2.1 Stable $\mathbb {Z}/\ell \mathbb {Z}$ -covers of a genus zero curve

Let k be an algebraically closed field of characteristic $p> 0$ , and let S be an irreducible scheme over k. Let $G=\mathbb {Z}/\ell \mathbb {Z}$ be a cyclic group of odd prime order $\ell \not = p$ . Let $G^*=G-\{0\}$ .

Let $\psi :Y \to S$ be a semi-stable curve. If $s \in S$ , let $Y_s$ denote the fiber of Y over s. Let $\mathrm {Sing}_S(Y)$ be the set of $z \in Y$ for which z is a singular point of the fiber $Y_{\psi (z)}$ .

A mark $R_\Xi $ on $Y/S$ is a closed subscheme of $Y-\mathrm {Sing}_S(Y)$ which is finite and étale over S. The degree of $R_\Xi $ is the number of points in any geometric fiber of $R_\Xi \to S$ . A marked semi-stable curve $(Y/S, R_\Xi )$ is stably marked if every geometric fiber of Y satisfies the following condition: for each irreducible component $Y_0$ of genus zero, $\# (Y_0 \cap (\mathrm {Sing}_S(Y) \cup R_\Xi )) \ge 3$ . (If the fibers of $Y/S$ have genus $1$ , we also assume that the degree of $R_\Xi $ is positive.)

Consider a G-action $\iota _0: G \hookrightarrow \mathrm {Aut}_S(Y)$ on Y. Let R denote the ramification locus of $Y \to Y/\iota _0(G)$ . The smooth ramification locus is $R_{\mathrm {sm}} := R - (R\cap \mathrm {Sing}_S(Y))$ . We say that $(Y/S,\iota _0)$ is a stable G-curve if $Y/S$ is a semi-stable curve, if $\iota _0: G \hookrightarrow \mathrm {Aut}_S(Y)$ is an action of G, if $R_{\mathrm {sm}}$ is a mark on $Y/S$ , and if $(Y/S,R_{\mathrm {sm}})$ is stably marked.

If $z \in \mathrm {Sing}_S(Y)$ , let $Y_{z,1}$ and $Y_{z,2}$ denote the two components of the formal completion of $Y_{\psi (z)}$ at z. A stable G-curve $(Y/S, \iota _0)$ is admissible if the following conditions are satisfied for every geometric point $z \in R\cap \mathrm {Sing}_S(Y)$ :

1. 1. $\iota _0(1)$ stabilizes each branch $Y_{z,i}$ ;

2. 2. z is a ramification point of the restriction of $\iota _0$ to $Y_{z,i}$ ; and

3. 3. the characters of the action of $\iota _0$ on the tangent spaces of $Y_{z,1}$ and $Y_{z,2}$ at z are inverses.

Suppose that $(Y/S, \iota _0)$ is an admissible stable G-curve. Then $Y/\iota _0(G)$ is also a stably marked curve [Reference Ekedahl8, Prop. 1.4]. The mark on $Y/\iota _0(G)$ is the smooth branch locus $B_{\mathrm {sm}}$ , which is the (reduced subscheme of) the image of $R_{\mathrm {sm}}$ under the morphism $Y \to Y/\iota _0(G)$ . Let n be the degree of $R_{\mathrm {sm}}$ (the number of smooth ramification points). We suppose from now on that $Y/\iota _0(G)$ has arithmetic genus $0$ . By the Riemann–Hurwitz formula, the arithmetic genus of each fiber of Y is

(1) $$ \begin{align} g=(n-2)(\ell-1)/2. \end{align} $$

2.2 The inertia type and signature type

Let s be a geometric point of S, and let a be a point of the fiber $R_{\mathrm {sm},s}$ . Then $G= \mathbb {Z}/\ell \mathbb {Z}$ acts on the tangent space of $Y_s$ at a via a character $\chi _a: G \to k^*$ . In particular, there is a unique choice of $\gamma _a \in (\mathbb {Z}/\ell \mathbb {Z})^*$ so that $\chi _a(1) = \zeta _\ell ^{\gamma _a}$ . We say that $\gamma _a$ is the canonical generator of inertia at a. The inertia type of $(Y/S,\iota _0)$ is the multiset ${\overline {\gamma }} = \{\gamma _{a} \mid a \in R_{\mathrm {sm},s}\}$ . It is independent of the choice of s. By Riemann’s existence theorem, there exists a cover $(Y, \iota _0)$ with inertia type $\{\gamma _{a} \mid a \in R_{\mathrm {sm},s}\}$ if and only if $\sum _{a \in R_{\mathrm {sm},s}} \gamma _a = 0 \in \mathbb {Z}/\ell \mathbb {Z}$ [Reference Völklein24, Th. 2.13].

A labeling of a mark $R_\Xi $ of degree n is a bijection $\eta $ between $\{1, \ldots , n\}$ and the irreducible components of $R_\Xi $ . A labeling of an admissible stable G-curve $(Y/S,\iota _0)$ is a labeling $\eta $ of $R_{\mathrm {sm}}$ . There is an induced labeling $\eta _0:\{1, \ldots , n\} \to B_{\mathrm {sm}}$ .

If $(Y/S,\iota _0,\eta )$ is a labeled G-curve, the class vector is the map of sets $\gamma :\{1, \ldots , n\} \to G^*$ such that $\gamma (i) = \gamma _{\eta (i)}$ . We write $\gamma = (\gamma (1), \ldots , \gamma (n))$ . If $\gamma $ is a class vector, we denote its inertia type by ${\overline {\gamma }}: G^* \to \mathbb {Z}_{\ge 0}$ where ${\overline {\gamma }}(h) = \#\{i \mid 1 \leq i \leq n, \ \gamma (i) = h\}$ for all $h\in G^*$ .

Let $\zeta _{\ell } \in k$ be a primitive $\ell $ th root of unity. The automorphism $\iota _0(1)$ induces an action on $H^0(Y_s, \Omega ^1)$ . Let $\mathcal {L}_i$ be the $\zeta _\ell ^i$ -eigenspace of $H^0(Y_s, \Omega ^1)$ , for $0 \leq i \leq \ell -1$ . There is an eigenspace decomposition:

$$\begin{align*}H^0(Y_s, \Omega^1)=\operatorname{\mathrm{\oplus}}\limits_{i=0}^{\ell-1} \mathcal{L}_i.\end{align*}$$

Let $s_i =\mathrm {dim}(\mathcal {L}_i)$ . Then $\mathcal {L}_0=\{0\}$ and $s_0=0$ since $Y/\iota _0(G)$ has genus $0$ . The signature type is $(s_1, \ldots , s_{\ell -1})$ . It is locally constant on S. For an integer t, let $\langle \frac {t}{\ell } \rangle = \frac {t}{\ell } - \lfloor \frac {t}{\ell } \rfloor $ denote the fractional part of $\frac {t}{\ell }$ .

Lemma 2.1. For $1 \leq i \leq \ell -1$ ,

$$ \begin{align*} s_i=-1 + \sum\limits_{j=1}^{n} \langle \frac{- i a_j}{\ell} \rangle. \end{align*} $$

2.3 Restrictions on the p-rank

Let $\mu _p$ be the kernel of Frobenius on ${\mathbb G}_m$ . The p-rank of a semi-abelian variety $A'$ over k is $f_{A'}=\mathrm {dim}_{{\mathbb F}_p}\mathrm {Hom}(\mu _p, A')$ . If $A'$ is an extension of an abelian variety A by a torus T, then $f_{A'}=f_A + \mathrm {rank}(T)$ .

