Branch points of split degenerate superelliptic curves II: on a conjecture of Gerritzen and van der Put

Throughout this paper, we assume that $K$ is a field equipped with a discrete valuation $v:K^{\times}\to\mathbb{Z}$, and we denote by $\mathbb{C}_{K}$ the completion of an algebraic closure of $K$. We fix a prime $p$ and a primitive $p$th root of unity $\zeta_{p}\in\mathbb{C}_{K}$ and assume throughout that we have $\zeta_{p}\in K$. (This will ensure, among other things, that each automorphism of $C/K$ as a cyclic $p$-cover of $\mathbb{P}_{K}^{1}$ is defined over $K$.) We adopt the convention of using the notation $\mathbb{P}_{K}^{1}$ both for the projective line with its structure as a variety over $K$ and for the set of $K$-points of $\mathbb{P}_{K}^{1}$, i.e. in a context that will appear frequently in this paper, we write $\mathbb{P}_{K}^{1}$ for the set $K\cup\{\infty\}$.

Mumford showed in his groundbreaking paper [8] that any curve $C/K$ (not necessarily superelliptic) of genus $g\geq 1$ can be realized as a quotient of a certain subset $\Omega\subset\mathbb{P}_{K}^{1}$ by the action of a free subgroup $\Gamma<\mathrm{PGL}_{2}(K)$ of $g$ generators via fractional linear transformations if and only if the the curve $C$ satisfies a property called split degenerate reduction (see [9, Definition 6.7] or [6, §IV.3]). The free subgroup $\Gamma<\mathrm{PGL}_{2}(K)$ must act discontinuously on $\mathbb{P}_{K}^{1}$ (i.e. the set of limit points under its action must not coincide with all of $\mathbb{P}_{K}^{1}$), and the subset $\Omega\subset\mathbb{P}_{K}^{1}$ such that $C$ can be uniformized as the quotient $\Omega/\Gamma$ coincides with the set of non-limit points. This main result on non-archimedean uniformization of curves is given as [8, Theorem 4.20] and [6, Theorems III.2.2, III.2.12.2, and IV.3.10]; many more details are contained in those sources. We comment that in the special case of $g=1$, after applying an appropriate automorphism of $\mathbb{P}_{K}^{1}$ we get $\Omega=\mathbb{P}_{K}^{1}\smallsetminus\{0,\infty\}=K^{\times}$ and that $\Gamma$ is generated by the fractional linear transformation $z\mapsto qz$ for some element $q\in K^{\times}$ of positive valuation, and thus we recover the Tate uniformization $C\cong K^{\times}/\langle q\rangle$ established in [10].

It is shown in [6, §9.2] and [12, §1] (for the $p=2$ case) and in [11, §2] (for general $p$) that given a prime $p$ and a split degenerate curve $C/K$ of genus $(p-1)g$ realized as such a quotient $\Omega/\Gamma$, the curve $C$ is superelliptic and a degree-$p$ cover of $\mathbb{P}_{K}^{1}$ if and only if $\Gamma$ is normally contained in a larger subgroup $\Gamma_{0}<\mathrm{PGL}_{2}(K)$ generated by $g+1$ elements $s_{0},\dots,s_{g}$ whose only relations are $s_{0}^{p}=\dots=s_{g}^{p}=1$.¹¹1In fact, the genus of every split degenerate $p$-cyclic cover of $\mathbb{P}_{K}^{1}$ is divisible by $p-1$, so all such curves arise in this way. In this situation, we have $[\Gamma_{0}:\Gamma]=p$ and $\Omega/\Gamma_{0}\cong\mathbb{P}_{K}^{1}$ so that the natural surjection $\Omega/\Gamma\twoheadrightarrow\Omega/\Gamma_{0}$ is just the degree-$p$ covering map $C\to\mathbb{P}_{K}^{1}$. Each order-$p$ element $s_{i}\in\mathrm{PGL}_{2}(K)$ fixes exactly $2$ points of $\mathbb{P}_{K}^{1}$, which we denote as $a_{i}$ and $b_{i}$.

