote a choice of norm on $K$ coming from this valuation. Let $R=\{x\in{}K|\mathrm{val}(x)\geq{}0\}$ be the valuation ring of $K$. This is a local ring with unique maximal ideal $M=\{x\in{}K|\mathrm{val}(x)>0\}$. Let $k=R/M$ denote the residue field of $K$. We are most interested in the field of $p$-adic numbers $\mathbb{Q}_{p}$, which unfortunately is not algebraically closed. (For this case, $R=\mathbb{Z}_{p}$, the ring of $p$-adic integers, and $k=\mathbb{F}_{p}$, the field with $p$ elements.) Therefore for theoretical purposes we will often consider $K=\mathbb{C}_{p}$, the complete algebraic closure of $\mathbb{Q}_{p}$. (In this case $R$ is much larger, and $k$ is the algebraic closure of $\mathbb{F}_{p}$.) In most of this paper, choosing elements of $\mathbb{C}_{p}$ that happen to be elements of $\mathbb{Q}_{p}$ as inputs for algorithms yields an output once again in $\mathbb{Q}_{p}$. This “$\mathbb{Q}_{p}$ in, $\mathbb{Q}_{p}$ out” property means we may take $K$ to be $\mathbb{Q}_{p}$ for our algorithmic purposes, while still considering $K=\mathbb{C}_{p}$ when more convenient for the purposes of theory. Much of the theory presented here works for other non-Archimedean fields, such as the field of Puiseux series $\mathbb{C}\{\!\{t\}\!\}$.

We recall some standard definitions and notation for $p$-adic numbers; for further background on the $p$-adics, see [Ho]. For a prime $p$, the $p$-adic valuation $\text{val}_{p}:\mathbb{Q}^{*}\rightarrow\mathbb{Z}$ is defined by $\text{val}_{p}\left(p^{v}\frac{m}{n}\right)=v$, where $m$ and $n$ are not divisible by $p$. The usual $p$-adic norm $|\cdot|_{p}$ on $\mathbb{Q}$ is defined for $a\in\mathbb{Q}^{*}$ by $|a|_{p}=\frac{1}{p^{\text{val}_{p}(a)}}$ and for $0$ by $|0|_{p}=0$. This means that large powers of $p$ are small in absolute value, and small powers of $p$ are large in absolute value. We will usually omit the subscript $p$ from both $|\cdot|_{p}$ and $\text{val}_{p}$.

The completion of $\mathbb{Q}$ with respect to the $p$-adic norm is denoted $\mathbb{Q}_{p}$, and is called the field of $p$-adic numbers. Each nonzero element $b$ of $\mathbb{Q}_{p}$ can be written uniquely as

where $v\in\mathbb{Z}$, $a_{v}\neq 0$ and $a_{n}\in\{0,1,\ldots,p-1\}$ for all $n$. The $p$-adic valuation and norm extend to this field, and such a sum will have $\text{val}(b)=v$ and $|b|=\frac{1}{p^{v}}$. In analog to decimal expansions, we will sometimes write

where the expression trails to the left since higher powers of $p$ are smaller in $p$-adic absolute value. We may approximate $b\in\mathbb{Q}_{p}$ by a finite sum

which will give an error of size at most $\frac{1}{p^{N+1}}$.

Consider the group $PGL(2,K)$, which acts on $\mathbb{P}^{1}(K)$ by treating elements as column vectors. That is, a matrix acts on the point $(a:b)\in\mathbb{P}^{1}(K)$ by acting on the vector $\left(\begin{smallmatrix}a\\ b\end{smallmatrix}\right)$ on the left. Viewed on an affine patch, the elements of this group act as fractional linear transformations. We are interested in the action of certain subgroups of $PGL(2,K)$ called Schottky groups, because a Schottky group minimally generated by $g\geq 2$ elements will give rise to a curve of genus $g$.

There are many equivalent definitions of Schottky groups, including the following useful characterization.