For an abelian variety A, the p-rank can also be defined as the integer $f_A$ such that the number of p-torsion points in $A(k)$ is $p^{f_A}$ . If A has dimension $g_A$ , then $0 \leq f_A \leq g_A$ . The p-rank of a stable curve Y is that of $\mathrm {Pic}^0(Y)$ .

Recall that $\ell \not = p$ is prime. Let e be the order of p in the multiplicative group $(\mathbb Z/ \ell \mathbb Z)^\times $ .

The p-rank of Y equals the stable rank of the Cartier operator $\mathbf {C}$ . If $\omega \in H^0(Y, \Omega ^1)$ , then $\mathbf {C}(\zeta _\ell ^{pi} \omega )=\zeta _\ell ^i\mathbf {C}(\omega )$ . Then $\mathbf {C}(\mathcal {L}_i) \subset \mathcal {L}_{\sigma (i)}$ where $\sigma $ is the permutation of $\mathbb {Z}/\ell \mathbb {Z}-\{0\}$ which sends i to $p^{-1}i$ . The cycle structure of $\sigma $ is determined by the splitting of p in $\mathbb {Z}[\zeta _{\ell }]$ . Recall that e is the order of p modulo $\ell $ . There are $(\ell -1)/e$ primes of $\mathbb {Z}[\zeta _\ell ]$ lying over p with residual degree e. Each orbit of $\mathbf {C}$ on $\{\mathcal {L}_i\}$ has cardinality e. Let O denote the set of orbits. The contribution to the p-rank from each of the e eigenspaces in an orbit $o \in O$ is bounded by the minimum of $s_i =\mathrm {dim}(\mathcal {L}_i)$ for $\mathcal {L}_i$ in o.

Bouw used these ideas to find an upper bound on the p-rank, which depends only on p, $\ell $ , and the inertia type ${\overline {\gamma }}$ ; it is

(2) $$ \begin{align} B({\overline{\gamma}}):= \sum_{o \in O} e \cdot \mathrm{min}\{s_i \mid \mathcal{L}_i \in o\}. \end{align} $$

3.1 Moduli spaces of stable $\mathbb {Z}/\ell \mathbb {Z}$ -covers

We define moduli functors on the category of schemes over k whose S-points represent the listed objects:

1. 1. $\overline {{\mathcal T}}_{\ell , g}$ : admissible stable G-curves $(Y/S, \iota _0)$ with $Y/S$ of genus g.

2. 2. ${\widetilde {{\mathcal T}}}_{\ell , g}$ : $(Y/S, \iota _0)$ as above, together with a labeling $\eta $ of the smooth ramification locus.

3. 3. ${\widetilde {{\mathcal T}}}_{\ell , g; t}$ : $(Y/S, \iota , \eta )$ as above, together with a mark $R_\Xi $ of degree t such that $(Y/S,R_\Xi )$ is stably marked.

Let ${\mathcal T}_{\ell ,g} \subset \overline {{\mathcal T}}_{\ell , g}$ be the sublocus representing smooth G-curves.

Let $\gamma : \{1, \ldots , n\} \to (\mathbb {Z}/\ell \mathbb {Z})^*$ be a class vector of length $n=n(\gamma )$ . By (1), $\gamma $ determines the genus $g=g(\gamma )=(n-2)(\ell -1)/2$ . Let ${\widetilde {{\mathcal T}}}_{\ell , \gamma } \subset {\widetilde {{\mathcal T}}}_{\ell ,g}$ be the substack for which $(Y/S,\iota _0, \eta )$ has class vector $\gamma $ . Let ${\overline {{\mathcal T}}}_{\ell , {\overline {\gamma }}} \subset {\overline {{\mathcal T}}}_{\ell ,g}$ be the substack for which $(Y/S,\iota _0)$ has inertia type ${\overline {\gamma }}$ .

If two class vectors $\gamma $ and $\gamma '$ yield the same inertia type, so that ${\overline {\gamma }} = {\overline {\gamma }}'$ , then there is a permutation $\varpi $ of $\{1, \ldots , n\}$ such that $\gamma ' = \gamma \circ \varpi $ . This relabeling of the branch locus yields an isomorphism ${\widetilde {{\mathcal T}}}_{\ell , \gamma } \simeq {\widetilde {{\mathcal T}}}_{\ell , {\gamma \circ \varpi }}$ . Suppose $\gamma $ and $\gamma '$ differ by an automorphism of G, so that there exists $\tau \in \mathrm {Aut}(G)$ such that $\gamma '= \tau \circ \gamma $ . This relabeling of the G-action yields an isomorphism ${\widetilde {{\mathcal T}}}_{\ell , \gamma } \simeq {\widetilde {{\mathcal T}}}_{\ell , \tau \circ \gamma }$ .

3.2 Clutching maps

We review the clutching maps $\kappa _{g_1,g_2}$ and $\lambda _{g_1,g_2}$ of [Reference Knudsen15]. Each of these is the restriction of a finite, unramified morphism between moduli spaces of labeled curves. They can be described in terms of their images on S-points for an arbitrary k-scheme S. We give explicit descriptions only for sufficiently general S-points and defer to [Reference Knudsen15] for complete definitions. A stable curve Y has compact type if its dual graph is a tree or, equivalently, if $\mathrm {Pic}^0(Y)$ is represented by an abelian scheme.

For $i=1,2$ , let $\gamma _i$ denote a class vector with length $n_i=n_i(\gamma _i)$ and let $g_i=g(\gamma _i)$ .

3.2.1 Clutching maps (compact type)

There is a closed immersion [Reference Knudsen15, Cor. 3.9]

$$\begin{align*}\kappa_{g_1,g_2}: {\overline{{\mathcal M}}}_{g_1; t_1} \times {\overline{{\mathcal M}}}_{g_2; t_2} \to {\overline{{\mathcal M}}}_{g_1+g_2; t_1+t_2-2}. \end{align*}$$

This clutching map glues two curves $Y_1/S$ and $Y_2/S$ together to form a curve $Y/S$ by identifying the last section of $Y_1$ and the first section of $Y_2$ in an ordinary double point.

As seen in [Reference Achter and Pries3, §2.3], the clutching map extends to the moduli space of labeled $\mathbb {Z}/\ell \mathbb {Z}$ -curves as follows. Let $g=g_1+g_2$ and $n=n_1+n_2-2$ and

$$ \begin{align*}\gamma=(\gamma_1(1), \ldots, \gamma_1(n_1-1), \gamma_2(2), \ldots, \gamma_2(n_2)).\end{align*} $$

If $(Y_i/S, \iota _{0,i}, \eta _i)$ is a labeled G-curve with class vector $\gamma _i$ , for $i=1,2$ , then the clutched curve $Y/S$ has genus g and admits a G-action $\iota _0$ and a labeling $\eta $ with class vector $\gamma $ . Moreover, $Y/S$ can be deformed to a smooth G-curve if and only if the G-action is admissible, that is, if and only if $\gamma _1(n_1)$ and $\gamma _2(1)$ are inverses [Reference Ekedahl8, Prop. 2.2]. In this situation, we write

$$ \begin{align*} \kappa_{g_1,g_2}: {\widetilde{{\mathcal T}}}_{\ell, \gamma_1} \times {\widetilde{{\mathcal T}}}_{\ell, \gamma_2} \to {\widetilde{{\mathcal T}}}_{\ell, \gamma}. \end{align*} $$

By [Reference Bosch, Lütkebohmert and Raynaud6, Ex. 9.2.8], ${\mathop{\mathrm{Pic}}}^0(Y) \simeq {\mathop{\mathrm{Pic}}}^0(Y_1) \times {\mathop{\mathrm{Pic}}}^0(Y_2)$ . In particular, the p-rank of Y is

(3) $$ \begin{align} f(Y) = f(Y_1)+f(Y_2). \end{align} $$

The signature type of $(Y/S, \iota _0)$ is the sum of those for $(Y_i/S, \iota _{0,i})$ .