As in [14], we call the subgroup $\Gamma_{0}<\mathrm{PGL}_{2}(K)$ discussed above a $p$-Whittaker group (see [14, Remark 2.12] for an explanation for this terminology). One can show that we have $a_{i},b_{i}\in\Omega$ for $0\leq i\leq g$: see 4.3 below. Writing $S=\{a_{0},b_{0},\dots,a_{g},b_{g}\}\subset\Omega$, it is easy to verify that the set-theoretic image of $S$ modulo the action of the $p$-Whittaker group $\Gamma_{0}$ coincides with the set of branch points $\mathcal{B}\subset\mathbb{P}_{K}^{1}\cong\Omega/\Gamma_{0}$. For $0\leq i\leq g$, let us write $\alpha_{i},\beta_{i}\in\mathcal{B}$ for the respective images of $a_{i},b_{i}\in\Omega$, so that $\mathcal{B}=\{\alpha_{0},\beta_{0},\dots,\alpha_{g},\beta_{g}\}$. By [11, Proposition 3.1(a)], the superelliptic curve $C$ has an equation of the form

where the term $(x-\alpha_{i})^{m_{i}}$ (resp. $(x-\beta_{i})^{p-m_{i}}$) in the product is replaced by $1$ if we have $\alpha_{i}=\infty$ (resp. $\beta_{i}=\infty$).

In the previous work [14], the author considered a construction proceeding in the other direction: after fixing a prime $p$, we begin with a subset $S\subset\mathbb{P}_{K}^{1}$ of cardinality $2g+2$ for some integer $g\geq 1$ and try to construct a superelliptic curve of genus $(p-1)g$ over $K$ which is uniformized using a Schottky group $\Gamma\lhd\Gamma_{0}=\langle s_{0},\dots,s_{g}\rangle$, where the fixed points $a_{i},b_{i}\in\mathbb{P}_{K}^{1}$ of each generator $s_{i}$ of the associated $p$-Whittaker group are the elements of $S$. It is by no means the case that an arbitrary $(2g+2)$-element set $S$ can be partitioned into pairs $\{a_{i},b_{i}\}$ which each consitute the set of fixed points of an order-$p$ automorphism $s_{i}\in\mathrm{PGL}_{2}(K)$ such that the group $\Gamma_{0}:=\langle s_{0},\dots,s_{j}\rangle$ is a $p$-Whittaker group that can be used to uniformize a superelliptic curve over $K$ (i.e. it may be the case that for every possible partition $S=\bigsqcup_{i=0}^{g}\{a_{i},b_{i}\}$ and every choice of order-$p$ automorphisms $s_{i}$ fixing $a_{i},b_{i}$, the elements $s_{i}$ may satisfy some group relations other than $s_{0}^{p}=\dots=s_{g}^{p}=1$, so that the subgroup of $\mathrm{PGL}_{2}(K)$ that they generate cannot be $p$-Whittaker).

Given a $(2g+2)$-element subset $S$, a partition $S=\bigsqcup_{i=0}^{g}\{a_{i},b_{i}\}$, and a choice of $s_{i}\in\mathrm{PGL}_{2}(K)$ of order $p$ which fixes $a_{i},b_{i}$ for $0\leq i\leq g$, we defined (in [14, Definition 1.1]) the associated subgroups $\Gamma\lhd\Gamma_{0}<\mathrm{PGL}_{2}(K)$ as $\Gamma_{0}=\langle s_{0},\dots,s_{g}\rangle$ and $\Gamma=\langle s_{0}^{j-1}s_{i}s_{0}^{j}\rangle_{0\leq i\leq g,1\leq j\leq p-1}$. If $\Gamma$ is a Schottky group (thus leading to the construction of a superelliptic curve) and $\Gamma_{0}$ cannot be generated by fewer than $g+1$ elements (which then ensures that $\Gamma$ is freely generated by the $(p-1)g$ elements of the form $s_{0}^{j-1}s_{i}s_{0}^{j}$), then we say (as in [14, Definition 1.2]) that $S$ is $p$-superelliptic. It turns out that if a set $S$ is $p$-superelliptic, then there is a unique partition $S=\bigsqcup_{i=0}^{g}\{a_{i},b_{i}\}$ such that the associated group $\Gamma$ is Schottky (regardless of the choice of order-$p$ automorphism $s_{i}\in\mathrm{PGL}_{2}(K)$ fixing $a_{i},b_{i}$, which is unique up to power by [14, Proposition 2.8(a)]²²2While replacing a generator $s_{i}$ by a prime-to-$p$ power does not change $\Gamma_{0}$, it does change $\Gamma$, and it affects the curve $C\cong\Omega/\Gamma$ by changing the integer $m_{i}$ appearing in its defining equation (2): see [11, Proposition 3.2].). Moreover, it is shown that the condition on $S$ of being clustered in $\frac{v(p)}{p-1}$-separated pairs (see 3.3 below) is necessary (though not sufficient in general: see [14, Example 2.18]) for $S$ to be $p$-superelliptic. The above two assertions are given as parts (b) and (a) respectively of [14, Remark 2.15].

