A combinatorial Li–Yau inequality and rational points on curves

Abstract

We present a method to control gonality of nonarchimedean curves based on graph theory. Let $k$ denote a complete nonarchimedean valued field. We first prove a lower bound for the gonality of a curve over the algebraic closure of $k$ in terms of the minimal degree of a class of graph maps, namely: one should minimize over all so-called finite harmonic graph morphisms to trees, that originate from any refinement of the dual graph of the stable model of the curve. Next comes our main result: we prove a lower bound for the degree of such a graph morphism in terms of the first eigenvalue of the Laplacian and some “volume” of the original graph; this can be seen as a substitute for graphs of the Li–Yau inequality from differential geometry, although we also prove that the strict analogue of the original inequality fails for general graphs. Finally, we apply the results to give a lower bound for the gonality of arbitrary Drinfeld modular curves over finite fields and for general congruence subgroups $\varGamma$ of $\varGamma (1)$ that is linear in the index $[\varGamma (1):\varGamma ]$, with a constant that only depends on the residue field degree and the degree of the chosen “infinite” place. This is a function field analogue of a theorem of Abramovich for classical modular curves. We present applications to uniform boundedness of torsion of rank two Drinfeld modules that improve upon existing results, and to lower bounds on the modular degree of certain elliptic curves over function fields that solve a problem of Papikian.

Appendix A: Other notions of gonality from the literature

In this appendix, we describe various other notions of graph gonality from the literature, and discuss the relation of stable gonality to these alternatives.

A.1. We first recall the notion of graph gonality from Caporaso [15], but we change the terminology to be compatible with [4, 5] and the current paper. For the convenience of the reader, we include a dictionary between the terminology in [15] and this paper in Table 2.

A morphism between two loopless graphs $G$ and $G'$ (denoted by $\varphi :G\rightarrow G'$) is a map

$$\begin{aligned} \varphi :{{\mathrm{V}}}(G)\cup {{\mathrm{E}}}(G)\rightarrow {{\mathrm{V}}}(G')\cup {{\mathrm{E}}}(G') \end{aligned}$$

such that $\varphi ({{\mathrm{V}}}(G))\subset {{\mathrm{V}}}(G')$, and for every edge $e\in {{\mathrm{E}}}(x,y)$, either $\varphi (e)\in {{\mathrm{E}}}(\varphi (x),\varphi (y))$ or $\varphi (e)\in {{\mathrm{V}}}(G')$ and $\varphi (x)=\varphi (y)=\varphi (e)$; together with, for every $e\in {{\mathrm{E}}}(G)$, a non-negative integer $r_\varphi (e)$, the index of $\varphi$ at $e$, such that $r_\varphi (e)=0$ if and only if $\varphi (e)\in {{\mathrm{V}}}(G')$.

Previously, in Definition 3.6, we only considered finite morphisms, which are morphisms that map edges to edges. The notions of harmonicity and degree that we introduced in Definition 3.6 make sense for morphisms, even if they are not finite. A harmonic morphism is called non-degenerate if $m_\varphi (v)\ge 1$ for every $v\in {{\mathrm{V}}}(G)$ (this is automatic if it is finite).

Table 2 Small dictionary of terminology

A.2. The gonality of a graph is defined to be

$$\begin{aligned} {{\mathrm{gon}}}(G)=\min \{\deg \varphi |\varphi&\text { a non-degenerate harmonic morphism}\\&\text {from } G \text { to a tree } T\}. \end{aligned}$$

Caporaso proves that the gonality of a complex nodal curve is bounded below by the gonality of any refinement of its intersection dual graph.

Lemma A.3

The stable gonality of a graph $G$ is equal to the minimum of the gonalities of all its refinements:

$$\begin{aligned} {{\mathrm{sgon}}}(G) = \min \{ {{\mathrm{gon}}}(G') | G' \text { is a refinement of } G\}. \end{aligned}$$

Proof

It suffices to prove that any non-degenerate harmonic morphism $\varphi :G \rightarrow T$ from a graph $G$ to a tree $T$ admits a refinement $\varphi ' :G' \rightarrow T'$ that is a finite harmonic morphism of the same degree as $\varphi$. Thus, let $e=(v_1,v_2) \in G$ denote an edge that is mapped to a vertex $\varphi (e)=x \in {{\mathrm{V}}}(T)$. Add an extra leaf $\ell$ to $T$ at $x$, subdivide $e$ into two edges $(v_1,m)$ and $(m,v_2)$, and map both $e_1$ and $e_2$ to $\ell$. Set $r_{\varphi '}(e_i)=m_\varphi (v_i)$ for $i=1,2$. Finally, add a leaf $\ell _w$ to all $w \in \varphi ^{-1}(x)$, map them all to $\ell$, and set $r_{\varphi '}(\ell _w)=m_\varphi (w)$. $\square$