For $1 \leq g_1 \leq g-1$ , let $\Delta _{g_1}[{\overline {{\mathcal T}}}_{\ell , {\overline {\gamma }}}]$ be the image of $\kappa _{g_1,g_2}$ in ${\overline {{\mathcal T}}}_{\ell , {\overline {\gamma }}}$ , where $g_2=g-g_1$ and $(\gamma _1, \gamma _2)$ ranges over the appropriate admissible pairs of class vectors.

3.2.2 Clutching maps (non-compact type)

In this case, let $g=g_1+g_2 + (\ell -1)$ and $n=n_1+n_2$ and $\gamma = (\gamma _1(1), \ldots , \gamma _1(n_1), \gamma _2(1), \ldots , \gamma _2(n_2))$ . The other clutching maps are

$$\begin{align*}\lambda_{g_1,g_2}: {\overline{{\mathcal T}}}_{\ell, {\overline{\gamma}}_1;1}\times {\overline{{\mathcal T}}}_{\ell, {\overline{\gamma}}_2;1} \to {\overline{{\mathcal T}}}_{\ell, {\overline{\gamma}}}.\end{align*}$$

To define $\lambda _{g_1,g_2}$ , consider a $\mathbb {Z}/\ell \mathbb {Z}$ -curve $(Y_i/S, \iota _{0,i})$ with a mark $P_i$ , for $i=1,2$ . One can glue these curves together to form a curve $Y/S$ by identifying the orbits of $P_1$ and $P_2$ in $\ell $ ordinary double points. (Specifically, identify $\iota _{0,1}(g)(P_1)$ and $\iota _{0,2}(g)(P_2)$ for $g \in G$ .) Then $Y/S$ admits a G-action $\iota _0$ and has inertia type ${\overline {\gamma }}$ .

Since $Y_1$ and $Y_2$ intersect in more than one point, the curve $Y/S$ has non-compact type. By [Reference Bosch, Lütkebohmert and Raynaud6, Ex. 9.2.8], ${\mathop{\mathrm{Pic}}}^0(Y)$ is an extension

$$ \begin{align*} 0 \to Z \to {\mathop{\mathrm{Pic}}}^0(Y) \to {\mathop{\mathrm{Pic}}}^0(Y_1) \times {\mathop{\mathrm{Pic}}}^0(Y_2) \to 0, \end{align*} $$

where Z is an $(\ell -1)$ -dimensional torus. Thus, Y has genus g and the p-rank of Y is

(4) $$ \begin{align} f(Y) = f(Y_1)+f(Y_2)+(\ell-1). \end{align} $$

There is an action of $\mathbb {Z}/\ell \mathbb {Z}$ on Z, and each of the nontrivial eigenspaces has dimension $1$ ; we define the signature type of Z to be $(1, \ldots , 1)$ . The signature type of $(Y/S, \iota )$ is the sum of those for $(Y_i/S, \iota _{0,i})$ and Z.

For $0 \leq g_1 \leq g-(\ell -1)$ , let $\Xi _{g_1}[{\overline {{\mathcal T}}}_{\ell , g}] \subset {\overline {{\mathcal T}}}_{\ell , g}$ be the image of $\lambda _{g_1,g_2}$ , where $g_2=g-g_1-(\ell -1)$ and $(\gamma _1, \gamma _2)$ ranges over the appropriate pairs of class vectors. Let $\Delta _0[{\overline {{\mathcal T}}}_{\ell , g}]$ be the union of $\Xi _{g_1}[{\overline {{\mathcal T}}}_{\ell , g}]$ for $0 \leq g_1 \leq g-(\ell -1)$ . Then $\Delta _0[{\overline {{\mathcal T}}}_{\ell , g}]$ is the set of moduli points of stable $\mathbb {Z}/\ell \mathbb {Z}$ -curves of genus g which are not of compact type.

3.3 Components and dimension of the boundary

The boundary of ${\overline {{\mathcal T}}}_{\ell , g}$ is $\delta {\mathcal T}_{\ell , g} = {\overline {{\mathcal T}}}_{\ell , g} - {\mathcal T}_{\ell , g}$ . If $g \geq 2$ , then $\delta {\mathcal T}_{\ell , g}$ is the union of $\Delta _i = \Delta _{i}[{\overline {{\mathcal T}}}_{\ell , g}]$ for $1 \leq i \leq g-1$ and $\Xi _i = \Xi _i[{\overline {{\mathcal T}}}_{\ell , g}]$ for $0 \leq i \leq g-(\ell -1)$ , some of which may be empty. Note that $\Delta _i$ and $\Delta _{g-i}$ (resp. $\Xi _i$ and $\Xi _{g-i-(\ell -1)}$ ) are the same substack of ${\overline {{\mathcal T}}}_{\ell , g}$ .

If ${\mathcal S}$ is a stack with a map ${\mathcal S} \to {\overline {{\mathcal T}}}_{\ell , g}$ , let $\Delta _i[{\mathcal S}]={\mathcal S} \times _{{\overline {{\mathcal T}}}_{\ell , g}} \Delta _i[{\overline {{\mathcal T}}}_{\ell ,g}]$ . So, $\Delta _i[{\widetilde {{\mathcal T}}}_{\ell , g}] = {\widetilde {{\mathcal T}}}_{\ell , g}\times _{{\overline {{\mathcal T}}}_{\ell , g}} \Delta _i$ . Similar notation is used for $\Xi _i$ .

Proof. Let W be an irreducible component of $\delta {\mathcal T}_{\ell , g}$ . There is an inertia type ${\overline {\gamma }}$ such that W is either a component of (i) $\Delta _i[{\mathcal T}_{\ell , {\overline {\gamma }}}]$ for some $0 \leq i \leq g-1$ or (ii) $\Xi _i[{\mathcal T}_{\ell , {\overline {\gamma }}}]$ for some $0 \leq i \leq g-(\ell -1)$ . By Lemma 3.1(5), it suffices to show that $\mathrm {dim}(W)=n-4$ .