The main goal of [14] is to find a method of determining whether a given $(2g+2)$-element subset $S\subset\mathbb{P}_{K}^{1}$ is $p$-superelliptic. An algorithm is provided (as [14, Algorithm 4.2]) in which a non-empty even-cardinality input subset $S\subset\mathbb{P}_{K}^{1}$ is transformed through a sequence of modifications called foldings (which are bijections $\phi:S\to S^{\prime}\subset\mathbb{P}_{K}^{1}$ satisfying certain properties) which do not affect the associated group $\Gamma$, checking at each step that the resulting set $S^{\prime}$ is clustered in $\frac{v(p)}{p-1}$-separated pairs, until eventually, in the case that the input set $S$ is $p$-superelliptic, it is transformed into a set $S^{\mathrm{min}}$ satisfying a property which is called optimality (see [14, Definition 3.12]) for which there are no more “good foldings” to be performed. It is shown in particular (as [14, Lemma 3.18]) that an optimal set is $p$-superelliptic.

Our main purpose in this paper is to compare a $p$-superelliptic set $S$ with its image $\mathcal{B}$ modulo the action of $\Gamma_{0}$ (the branch points of the resulting superelliptic curve), specifically in terms of the combinatorial data of the distances (under the metric induced by the discrete valuation $v$) between elements of $S$ and the corresponding elements in $\mathcal{B}$. In other words, we are interested in comparing the cluster data of $S$ with that of its modulo-$\Gamma_{0}$ image $\mathcal{B}$ (see 2.3 below). The main inspiration for this general question is a conjecture posed by Gerritzen and van der Put on page 282 of their book [6], which may be paraphrased in the language of clusters (as 2.7 below) as essentially stating that a subset $\mathfrak{s}\subset S$ is a cluster if and only its image modulo the action of $\Gamma_{0}$ is a cluster of $\mathcal{B}$. This conjecture was formulated only in a context where $p=2$ and the residue characteristic of $K$ is not $2$.

We are able to resolve this conjecture by showing that it is false in general as stated (see 2.8 below) but that it is true when $S$ satisfies the condition of optimality discussed in the previous subsection. (We have shown as [14, Corollary 3.23, Remark 3.27] that in the special case studied by Kadziela in his dissertation [7, Chapters 5, 6], the set $S$ is optimal, thus recovering Kadziela’s result that the conjecture holds in this case.) We assert this in the more general context of removing all conditions on $p$ and on the residue characteristic of $K$. We are moreover able to directly compare the cluster data of the optimal set $S$ with that of its image $\mathcal{B}$. When $p$ is not the residue characteristic of $K$, this comparison is simple to state; otherwise, a general formula relating the relative depth of a cluster of $S$ with that of its image cluster of $\mathcal{B}$ is too unpleasant to write down here, but such a formula becomes simple in certain cases such as when no cluster of $S$ is itself the union of $\geq 2$ even-cardinality sub-clusters. Our main result (in the cases that are not too cumbersome to write down) is summed up in the following theorem.

We now present our main result not in terms of clusters but in the language of convex hulls in the Berkovich projective line, which removes any need to restrict to certain hypotheses on a given even-cardinality cluster $\mathfrak{s}$ or to assume that $\infty$ lies in either $S$ or $\mathcal{B}$.

The fact that 1.3 implies 1.1 will be given as 3.12 below, while the results in §3.2 will show that 1.3 allows us to compute the relative depths of clusters of $\mathcal{B}$ in the cases not covered by the statement of 1.1, so that 1.3 is stronger than 1.1. We also mention that, as discussed in §1.2, one may use [14, Algorithm 4.2] to turn any $p$-superelliptic set into an optimal set; therefore, 1.3 enables us, given any $p$-superelliptic set $S$, to determine the metric graph isomorphism type of $\Sigma_{\mathcal{B}}$ (or, equivalently, the cluster data of $\mathcal{B}$ by the results of §3.2,3.3 below), where $\mathcal{B}\subset\mathbb{P}_{K}^{1}$ is the set of branch points of the superelliptic curve associated to $S$.