Let $\Gamma$ be a Schottky group minimally generated by $\gamma_{1},\ldots,\gamma_{g}$. The above proposition implies that each element $\gamma{}\in\Gamma$ can be written as a unique shortest product $h_{1}h_{2}\dotsb{}h_{k}$, where each $h_{i}\in{}\{\gamma{}_{1},\dotsc{},\gamma{}_{g},\gamma{}_{1}^{-1},\dotsc{},\gamma{}_{g}^{-1}\}$. This product is called the reduced word for $\gamma$.

Let $\Sigma$ be the set of points in $\mathbb{P}^{1}(K)$ that are fixed points of elements of $\Gamma$ or limit points of the fixed points. The group $\Gamma$ acts nicely on $\Omega:=\mathbb{P}^{1}(K)\setminus\Sigma$; for this reason we will sometimes refer to $\Sigma$ as the set of bad points for $\Gamma$.

In a companion paper to [Mu1] (see [Mu2]), Mumford also considered abelian varieties over non-Archimedean fields. He showed that these could be represented as $(K^{*})^{g}/Q$, where $Q\in(K^{*})^{g\times g}$ is called a period matrix for the abelian variety, and represents the multiplicative subgroup generated by its columns.

Since their initial appearance in the 1970s, a rich theory behind Mumford curves has been developed, largely in the 1980s in such works as [GP]. However, prior to the work in this paper there have been few numerical algorithms for working with them (an exception being a treatment of hyperelliptic Mumford curves, mostly genus $2$, in [Ka] from 2007). We have designed and implemented algorithms that accomplish Mumford curve-based tasks over $\mathbb{Q}_{p}$ previously absent from the realm of computation, and have made many seemingly theoretical and opaque objects hands-on and tractable.

After discussing in Section 2 a technical hypothesis (“good position”) for the input for our algorithms, we present our main algorithms in Section 3. They accomplish the following tasks, where we denote $\Omega/\Gamma$ by $C$:

• •

  Given a Schottky group $\Gamma$, find a period matrix $Q$ for the abelian variety $\text{Jac}(C)$ (Algorithm 3.3).

• •

  Given a Schottky group $\Gamma$, find a triple $(G,\ell,h)$, where

  • –

    $G$ is a graph,

  • –

    $\ell$ is a length function on $G$ such that the metric graph $(G,\ell)$ is the abstract tropical curve which is a skeleton of $C^{an}$ (the analytification of $C$), and

  • –

    $h$ is a natural equivalence $h:R^{g}\rightarrow G$ from the rose graph on $g$ petals;

  this data specifies a point in the tropical Teichmüller space described in [CMV] (Algorithm 3.9).

• •

  Given a Schottky group $\Gamma$, find points in a canonical embedding of the curve $C$ into $\mathbb{P}^{g-1}$ (Algorithm 3.13).

In Section 4, we present an algorithm to achieve the “good position” hypothesis that allows the other algorithms to run efficiently, which in doing so verifies that the input group is Schottky (or proves that the group is not Schottky). This is the most important result of this paper, as the algorithms in Section 3 rely heavily upon it.

We take advantage of a property that makes non-Archimedean valued fields like $\mathbb{Q}_{p}$ special: $|x+y|\leq{}\max\{|x|,|y|\}$. As a result, the error does not accumulate in the computation. Thus we avoid a dangerous hazard present in doing numerical computation over $\mathbb{R}$ or $\mathbb{C}$. The computational problems are hard in nature. Efficient computation for similar problems is not common in the literature even for genus $2$ case. Our algorithms are capable of solving genus $2$ and some genus $3$ examples on a laptop in reasonable time (several minutes). However, they are less efficient for larger cases. The reason is that the running time grows exponentially as the requirement on the precision of the output (in terms of the number of digits) grows. One of the future goals is to find a way to reduce the running time for the algorithms.