Case (i): In this case, a generic point of W is the moduli point of a singular curve Y with two irreducible components $Y_1$ and $Y_2$ intersecting in one ordinary double point y. Let ${\overline {\gamma }}_i$ be the inertia type of the restriction of the $\mathbb {Z}/\ell \mathbb {Z}$ -action to $Y_i$ , and let $n_i=n({\overline {\gamma }}_i)$ . Then $n_1+n_2 -2 = n$ since y is a ramification point for the two restrictions. So

$$\begin{align*}\mathrm{dim}(W)=\dim({\mathcal T}_{\ell, {\overline{\gamma}}_1}) + \dim({\mathcal T}_{\ell, {\overline{\gamma}}_2})=(n_1-3)+(n_2-3)=n_1+n_2-6=n-4.\end{align*}$$

Case (ii): In this case, a generic point of W is the moduli point of a singular curve Y with two irreducible components $Y_1$ and $Y_2$ , of genera i and $g-i-(\ell -1)$ intersecting at one unramified $\mathbb {Z}/\ell \mathbb {Z}$ -orbit. Let ${\overline {\gamma }}_i$ be the inertia type of the restriction of the $\mathbb {Z}/\ell \mathbb {Z}$ -action to $Y_i$ , and let $n_i=n({\overline {\gamma }}_i)$ . Then $n_1+n_2=n$ . There is a one-dimensional choice of an unramified orbit on each of $Y_1$ and $Y_2$ . So,

$$\begin{align*}\mathrm{dim}(W)=\mathrm{dim}({\mathcal T}_{\ell, {\overline{\gamma}}_1}) + 1 + \mathrm{dim}({\mathcal T}_{\ell, {\overline{\gamma}}_2}) + 1= (n_1-3)+(n_2-3)+2 =n-4. \\[-38pt] \end{align*}$$

The next result is used to find an upper bound on the dimension of the p-rank strata.

Proposition 3.3. If ${\mathcal S} \subset {\overline {{\mathcal T}}}_{\ell ,g}$ has the property that ${\mathcal S}$ intersects $\Delta _i$ , then

$$\begin{align*}\mathrm{dim}({\mathcal S}) \leq \mathrm{dim}(\Delta_i[{\mathcal S}]) +1.\end{align*}$$

Proof. A smooth proper stack has the same intersection-theoretic properties as a smooth proper scheme [Reference Vistoli23, p. 614]. In particular, if two closed substacks of ${\overline {{\mathcal T}}}_g$ intersect, then the codimension of their intersection is at most the sum of their codimensions. Now, $\Delta _i[{\overline {{\mathcal T}}}_{\ell ,g}]$ is a closed substack of ${\overline {{\mathcal T}}}_{\ell ,g}$ . It suffices to consider the case that ${\mathcal S}$ is closed. Thus,

$$\begin{align*}\mathrm{codim}(\Delta_i[{\mathcal S}], {\overline{{\mathcal T}}}_{\ell,g}) \leq \mathrm{codim}(\Delta_i, {\overline{{\mathcal T}}}_{\ell,g}) + \mathrm{codim}({\mathcal S}, {\overline{{\mathcal T}}}_{\ell,g}).\end{align*}$$

The result follows from Lemma 3.2 since $\mathrm {codim}(\Delta _i, {\overline {{\mathcal T}}}_{\ell , g})=1$ .

3.4 The p-rank stratification

If A is a semi-abelian scheme over a Deligne–Mumford stack ${\mathcal S}$ , then there is a stratification ${\mathcal S} = \cup {\mathcal S}^{f}$ by locally closed reduced substacks such that $s \in {\mathcal S}^f(k)$ if and only if $f(A_s) =f$ [Reference Katz14, Th. 2.3.1] (see also [Reference Achter and Pries4, Lem. 2.1]). For example, ${{\mathcal T}}_{\ell , g}^f$ is the locally closed reduced substack of ${{\mathcal T}}_{\ell ,g}$ whose points represent smooth $\mathbb {Z}/\ell \mathbb {Z}$ -curves of genus g with p-rank f.

We use the following notation for the p-rank f stratum of the boundary: $\Delta _i[{\overline {{\mathcal T}}}_{\ell , g}]^f:=(\Delta _i[{\overline {{\mathcal T}}}_{\ell , g}])^f$ . These strata are easy to describe using the clutching maps. First, if $1 \leq i \leq g-1$ , then (3) implies that $\Delta _i[{\overline {{\mathcal T}}}_{\ell , g}]^f$ is the union of the images of ${\widetilde {{\mathcal T}}}_{\ell , i}^{f_1} \times {\widetilde {{\mathcal T}}}_{\ell , g-i}^{f_2}$ under $\kappa _{i, g-i}$ as $(f_1,f_2)$ ranges over all pairs (satisfying Lemma 2.2) such that

$$ \begin{align*} 0 \leq f_1 \leq i,\ 0 \leq f_2 \leq g-i, \text{ and } f_1+f_2=f. \end{align*} $$

Second, if $f \geq 2$ and $0 \leq i \leq g-(\ell -1)$ , then (4) implies that $\Xi _i[{\overline {{\mathcal T}}}_{\ell , g}]^f$ is the union of the images of ${\overline {{\mathcal T}}}_{\ell , i;1}^{f_1} \times {\overline {{\mathcal T}}}_{\ell , g-(\ell -1)-i;1}^{f_2}$ under $\lambda _{i, g-(\ell -1)-i}$ as $(f_1,f_2)$ ranges over all pairs (satisfying Lemma 2.2) such that

$$ \begin{align*} 0 \leq f_1 \leq i,\ 0 \leq f_2 \leq g-(\ell-1)-i, \text{ and } f_1+f_2=f-(\ell-1). \end{align*} $$

3.5 Shimura varieties

We briefly review some notation about Shimura varieties that we need in §§5.3 and 6.2. We refer to [Reference Li, Mantovan, Pries and Tang17, §3.3] for a longer explanation.

Recall that ${\overline {{\mathcal T}}}_{\ell , {{\overline {\gamma }}}}$ is the moduli space of $\mathbb {Z}/\ell \mathbb {Z}$ -covers $Y \to {\mathbb P}^1$ with inertia type ${\overline {\gamma }}$ . By Lemma 3.1, ${\overline {{\mathcal T}}}_{\ell , {{\overline {\gamma }}}}$ is irreducible and has dimension $n({\overline {\gamma }})-3$ .

5.1 Notation for trielliptic covers

Suppose $(Y/S, \iota _0)$ is a smooth trielliptic curve. The $\mathbb {Z}/3 \mathbb {Z}$ -cover $\psi :Y \to \mathbb {P}^1$ has an equation of the form:

(6) $$ \begin{align} y^3=\prod_{i=1}^{d_1} (x-\alpha_i) \prod_{i=1}^{d_2} (x-\beta_i)^2. \end{align} $$

Without loss of generality, we assume that $\psi $ is not branched at $\infty $ . The number of branch points of $\psi $ is $n=d_1+d_2$ and the genus of Y is $g=d_1+d_2-2$ . The inertia type of $\psi $ is ${{\overline {\gamma }}} =(\underbrace {1,\ldots ,1}_{d_1},\underbrace {2,\ldots ,2}_{d_2})$ .

There is a $\mathbb {Z}/3 \mathbb {Z}$ -eigenspace decomposition $H^0(Y, \Omega ^1_Y)= \mathcal {L}_1 \oplus \mathcal {L}_2$ where $\omega \in \mathcal {L}_i$ if $\zeta _3 \circ \omega = \zeta _3^i \omega $ . The signature type of $(Y/S, \iota _0)$ is $(r,s)$ where $r=\dim (\mathcal {L}_1)$ and $s=\dim (\mathcal {L}_2)$ .