We postpone the proof of our main result to §4 and use §2,3 to establish three different ways to discuss the non-archimedean combinatorial properties of a finite subset $A\subset\mathbb{P}_{K}^{1}$ and how to translate between them, in order to understand the relationship between the conjecture as originally stated by Gerritzen and van der Put and our two variants of it in the form of Theorems 1.1 and 1.3. These three aspects of the set $A$ are its position, its cluster data, and the metric graph properties of its convex hull in the Berkovich projective line. Each of these frameworks has its advantages and its disadvantages. Looking at the position of $A$ is the framework used by Gerritzen and van der Put in many places in their book [6] and is directly tied to constructing useful models of the projective line, but position conveys less information than cluster data and the convex hull do (specifically, the depths of clusters and the lengths of segments of the convex hull are not reflected by the position). The language of cluster data has recently become popular as it relates to many applications involving the arithmetic of hyperelliptic and superelliptic curves ([3, 4] being the originating examples), and cluster data is easy to immediately compute, but it has the downside of being somewhat affected by automorphisms of the projective line (albeit in a way that can be described by easy formulas) while position and metric graph properties of the convex hull are independent of the chosen coordinate of the projective line; it is for this reason that our results stated in terms of clusters often require an extra hypothesis to be articulated succinctly. Moreover, some hypotheses and results are cumbersome to describe in terms of clusters, which is why only certain cases are fully described in the statement of 1.1. Viewing a set $A$ through its convex hull and studying its metric graph properties, while less accessible of a framework than that of position or cluster data, is the most powerful both in terms of ease of stating hypotheses and results and ease of proving them directly. It is for this reason that the result that we will directly prove is 1.3 and our methods of proving it mainly involve convex hulls and other subspaces of the Berkovich projective line.

In §2, we examine the aforementioned conjecture of Gerritzen and van der Put, putting it as precisely as possible by rigorously defining what they mean by “position” of a finite subset of $\mathbb{P}_{K}^{1}$ (in §2.1) and then translating their conjecture into the language of clusters (in §2.2). We then show in §2.3 that without adding the hypothesis that the set $S$ is optimal as in 1.1, the conjecture does not hold.

The object of §3 is then to reframe everything in terms of the convex hull $\Sigma_{S}$ of the subset $S\subset\mathbb{P}_{K}^{1}$ as a subset of the Berkovich projective line $\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$ consisting of points of Type I, culminating in a proof that the “Berkovich version” of our main result (1.3) implies the “cluster version” of our main result (1.1). This is done by introducing (a slightly simplified version of) the Berkovich projective line $\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$ in §3.1, studying convex hulls in $\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$ in §3.2, and directly relating the metric graph structure of the convex hull of $A$ to its cluster data in §3.3.

The actual proof of 1.3 takes up §4, which is the heart of this paper and broken up into five subsections, the first two of which provide background information about the action of $\mathrm{PGL}_{2}(K)$ on convex hulls and the map $\pi:\Omega\twoheadrightarrow\Omega/\Gamma\cong C$ as an explicit theta function, and the last three of which break the proof of 1.3 into three parts. Along the way, we will prove a result (4.10) which approximates the outputs of $\pi$ on a certain subset of its domain $\Omega$, which is interesting in its own right.

We finish the paper with a corollary to our main result which describes a property that generally holds for the branch points of a split degenerate $p$-cyclic cover of the projective line.

The author is grateful to Christopher Rasmussen for helpful discussions that took place during the process of developing these results. The author would also like to thank Robert Benedetto for conversations which enabled him to frame the main result in terms of map on subsets of the Berkovich projective line which is naturally induced by the uniformizing map of the superelliptic curve in the proof of 1.3 and to make the proof more rigorous.

Although the term “position” is not defined in either [6] or [7] in very precise language, one can understand the meaning to be as follows. Let $R\subset K$ be the ring of integers, and let $k$ be the residue field of the local ring $R$. Given an ordered $3$-element subset $\underline{z}=\{z_{0},z_{1},z_{\infty}\}\subset\mathbb{P}_{K}^{1}$, there is a unique automorphism $\gamma_{\underline{z}}\in\mathrm{PGL}_{2}(K)$ which sends $z_{i}$ to $i\in\mathbb{P}_{K}^{1}$ for $i=0,1,\infty$. Composing this with the reduction map $R\to k$, we get a map $\bar{\gamma}_{\underline{z}}:\mathbb{P}_{K}^{1}\to\mathbb{P}_{k}^{1}$. In [6, §I.2], Gerritzen and van der Put define a tree $T(A)$ whose vertices correspond to equivalence classes of ordered $3$-element subsets $\underline{z}$ of $A$, where subsets $\underline{z},\underline{w}\subset\mathbb{P}_{K}^{1}$ are equivalent if the automorphism $\bar{\gamma}_{\underline{w}}\bar{\gamma}_{\underline{z}}^{-1}\in\mathrm{PGL}_{2}(k)$ is invertible. Writing $|T(A)|$ for the set of vertices of the graph $T(A)$, let us define the map