Other future goals for Mumford curve algorithms (detailed in Section 5) include natural reversals of the algorithms in Section 3. We are also interested in a particular family of Schottky groups called Whittaker groups, defined in Subsection 5.2. These are the Schottky groups that give rise to hyperelliptic Mumford curves. Some computations for genus $2$ curves arising from Whittaker groups were done in [Ka], including computation of Jacobians and finding group representations from ramification points. Two desirable algorithms in this area include:

• •

  Given a Whittaker group $W$, find an affine equation for $\Omega/W$.

• •

  Given a totally split hyperelliptic curve $C$, find a Whittaker group $W$ such that $C\cong\Omega/W$.

The first can be accomplished if a particular presentation of $W$ is available, and a brute force algorithm in [Ka] can compute the second if the ramification points of $C$ are in a nice position. Future work removing these requirements and improving efficiency would make hyperelliptic Mumford curves very easy to work with computationally.

This section introduces good fundamental domains and the notion of good position for generators, both of which will play key roles in our algorithms for Mumford curves. Our main algorithms in Section 3 require as input Schottky generators in good position, without which the rate of convergence of approximations will drop drastically. For our method of putting generators into good position, see Section 4.

We start with the usual projective line $\mathbb{P}^{1}$, then discuss the analytic projective line $(\mathbb{P}^{1})^{an}$. Our treatment of good fundamental domains follows Gerritzen and van der Put [GP]. The notion is also discussed by Kadziela [Ka]. The introduction to the analytic projective line follows Baker, Payne and Rabinoff [BPR].

The open balls generate a topology on $\mathbb{P}^{1}$. Both open balls and closed balls are simultaneously open and closed in this topology, as is the case for any non-Archimedean field due to the ultrametric inequality $|x+y|\leq\max\{|x|,|y|\}$. Let $|K^{\times}|$ denote the image of $K^{\times}$ under $|\cdot|$. If $r\in{}|K^{\times{}}|$, the open ball and the closed ball are distinguished by whether there exist two points $x,y$ in the ball such that $|x-y|$ equals the diameter. The complement of an open ball is a closed ball, and vice versa.

The definition above implies that $\gamma{}_{i}(\mathbb{P}^{1}\backslash{}B_{i}^{\prime+})=B_{i}$ and $\gamma{}_{i}^{-1}(\mathbb{P}^{1}\backslash{}B_{i}^{+})=B_{i}^{\prime}$ for all $i$.

The following lemma follows from Definition 2.2 by induction (see [Ka, Theorem 6.2]).

For a fixed set of generators of $\Gamma{}$, there need not exist a good fundamental domain. If there exists a good fundamental domain for some set of free generators of $\Gamma{}$, we say that the generators are in good position. Gerritzen and van der Put [GP, §I.4] proved that there always exists a set of generators in good position. They also proved the following desirable properties for good fundamental domains.

The statements (2), (3), and (4) imply that $\Omega{}/\Gamma{}$ can be obtained from $F$ by glueing the boundary of $F$. More specifically, $B_{i}^{+}\backslash{}B_{i}$ is glued with $B_{i}^{\prime+}\backslash{}B_{i}^{\prime}$ via the action of $\gamma{}_{i}$. We have designed the following subroutine, which takes any point $p$ in $\Omega{}$ and finds a point $q$ in $F$ such that they are equivalent modulo the action of $\Gamma{}$. This subroutine is useful in developing the algorithms in Section 3 and 4.

We can extend the definition of good fundamental domains to the analytic projective line $(\mathbb{P}^{1})^{an}$. In general, analytification of an algebraic variety is defined in terms of multiplicative seminorms. For our special case $(\mathbb{P}^{1})^{an}$, there is a simpler description. As detailed in [Ba], $(\mathbb{P}^{1})^{an}$ consists of four types of points:

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  Type 1 points are just the usual points of $\mathbb{P}^{1}$.

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  Type 2 points correspond to closed balls $B(a,r)^{+}$ where $r\in{}|K^{\times{}}|$.