If $(Y/S, \iota _0)$ is a trielliptic curve, then so is $(Y/S, \iota ^{\prime }_0)$ where $\iota ^{\prime }_0(1)=\iota _0(2)$ . Replacing $\iota _0$ with $\iota ^{\prime }_0$ exchanges the values of $d_1$ and $d_2$ and the values of r and s.

The next result follows from Lemma 2.1

Lemma 5.3. There is a bijection between trielliptic signatures $(r,s)$ for g and inertia types of $\mathbb {Z}/3 \mathbb {Z}$ -Galois covers of $\mathbb {P}^1$ of genus g given by the formulae

$$\begin{align*}d_1=2r-s+1, \ d_2=2s-r+1.\end{align*}$$

In other words, $r=(2d_1+d_2-3)/3$ and $s=(d_1+2d_2-3)/3$ .

5.2 Components of the moduli space and maximal p-rank

Let $(r,s)$ be a trielliptic signature for g. Let ${\overline {\gamma }}$ be the inertia type given by ${\overline {\gamma }}(i) = d_i$ where $d_i$ are as in Lemma 5.3. Let f be an integer $0 \leq f \leq g$ satisfying the conditions of Lemma 2.2, namely, f is even if $p \equiv 2 \bmod 3$ and $f \not = g-1$ if $p \equiv 1 \bmod 3$ .

Consider the moduli space ${\mathcal T}_{(r,s)}={\mathcal T}_{3, {\overline {\gamma }}}$ of smooth trielliptic curves with signature $(r,s)$ and inertia type ${\overline {\gamma }}$ . For f as above, let ${\mathcal T}_{(r,s)}^f$ denote the p-rank f stratum of ${\mathcal T}_{(r,s)}$ . Similarly, define ${\overline {{\mathcal T}}}^f_{(r,s)}$ by replacing the word smooth by stable.

The next result is a special case of Proposition 4.1.

We first consider the case of maximal p-rank. Define $B(r,s)=g$ if $p \equiv 1 \bmod 3$ and $B(r,s)=2\min \{r,s\}$ if $p \equiv 2 \bmod 3$ . By Theorem 2.3, the p-rank of a trielliptic curve of signature $(r,s)$ satisfies $f \leq B(r,s)$ .

5.3 Base cases

Moonen proved there are exactly 20 families of cyclic covers of ${\mathbb P}^1$ that are special, meaning that the image of the family under the Torelli morphism is open and dense in the associated unitary Shimura variety; these are listed as $M[1]$ – $M[20]$ in [Reference Moonen19, Table 1]. In [Reference Li, Mantovan, Pries and Tang17, §§4–6], the authors computed the Newton polygons occurring on these families. In [Reference Li, Mantovan, Pries and Tang17, Th. 5.11] and [Reference Li, Mantovan, Pries and Tang16, Th. 7.1], they proved that each of these Newton polygons occurs for the Jacobian of a smooth curve in the family, except possibly the supersingular ones when p is small.

For trielliptic covers, there are three families that are special: $M[3]$ , $M[6]$ , and $M[10]$ . Since the p-rank is an invariant of the Newton polygon, we can find the dimension of the p-rank strata of these families. When the Newton polygon is supersingular (which happens only when $f=0$ ), we can remove the requirement that $p>>0$ in all but one case.

The results below for the signature $(r,s)$ are also true for the signature $(s,r)$ .

For the family $M[10]$ , we consider the unitary Shimura variety $\mathrm {Sh}$ attached to the data of $\ell =3$ and the signature type $(3,1)$ . As in §3.5, let $\Sigma $ be the irreducible component of $\mathrm {Sh}$ which contains the Torelli locus.

5.4 Trielliptic curves whose p-rank is not maximal

The next result extends Proposition 5.6 by showing, for each prime $p \geq 5$ , that there exist trielliptic curves of each signature type $(r,s)$ whose p-rank is not the maximum $B(r,s)$ . Recall that $B(r,s) = 2 \mathrm {min}\{r,s\}$ when $p \equiv 2 \bmod 3$ and $B(r,s)=r+s$ when $p \equiv 1 \bmod 3$ .

Proof. Recall that $\mathrm {dim}({\mathcal T}_{(r,s)})=g-1$ and the generic geometric point of ${\mathcal T}_{(r,s)}$ represents a trielliptic curve with p-rank $B(r,s)$ by Proposition 5.6. If ${\mathcal T}_{(r,s)}^{B(r,s)-2}$ is non-empty, then, by definition, each of its generic geometric points represents a smooth trielliptic curve with signature $(r,s)$ and p-rank $f=B(r,s)-2$ . Furthermore, if ${\mathcal S}$ is one of the irreducible components of ${\mathcal T}_{(r,s)}^{B(r,s)-2}$ , then $\mathrm {dim}({\mathcal S}) \leq g-2$ and Proposition 5.5 implies that $\mathrm {dim}({\mathcal S}) \geq g-2$ , so $\mathrm {dim}(S)=g-2$ .

It thus suffices to prove that ${\mathcal T}_{(r,s)}^{B(r,s)-2}$ is non-empty. Without loss of generality, suppose $r \leq s$ . The proof is by induction on r.

If $r=1$ , then $1 \leq s \leq 3$ . If $r=1$ and $s=1$ , then $B(1,1)-2=0$ and the result follows from Lemma 5.7(1) (deferred to Lemma 7.1(1) when $p \equiv 2 \bmod 3$ ). If $r=1$ and $s=2$ , then $B(1,2)-2=1$ when $p \equiv 1 \bmod 3$ and $B(1,2)-2=0$ when $p \equiv 2 \bmod 3$ and the result follows from Lemma 5.7(2). If $r=1$ and $s=3$ , then $B(1,3)-2=2$ when $p \equiv 1 \bmod 3$ and $B(1,3)-2=0$ when $p \equiv 2 \bmod 3$ , and the result follows from Lemma 5.7(3).

Now, suppose $2 \leq r \leq s$ .

Case 1: Suppose $s \leq 2r$ . Then $(r-1,s-1)$ is a valid trielliptic signature. Note that $B(r-1,s-1)=B(r,s)-2$ . Let ${\mathcal S}_1$ be an irreducible component of ${\mathcal T}_{(1,1)}^0$ , which is non-empty when $p \geq 5$ by Lemma 5.7(1). Let ${\mathcal S}_2$ be an irreducible component of ${\mathcal T}_{(r-1,s-1)}^{B(r-1,s-1)}$ , which is non-empty by Proposition 5.6. Consider an irreducible component ${\widetilde {\mathcal S}}_1$ of ${\widetilde {{\mathcal T}}}_{(1,1)}^0$ lying above ${\mathcal S}_1$ and an irreducible component ${\widetilde {\mathcal S}}_2$ of ${\widetilde {{\mathcal T}}}_{(r-1,s-1)}^{B(r-1,s-1)}$ lying above ${\mathcal S}_2$ .

When $(r,s)=(1,1)$ , then $d_1=d_2=2$ are both positive. Thus, without loss of generality, we can choose ${\widetilde {\mathcal S}}_1$ (the labeling of the ramification points) so that the clutching situation below is admissible:

$$\begin{align*}\kappa_{2, g-2}:{\widetilde{{\mathcal S}}}_1 \times {\widetilde{{\mathcal S}}}_2 \to {\overline{{\mathcal T}}}_{(r,s)}^{B(r,s) - 2}.\end{align*}$$

Let $K=\kappa _{2, g-2}({\widetilde {{\mathcal S}}}_1 \times {\widetilde {{\mathcal S}}}_2)$ . By construction, K is contained in $\Delta _2[{\overline {{\mathcal T}}}_{(r,s)}^{B(r,s) - 2}]$ .