The following elementary fact, a “trivial” spectral bound on the gonality, does not seem to have been observed before:

Proposition A.4

The gonality of a graph $G$ is bounded below by the edge-connectivity (viz., the number of edges that need to be removed from the graph in order to disconnect it):

$$\begin{aligned} {{\mathrm{gon}}}(G) \ge \eta (G). \end{aligned}$$

If $G$ is a simple graph (i.e., without multiple edges), unequal to a complete graph, then

$$\begin{aligned} {{\mathrm{gon}}}(G) \ge \lambda _G. \end{aligned}$$

Proof

Let $\varphi :G \rightarrow T$ denote a harmonic non-degenerate morphism. Choose any edge $e \in {{\mathrm{E}}}(T)$. Since removing $e$ from $T$ disconnects it, $\varphi ^{-1}(e)$ is a set of edges of $G$ whose removal disconnects $G$. Hence

$$\begin{aligned} {{\mathrm{gon}}}(G) \ge |\varphi ^{-1}(e)| \ge \eta (G). \end{aligned}$$

For a simple graph which is not complete, the bound

$$\begin{aligned} \eta (G) \ge {\lambda _G} \end{aligned}$$

is one of the inequalities of Fiedler [28], 4.1 & 4.2]. $\square$

Remark A.5

The “trivial” spectral bound in the above proposition is not very useful in practice, since it does not contain a “volume” term (like the Li–Yau inequality). Also, since every graph acquires edge connectivity two or one by refinements, the lower bound in the proposition trivializes under refinements (which are required by the reduction theory of morphisms).

A.6. (Relation with divisorial gonality). Another notion of gonality of graphs $G$ and, more generally, of metric graphs $\varGamma$ was introduced by Baker in [8], defined as the minimal degree $d$ for which there is a $g_d^1$ on $\varGamma$ (in analogy to the definition from algebraic geometry). Following Caporaso, we call this gonality of graphs divisorial gonality. In [14], Caporaso has proven a Brill–Noether upper bound for divisorial gonality. For a fixed unmetrized graph, the relation between gonality and divisorial gonality is studied in [15], especially Examples 2.18, 2.19 and Corollary 3.2.

Since the reduction of a stable curve is naturally a metric graph (cf. [8]), one should not ignore the metric in connection with gonality of curves. Baker has proven that the gonality of a curve $X$ is larger than or equal to the divisorial gonality of its metric reduction graph [8, Cor. 3.2].

Also, the stable gonality of a graph is larger than or equal to its stable divisorial gonality (i.e., the minimum of the divisorial gonality of all refinements).

The banana graph $B_n$ has divisorial and stable gonality $2$ but edge connectivity $n$ (cf. Table 3), showing that an equality analogous to the one in Proposition A.4 cannot hold for divisorial or stable gonality. Dion Gijswijt remarked that $\mathrm {dgon}(G) \ge \min \{ |G|, \eta (G) \}.$ With van Dobben de Bruyn, he has also proven that the divisorial gonality of a graph is larger than or equal to its treewidth (unpublished, but some preliminaries can be found in [60]), but the entries in Table 3 show that the inequality can be strict. Lower bounds on treewidth imply such bounds on divisorial gonality (e.g., [10, 58]).

Table 3 Some graphs and their invariants, including gonalities

A.7. It seems that our notion of stable gonality of a graph coincides with the notion of gonality introduced in [4] from the viewpoint of tropical geometry. The connection between tropical curves and metric graphs can already be found in Mikhalkin [41], and the notion of harmonic morphism of metric graphs in Anand [6].

A.8. We have collected some sample values in Table 3. As above, $\lambda _G$ is the first eigenvalue of $L_G$, and $\lambda _G^\sim$ is the first eigenvalue of the normalized Laplacian $L_G^\sim$; $\eta (G)$ is the edge connectivity, $\varDelta _G$ the maximal vertex degree, ${{\mathrm{\mathrm {vol}}}}(G)$ is the volume of the graph, $\mathrm {tw}(G)$ its treewidth; ${{\mathrm{gon}}}(G)$ is the gonality, $\mathrm {dgon}(G)$ is the divisorial gonality, and $\mathrm {sgon}(G)$ is the stable gonality of $G$. We leave out the lengthy but elementary calculations (for the divisorial gonality of $K_n$, we refer to [8, 3.3]).