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  Type 3 points correspond to closed balls $B(a,r)^{+}$ where $r\notin{}|K^{\times{}}|$.

• •

  Type 4 points correspond to equivalence classes of sequences of nested closed balls $B_{1}^{+}\supset{}B_{2}^{+}\supset{}\dotsb{}$ such that their intersection is empty.

There is a metric on the set of Type 2 and Type 3 points, defined as follows: let $P_{1}$ and $P_{2}$ be two such points and let $B(a_{1},r_{1})^{+}$ and $B(a_{2},r_{2})^{+}$ be the corresponding closed balls.

• (1)

  If one of them is contained in the other, say $B(a_{1},r_{1})^{+}$ is contained in $B(a_{2},r_{2})^{+}$, then the distance $d(P_{1},P_{2})$ is $\log_{p}(r_{2}/r_{1})$.

• (2)

  In general, there is a unique smallest closed ball $B(a_{3},r_{3})^{+}$ containing both of them. Let $P_{3}$ be the corresponding point. Then, $d(P_{1},P_{2})$ is defined to be $d(P_{1},P_{3})+d(P_{3},P_{2})$.

The metric can be extended to Type 4 points.

This metric makes $(\mathbb{P}^{1})^{an}$ a tree with infinite branching, as we now describe. There is a unique path connecting any two points $P_{1}$ and $P_{2}$. In case (1) above, the path is defined by the isometry $t\mapsto{}B(a_{1},p^{t})^{+}$, $t\in{}[\log{}(r_{1}),\log{}(r_{2})]$. It is straightforward to check that $B(a_{1},r_{2})^{+}=B(a_{2},r_{2})^{+}$. In case (2) above, the path is the concatenation of the paths from $P_{1}$ to $P_{3}$ and from $P_{3}$ to $P_{2}$. Then, Type 1 points become limits of Type 2 and Type 3 points with respect to this metric. More precisely, if $x\neq{}\infty{}$, then it lies at the limit of the path $t\mapsto{}B(x,p^{-t})^{+}$, $t\in{}[0,+\infty{})$. Type 1 points behave like leaves of the tree at infinity. For any two Type 1 points $x,y$, there is a unique path in $(\mathbb{P}^{1})^{an}$ connecting them, which has infinite length.

An analytic open ball $B(a,r)^{an}$ is a subset of $(\mathbb{P}^{1})^{an}$ whose set of Type 1 points is just $B(a,r)$ and whose Type 2, 3, and 4 points correspond to closed balls $B(a^{\prime},r^{\prime})^{+}\subset{}B(a,r)$ and the limit of sequences of such closed balls. An analytic closed ball is similar, with $B(a,r)$ replaced with $B(a,r)^{+}$. Just as in the case of balls in $\mathbb{P}^{1}$, the analytic closed ball $(B^{+})^{an}$ is not the closure of $B^{an}$ in the metric topology of $(\mathbb{P}^{1})^{an}$. The complement of an analytic open ball is an analytic closed ball, and vice versa. In an analytic closed ball $(B(a,r)^{+})^{an}$ such that $r\in{}|K^{\times}{}|$, the Gaussian point is the Type 2 point corresponding to $B(a,r)^{+}$. An analytic annulus is $B\backslash{}B^{\prime}$, where $B$ and $B^{\prime}$ are analytic balls such that $B^{\prime}\subsetneqq{}B$. If $B$ is an analytic open (resp. closed) ball and $B^{\prime}$ is an analytic closed (resp. open) ball, then $B\backslash{}B^{\prime}$ is an analytic open annulus (resp. analytic closed annulus). A special case of analytic open annulus is the complement of a point in an analytic open ball.

Any element of $PGL(2,K)$ sends open balls to open balls and closed balls to closed balls. Thus, there is a well defined action of $PGL(2,K)$ on $(\mathbb{P}^{1})^{an}$.