Let W be an irreducible component of ${\overline {{\mathcal T}}}_{(r,s)}^{B(r,s) - 2}$ which contains K. By the same reasoning as the first paragraph of the proof, $\mathrm {dim}(W) = g-2$ . On the other hand, since $\mathrm {dim}({\widetilde {{\mathcal S}}}_1)=0$ and $\mathrm {dim}({\widetilde {{\mathcal T}}}_{(r-1,s-1)}) = g-3$ , it follows that

$$\begin{align*}\mathrm{dim}(K) = \mathrm{dim}({\widetilde{{\mathcal S}}}_1) + \mathrm{dim}(\tilde{{\mathcal T}}_{(r-1,s-1)}) = g-3.\end{align*}$$

Thus, the generic point of W is not contained in K.

By construction, the generic point of ${\mathcal S}_1$ represents a smooth curve. The generic point of ${\overline {{\mathcal T}}}_{(r-1,s-1)}^{B(r-1,s-1)}$ represents a smooth curve by Proposition 5.6 and Lemma 3.2. So the generic point of W is not contained in any other boundary component. Thus, the generic point of W represents a smooth curve and ${\mathcal T}_{(r,s)}^{B(r,s)-2}$ is non-empty, with irreducible components of dimension $g-2$ .

Case 2: Suppose $s = 2r+1$ . Then $(r-1,s-2)$ is a valid trielliptic signature. Note that $B(r-1,s-2)=2(r-1)$ when $p \equiv 2 \bmod 3$ and $B(r-1,s-2)=g-3$ when $p \equiv 1 \bmod 3$ . Let $f'=0$ when $p \equiv 2 \bmod 3$ and $f'=1$ when $p \equiv 1 \bmod 3$ . Then $f'+ B(r-1,s-2)=B(r,s)-2$ .

Let ${\mathcal S}_1$ be an irreducible component of ${\mathcal T}_{(1,2)}^{f'}$ , which is non-empty by Lemma 5.7(2). Let ${\mathcal S}_2$ be an irreducible component of ${\mathcal T}_{(r-1,s-2)}^{B(r-1,s-2)}$ , which is non-empty by Proposition 5.6.

When $(r,s)=(1,2)$ , then $d_1=1$ and $d_2=4$ , which are both positive. We repeat the argument above, making an admissible clutching of the following form:

$$\begin{align*}\kappa_{3, g-3}:{\widetilde{{\mathcal S}}}_1 \times {\widetilde{{\mathcal S}}}_2 \to {\overline{{\mathcal T}}}_{(r,s)}^{B(r,s) - 2}.\end{align*}$$

The rest of the proof is the same.

5.5 Existence of trielliptic curves with given p-rank

In this section, suppose $p \equiv 2 \bmod 3$ . In this case, the necessary conditions on the p-rank are that f is even and $0 \leq f \leq 2\mathrm {min}(r,s)$ . For every signature type and for all odd $p \equiv 2 \bmod 3$ , we prove that every p-rank f satisfying the necessary conditions occurs for a smooth trielliptic curve of that signature in characteristic p. If ${\mathcal S}$ is an irreducible component of ${\overline {{\mathcal T}}}_{(r,s)}^f$ , recall from Proposition 5.5 that $\mathrm {dim}({\mathcal S}) \geq \mathrm {max}\{r,s\} -1 + f/2$ .

Proof. The first statement about the existence of the trielliptic curve with signature type $(r,s)$ and p-rank f is equivalent to the statement that ${\mathcal T}_{(r,s)}^f$ is non-empty.

To prove this, without loss of generality, suppose $r \leq s$ . The proof is by induction on r, with the result being true for $r=1$ by Proposition 5.6 when $f=2$ and Lemma 5.7 when $f=0$ . Suppose the result is true for all trielliptic signatures $(r_1,s_1)$ with $1 \leq r_1 < r$ .

Let $(r_2,s_2)$ be either (i) $(1,2)$ or (ii) $(1,1)$ , with choice (i) mandated if $s=2r+1$ and choice (ii) mandated if $s=r$ . Let $g_2=r_2+s_2$ . Let $r_1=r-r_2$ and $s_1=s-s_2$ , and $g_1=r_1+s_1$ . Note that $(r_1,s_1)$ is a trielliptic signature for $g_1$ and $r_1 \leq s_1$ .

By the hypothesis, $0 \leq f \leq 2r$ is even. Let $f_2$ be either (a) 2 or (b) 0, with choice (a) mandated if $f=2r$ and choice (b) mandated if $f=0$ . Let $f_1=f-f_2$ . Then $0 \leq f_1 \leq 2r_1$ and $f_1$ is even.

It follows that ${\mathcal T}_{(r_i,s_i)}^{f_i}$ is non-empty and contains an irreducible component ${\mathcal S}_i$ with $\dim ({\mathcal S}_i) = s_i-1 +f_i/2$ (by the inductive hypothesis when $i=1$ , Propositions 5.6 and 5.9 when $i=2$ ). One can add a labeling of the smooth ramification points by choosing an irreducible component $\tilde {{\mathcal S}}_i$ of $\tilde {{\mathcal T}}_{(r_i,s_i)}$ above ${\mathcal S}_i$ .

By construction, $K=\kappa _{g_1, g_2}(\tilde {{\mathcal S}}_1 \times \tilde {{\mathcal S}}_2)$ is contained in $\overline {{\mathcal T}}_{(r,s)}^f$ and

$$ \begin{align*}\mathrm{dim}(K)=\mathrm{dim}({\mathcal S}_1) + \mathrm{dim}({\mathcal S}_2) = s - 2 +f/2.\end{align*} $$

Then K is contained in a component W of $\overline {{\mathcal T}}_{(r,s)}^f$ . By Proposition 5.5, $\mathrm {dim}(W) \geq s-1 + f/2$ . By Proposition 3.3, $\mathrm {dim}(W) \leq s-1 + f/2$ . Thus, $\mathrm {dim}(W) = s-1 + f/2$ .

Finally, the generic point of W is not contained in K. Since the generic points of ${\mathcal S}_1$ and ${\mathcal S}_2$ represent smooth curves (this requires the hypothesis $p \not = 2$ for case (ii)), the generic point of W is not contained in any other boundary component. Thus, the generic point of W represents a smooth curve. It follows that ${\mathcal S}=W \cap {\mathcal T}_{(r,s)}^f$ is open and dense in W and thus is non-empty with dimension $s-1 + f/2$ .

6.1 Balanced degenerations

In this section, we introduce balanced degenerations which are helpful for finding an upper bound for the dimension of irreducible components. The reason for the balanced condition is that $B(r,s)$ is not additive in general. When $p \equiv 2 \bmod 3$ and $r \leq s$ , then $B(r,s)=B(r_1,s_1)+B(r_2,s_2)$ if and only if $r_1 \leq s_1$ and $r_2 \leq s_2$ .