to be the product of the maps $\bar{\gamma}_{\underline{z}}$, where $\underline{z}$ ranges over a set of representatives of each equivalence class of ordered $3$-element subsets $\underline{z}$ of $A$. The image $R_{A}(\mathbb{P}_{K}^{1})$ is a curve over $k$ whose components are all isomorphic to $\mathbb{P}_{k}^{1}$ and intersect only at ordinary double points, none of which is the image of a point in $A$; the components correspond to the vertices of the graph $T(A)$. See [6, §I.4.2] for more details and proofs of these assertions. (We moreover note that the image of $R_{A}$ is in fact the special fiber of a model of $\mathbb{P}_{R}^{1}$ which is minimal with respect to the condition that the images of the $R$-points of this model extending the points in $A$ do not intersect in the special fiber.) We may now define “position” as follows.

It is clear directly from definitions that the position of a subset $A\subset\mathbb{P}_{K}^{1}$ does not change after applying an automorphism in $\mathrm{PGL}_{2}(K)$ to the whole subset. Now the conjecture of Gerritzen and van der Put may be presented as follows.

We introduce the language of clusters and cluster data, following its use in [4], below.

In light of 2.6 above, and keeping in mind that a given element in a finite subset $A\subset\mathbb{P}_{K}^{1}$ may be moved to $\infty$ by applying an appropriate fractional linear transformation to $A$, 2.2 may be restated in the language of cluster data as follows.

Gerritzen and van der Put show in [6, §IX.2.5] that their conjecture holds as stated when $g=1$, and they claim, without showing explicit calculations, that they have confirmed by checking each possible position of a $6$-element subset $S\subset\mathbb{P}_{K}^{1}$ that their conjecture holds also when $g=2$. However, in the following remark we show that there are counterexamples 2.7 even when $g=2$.

The Berkovich projective line $\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$ over an algebraic closure of a discrete valuation field $K$ is a type of rigid analytification of the projective line $\mathbb{P}_{\mathbb{C}_{K}}^{1}$ and is typically defined in terms of multiplicative seminorms on $\mathbb{C}_{K}[x]$ as in [1, §1] and [2, §6.1]. Points of $\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$ are identified with multiplicative seminorms which are each classified as Type I, II, III, or IV. For the purposes of this paper, as in [14], we may safely ignore points of Type IV and need only adopt a fairly rudimentary construction which does not directly involve seminorms.

As is discussed in [14, Remark 2.2], the space $\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$ is path-connected, and there is a unique non-backtracking path between any pair of points in $\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$. This allow us to set the following notation. Below we denote the image in $\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$ of the non-backtracking path between two points $\eta,\eta^{\prime}\in\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$ by $[\eta,\eta^{\prime}]\subset\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$, and we will often refer to this image itself as “the path” from $\eta$ to $\eta^{\prime}$; note that with this notation we have $[\eta,\eta^{\prime}]=[\eta^{\prime},\eta]$. The above observations imply that, given a point $\eta\in\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$ and a subspace $\Lambda\in\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$, there is a (unique) point $\xi\in\Lambda$ such that every path from $\eta$ to a point in $\Lambda$ contains $\xi$; we will often speak of “the closest point in $\Lambda$ to $\eta$” in referring to this point $\xi$ (note that we will use this language even in the case that the distance from $\eta$ to any point in $\Lambda$ is infinite, as in when $\eta\notin\Lambda$ is of Type I). In a similar way, if $\Lambda,\Lambda^{\prime}\in\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$ are subspaces, we will speak of “the closest point in $\Lambda$ to $\Lambda^{\prime}$” (and vice versa). Given a point $\eta\in\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$ and subspaces $\Lambda,\Lambda^{\prime}\in\mathbb{P}_{\mathbb{C}_{K}}^{1,\mathrm{an}}$, we write $\delta(\eta,\Lambda)$ (resp. $\delta(\Lambda,\Lambda^{\prime})$) for the distance between $\eta$ and the closest point in $\Lambda^{\prime}$ to $\eta$ (resp. between the closest point in $\Lambda$ to $\Lambda^{\prime}$ and the closest point in $\Lambda^{\prime}$ to $\Lambda$).