Definition 2.8 implies that $\gamma{}_{i}((\mathbb{P}^{1})^{an}\backslash{}(B_{i}^{\prime+})^{an})=B_{i}^{an}$ and $\gamma{}_{i}^{-1}((\mathbb{P}^{1})^{an}\backslash{}(B_{i}^{+})^{an})=B_{i}^{\prime an}$.

We now argue that there is a one-to-one correspondence between good fundamental domains in $\mathbb{P}^{1}$ and good fundamental domains in $(\mathbb{P}^{1})^{an}$. (This fact is well-known, though seldom explicitly stated in the literature; for instance, it’s taken for granted in the later chapters of [GP].) If $\mathbb{P}^{1}\backslash{}(B_{1}\cup{}\dotsb{}\cup{}B_{g}\cup{}B_{1}^{\prime}\cup{}\dotsb{}\cup{}B_{g}^{\prime})$ is a good fundamental domain in $\mathbb{P}^{1}$, then $(\mathbb{P}^{1})^{an}\backslash{}(B_{1}^{an}\cup{}\dotsb{}\cup{}B_{g}^{an}\cup{}B_{1}^{\prime an}\cup{}\dotsb{}\cup{}B_{g}^{\prime an})$ is a good fundamental domain in $(\mathbb{P}^{1})^{an}$. Indeed, since the closed balls $B_{1}^{+},\dotsc{},B_{g}^{+},B_{1}^{\prime+},\dotsc{},B_{g}^{\prime+}$ are disjoint, and the corresponding analytic closed balls consist of points corresponding to closed balls contained in $B_{1}^{+},\dotsc{},B_{g}^{+},B_{1}^{\prime+},\dotsc{},B_{g}^{\prime+}$ and their limits, the analytic closed balls are also disjoint. Conversely, if $(\mathbb{P}^{1})^{an}\backslash{}(B_{1}^{an}\cup{}\dotsb{}\cup{}B_{g}^{an}\cup{}B_{1}^{\prime an}\cup{}\dotsb{}\cup{}B_{g}^{\prime an})$ is a good fundamental domain in $(\mathbb{P}^{1})^{an}$, then $\mathbb{P}^{1}\backslash{}(B_{1}\cup{}\dotsb{}\cup{}B_{g}\cup{}B_{1}^{\prime}\cup{}\dotsb{}\cup{}B_{g}^{\prime})$ is a good fundamental domain in $\mathbb{P}^{1}$, because the classical statement can be obtained from the analytic statement by considering only Type 1 points. This correspondence allows us to abuse notation by not distinguishing the classical case and the analytic case. Theorem 2.5 is also true for analytic good fundamental domains.

Another analytic object of interest to us is the minimal skeleton of the analytification of the genus $g$ curve $\Omega/\Gamma{}$ (a task that is part of Algorithm 3.9), so we close this section with background information on this object. The following definitions are taken from Baker, Payne and Rabinoff [BPR], with appropriate simplification.

Work towards algorithmic computation of semistable models is discussed in such works as [AW, §1.2] and [BW, §3.1], though such computation is in general a hard problem.

Semistable models are related to skeleta in the following way: take a semistable model $X$ of $C$. Associate a vertex for each irreducible component of $X_{k}$. For each ordinary intersection of two irreducible components in $X_{k}$, connect an edge between the two corresponding vertices. The resulting graph is combinatorially a skeleton of $C^{an}$.

If we have a Schottky group $\Gamma=\left<\gamma_{1},\ldots,\gamma_{g}\right>$ in terms of its generators, there are many objects we wish to compute for the corresponding curve $\Omega/\Gamma$, such as the Jacobian of the curve, the minimal skeleton of the analytification of the curve, and a canonical embedding for the curve. In this section we present algorithms for numerically computing these three objects, given the input of a Schottky group with generators in good position. For an algorithm that puts arbitrary generators of a Schottky group into good position, see Section 4.