In Definition 6.1, we implicitly require that $(r_1,s_1)$ and $(r_2,s_2)$ are trielliptic signatures, that $r_1+r_2=r$ and $s_1+s_2=s$ , that $f_i$ are even with $0 \leq f_i \leq 2r_i$ , and that $f_1 + f_2 \leq f$ .

Proposition 6.2. Suppose S has a balanced degeneration to $\Delta ((r_1,s_1)^{f_1}, (r_2, s_2)^{f_2})$ . Suppose, for $i=1,2$ , that

(7) $$ \begin{align} \mathrm{dim}(\tilde {\mathcal T}^{f_i}_{(r_i,s_i)})=\mathrm{dim}({\overline{{\mathcal T}}}^{f_i}_{(r_i,s_i)})=s_i-1+f_i/2. \end{align} $$

Then $\dim (S)= s-1 +f/2$ and S contains $\kappa (S_1 \times S_2)$ , where $S_i$ denotes a component of $\tilde {\mathcal T}^{f_i}_{(r_i,s_i)}$ for $i=1,2$ .

Proof. By Theorem 5.5, $\dim (S) \geq s - 1 +f/2$ . By Proposition 3.3,

$$\begin{align*}\dim(S) \leq \dim(\tilde {\mathcal T}^{f_1}_{(r_1,s_1)})+ \dim(\tilde {\mathcal T}^{f_2}_{(r_2,s_2)}) +1.\end{align*}$$

Since the degeneration is balanced, $r_i \leq s_i$ . By hypothesis,

$$\begin{align*}\mathrm{dim}(\tilde {\mathcal T}^{f_i}_{(r_i,s_i)})=\mathrm{dim}({\overline{{\mathcal T}}}^{f_i}_{(r_i,s_i)})=s_i-1+f_i/2.\end{align*}$$

So,

$$\begin{align*}\dim(S) \leq (s_1-1 +f_1/2) + (s_2-1+f_2/2)+1 \leq s-1 +f/2.\end{align*}$$

Thus, $\dim (S) = s-1+f/2$ . Furthermore, S contains $\kappa (S_1 \times S_2)$ in order for equality to hold in the dimension count.

There is an analogous result for the other boundary components, which we do not need in this paper. In this case, we say S degenerates to $\Xi ((r_1,s_1)^{f_1}, (r_2, s_2)^{f_2})$ if S intersects $\lambda ({\overline {{\mathcal T}}}^{f_1}_{(r_1,s_1);1} \times \tilde {\mathcal T}^{f_2}_{(r_2,s_2);1})$ . In this case, we allow $(r_1,s_1)=(0,0)$ to be a valid trielliptic signature, and require that $r_1+r_2=r-1$ and $s_1+s_2=s-1$ , and $f_1 + f_2 \leq f-2$ .

Proof. The proof is almost the same as for Proposition 6.2. For a $\Xi $ -degeneration, recall that $s_1+s_2=s-1$ , $r_1+r_2=r-1$ , and $f_1+f_2 \leq f-2$ . Marking an orbit increases the dimension by 1, so $\mathrm {dim}({\overline {{\mathcal T}}}^{f_1}_{(r_1,s_1);1})=s_i+f_i/2$ . Then

$$\begin{align*}\dim(S) \leq \dim({\overline{{\mathcal T}}}^{f_1}_{(r_1,s_1);1})+ \dim({\overline{{\mathcal T}}}^{f_2}_{(r_2,s_2);1}) +1,\end{align*}$$

so

$$\begin{align*}\dim(S) \leq (s_1 +f_1/2) + (s_2+f_2/2)+1 \leq s-1 +f/2. \\[-38pt] \end{align*}$$

6.2 A partial generalization of Proposition 5.9

When $f=B(r,s)-2$ , then the p-rank f stratum has codimension $1$ in ${\mathcal T}_{(r,s)}$ , by Proposition 5.9. When $p \equiv 2 \bmod 3$ is odd, by Theorem 5.11, ${\mathcal T}_{(r,s)}^f$ is non-empty for each $0 \leq f \leq 2\mathrm {min}\{r,s\}$ with f even.

When $p \equiv 2 \bmod 3$ is odd, we would like to generalize Proposition 5.9 by showing that every component of the p-rank $f=B(r,s)-4$ stratum has codimension $2$ in ${\mathcal T}_{(r,s)}$ . One reason this is hard to show is because it is not known whether the p-rank strata are nested in each other; specifically, it is not known whether every component of the $f=B(r,s)-4$ stratum is contained in the closure of the $f=B(r,s)-2$ stratum.

In the next result, we are able to extend Proposition 5.9 in this desired way but only under the strong restriction that $\mathrm {min}\{r,s\}=2$ .

In the rest of the section, we prove Corollary 6.4. By symmetry, it suffices to suppose $r=2$ ; then $s=2,3,4,5$ , and we handle these cases separately.

Corollary 6.4 is an application of Theorem 4.3, which we restate in the trielliptic context: suppose ${\mathcal S}$ is an irreducible component of the p-rank $0$ stratum ${\overline {{\mathcal T}}}_{(r,s)}^0$ of ${\overline {{\mathcal T}}}_{(r,s)}$ ; let $\sigma = \mathrm {dim}({\mathcal S})$ ; then there exists $\eta \in {\mathcal S}$ such that the curve $Y_\eta $ of compact type represented by $\eta $ is reducible, with at least $\sigma + 1$ components, such that the $\mathbb {Z}/3 \mathbb {Z}$ -action stabilizes and acts nontrivially on each component.

6.2.1 The case $r=2$ and $s=3$

Lemma 6.5. If S is an irreducible component of ${\overline {{\mathcal T}}}_{(2,3)}^0$ , then $\dim (S)=2$ .

Proof. When the signature is $(2,3)$ , the inertia type is ${\overline {\gamma }}=(1,1,2,2,2,2,2)$ .

By Theorem 5.5, $\dim (S) \geq 2$ . Since ${\mathcal T}_{(2,3)}$ is affine, S intersects either $\Delta _1=\Delta _4$ or $\Delta _2=\Delta _3$ . If S intersects $\Delta _2$ , then S has a balanced degeneration to $\Delta ((1,1)^0, (1,2)^0)$ . By Lemma 5.7, the hypothesis in (7) is true and so $\dim (S)=2$ by Proposition 6.2.

We assume that S does not intersect $\Delta _2$ and that $\dim (S) \geq 3$ and find a contradiction. By Theorem 4.3, S contains a point $\eta $ representing a curve $Y_\eta $ with at least four components. Since $Y_\eta $ has genus $5$ , it has three components of genus $1$ and one component $Y_0$ of genus $2$ (which is possibly reducible).

In the dual graph of $Y_\eta $ , we replace the vertex representing $Y_0$ by two vertices connected by a marked edge. This is illustrated in Figure 1: the schematic represents the four components of the curve, with the branch points marked by their canonical generators of inertia (the admissible condition implies that the two canonical generators of inertia are inverses at each ordinary double point); the schematic in Figure 2 represents the dual graph of Y.

Figure 1

Picture of the singular curve $Y_\eta $

Figure 2

The dual graph of $Y_\eta $

The moduli point of Y is in $\kappa (\tilde {\mathcal T}^0_{(1,0)} \times \tilde {\mathcal T}^0_{(1,3)})$ , but this is not a balanced degeneration. Note that $\mathrm {dim}(\tilde {\mathcal T}^0_{(1,0)})=0$ and $\mathrm {dim}(\tilde {\mathcal T}^0_{(1,3)})=2$ by Lemma 5.7.

By Proposition 3.3, $\mathrm {dim}(S) \leq 3$ . If $\mathrm {dim}(S) = 2$ , the proof is complete. If $\mathrm {dim}(S) =3$ , then there is a component $S_1$ of $\tilde {\mathcal T}^0_{(1,0)}$ and a component $S_2$ of $\tilde {\mathcal T}^0_{(1,3)}$ such that S contains $\kappa (S_1 \times S_2)$ .

Consider the unitary Shimura variety $\mathrm {Sh}$ attached to the data of $\ell =3$ and the signature type $(1,3)$ ; let $\Sigma $ be the irreducible component of $\mathrm {Sh}$ which contains the Torelli locus; and let $\Sigma ^0$ denote the p-rank $0$ stratum of $\Sigma $ . By Proposition 5.8, $\Sigma ^0$ is irreducible.

Consider a (labeled) tree $\eta '$ of four elliptic curves (with dual graph in Figure 3); this is a singular trielliptic curve with signature type $(1,3)$ and p-rank $0$ . Its Jacobian is represented by a point of $\Sigma ^0$ , and thus by a point in the closure of $S_2$ . This implies that the point $\kappa (S_1 \times \eta ')$ is in $\Delta _2[S]$ , which is a contradiction.

Figure 3

A point $\eta '$ of ${\overline {{\mathcal T}}}^0_{1,3}$

6.2.3 The case $(r,s)=(2,5)$

Lemma 6.7. If S is an irreducible component of ${\overline {{\mathcal T}}}_{(2,5)}^0$ , then $\dim (S)=4$ .

Proof. By Theorem 5.5, $\dim (S) \geq 4$ . We assume $\dim (S) \geq 5$ and find a contradiction. By Theorem 4.3, there exists a point $\eta \in S$ such that the curve $Y=C_\eta $ has at least six components. Since Y has genus $7$ , it has five components of genus $1$ and one component of genus 2 (which is possibly reducible).

In the dual graph of Y, we replace the vertex representing the curve of genus $2$ by two vertices connected by a marked edge. One possibility for the dual graph is seen in Figure 4.

Figure 4

The dual graph from the proof of Lemma 6.7.

Regardless of the location of the marked edge, $\eta $ is in $\kappa (\tilde {\mathcal T}^0_{(1,2)} \times \tilde {\mathcal T}^0_{(1,3)})$ . Thus, S has a balanced degeneration to $\Delta ((1,2)^0, (1,3)^0)$ . By Proposition 6.2(1), $\dim (S) \leq 4$ , which gives a contradiction. Hence, $\dim (S) = 4$ .

7.1 A computational approach to the base case

In this section, we prove that ${\mathcal T}^0_{(1,1)}$ is non-empty when $p \equiv 2 \bmod 3$ is odd; this completes the proof of Lemma 5.7(1).

Suppose $\psi :Y \to {\mathbb P}^1$ is a $\mathbb {Z}/\ell \mathbb {Z}$ -cover. In [Reference Elkin9, §§2 and 3], Elkin determines formulas for the action of the Cartier operator $\mathbf {C}$ on $H^0(Y, \Omega ^1)$ . The Cartier operator is a semi-linear operator, and the p-rank is the rank of its gth iterate. For small $\ell $ , g, and p, it is possible to compute the p-rank from Elkin’s work. We note that there are several choices involved in setting up the computation of iterates of the Cartier operator; these have led to mistakes in the literature, and the reader is advised to consult [Reference Achter and Howe2] to avoid repeating these.

We restrict to the case that $\ell =3$ . Without loss of generality, we can suppose that $\psi $ is not branched above $\infty $ . As in (6), there is an equation $y^3 = p_1(x)p_2(x)^2$ for $\psi $ , where $p_1(x), p_2(x) \in k[x]$ are square-free, monic, and relatively prime. Let $d_1=\mathrm {deg}(p_1(x))$ and $d_2=\mathrm {deg}(p_2(x))$ . From Lemma 5.3, recall that the trielliptic signature for $\psi $ is given by $(r,s)$ where $r=(2d_1+d_2-3)/3$ and $s = (d_1+2d_2-3)/3$ . Moreover, $g=d_1+d_2 - 2$ .

We restrict to the case $(r,s)=(1,1)$ and $g=2$ . Then $H^0(Y, \Omega ^1) = \mathcal {L}_1 \oplus \mathcal {L}_2$ where

$$\begin{align*}\mathcal{L}_2=\langle w_{1,1}:=dx/y \rangle, \ \mathcal{L}_1 = \langle w_{2,1}:=p_2(x) dx/y^2 \rangle.\end{align*}$$

We suppose that $p \equiv 2 \bmod 3$ ; then the Cartier operator $\mathbf {C}$ permutes $\mathcal {L}_1$ and $\mathcal {L}_2$ .

Let $e_1=(2p-1)/3$ and $e_2 = (p-2)/3$ . Note that $e_1+e_2=p-1$ . Let

$$\begin{align*}h_1(x)=p_1(x)^{e_1} p_2(x)^{e_2}, \ h_2(x)=p_1(x)^{e_2}p_2(x)^{e_1}.\end{align*}$$

When $r=s=1$ , then $\mathrm {deg}(h_1(x)) = \mathrm {deg}(h_2(x)) = 2p-2$ . In this case, for $i=1,2$ , there is only one monomial $x^e$ in $h_i(x)$ such that $e \equiv -1 \bmod p$ ; let $A_i$ be the coefficient of $x^{p-1}$ in $h_i(x)$ . By a special case of [Reference Elkin9, Th. 3.4],

(8) $$ \begin{align} \mathbf{C}(\omega_{1,j}) =A_1 \omega_{2,1}, \ \mathbf{C}(\omega_{2,j}) =A_2 \omega_{1,1}. \end{align} $$

Elkin provides similar formulas for any prime degree $\ell $ and inertia type ${\overline {\gamma }}$ ; unfortunately, this does not provide a viable way to study the p-rank strata when $g> 2$ and p is large for any $\ell $ , because the equations are algebraically complicated.

7.2 Possibility of components fully contained in the boundary

Here is an important fact about the p-rank stratification of the hyperelliptic locus. When p is odd, the generic geometric point of each irreducible component of ${\overline {{\mathcal H}}}^f_g$ represents a smooth hyperelliptic curve, by [Reference Achter and Pries5, Lem. 3.1].

The analogue of this fact is not true in general for the trielliptic locus. For example, when $p=2$ , then every point of ${\overline {{\mathcal T}}}^0_{(1,1)}$ represents a singular trielliptic curve by Lemma 7.1(2). This is the key reason for the restriction that p is odd in Proposition 5.9 and Theorem 5.11.

We give some other examples where this type of problem may occur.

The reason for the problem in Example 7.2 is that the pair $(1,0)$ and $(1,2)$ is not balanced, as in Definition 6.1. The inductive step for producing smooth trielliptic curves of larger genus and given p-rank works only when the pair of signatures is balanced.

More generally, the problem seen in Example 7.2 arises when $p \equiv 2 \bmod 3$ is odd if $r \leq s \leq 2r-1$ and $f \leq 2r-2$ and when $p=2$ if $3 \leq r \leq s \leq 2r -2$ .