Newton polygons arising from special families of cyclic covers of the projective line

Abstract

By a result of Moonen, there are exactly 20 positive-dimensional families of cyclic covers of the projective line for which the Torelli image is open and dense in the associated Shimura variety. For each of these, we compute the Newton polygons, and the $\mu$-ordinary Ekedahl–Oort type, occurring in the characteristic p reduction of the Shimura variety. We prove that all but a few of the Newton polygons appear on the open Torelli locus. As an application, we produce multiple new examples of Newton polygons and Ekedahl–Oort types of Jacobians of smooth curves in characteristic p. Under certain congruence conditions on p, these include: the supersingular Newton polygon for genus 5, 6, 7; fourteen new non-supersingular Newton polygons for genus 5–7; eleven new Ekedahl–Oort types for genus 4–7 and, for all $g \ge 6$, the Newton polygon with p-rank $g-6$ with slopes 1 / 6 and 5 / 6.

Appendix: Bounding the number of irreducible components of the basic locus of simple Shimura varieties

In this section, we study the basic locus of a certain type of unitary Shimura variety. Under some natural restrictions on the prime p, we prove that the number of irreducible components of the basic locus of its reduction modulo p is unbounded as p goes to $\infty$.

1.1 Notation

Let E be a CM field, Galois over ${\mathbb {Q}}$, and F its maximal totally real subfield. Recall from Sect. 3.1, that $V={\mathbb {Q}}^{2g}$ has a standard symplectic form $\Psi$. There is an action of E on V which is compatible with $\Psi$. In other words, if we view V as an E-vector space, the symplectic form on V naturally induces an E / F-Hermitian form $\psi$ on V.

Using the notation in Sect. 3.2, we consider the PEL-Shimura datum $(H_B, h)$ for $B=E$. In the following, we write $G=H_E$. Note that we have an exact sequence

$$\begin{aligned} 1\rightarrow {{\,\mathrm{Res}\,}}^F_{\mathbb {Q}}U(V,\psi )\rightarrow G\rightarrow {\mathbb {G}}_m \rightarrow 1 \end{aligned}$$

where $U(V,\psi )$ is the unitary group over F with respect to $\psi$. For example, in the notation of Sect. 3.2, when $E=K_m$ is a cyclotomic field, for each embedding $\tau :K_m\rightarrow {\mathbb {C}}$, then the signature of $\psi$ at the real place of F induced by $\tau$ is $({\mathfrak {f}}(\tau ), {\mathfrak {f}}(\tau ^*))$. Note that the reflex field of the Shimura datum (G, h) is contained in E.

For any rational prime p, we fix a prime ${\mathfrak {p}}$ of E above p. In the following, with some abuse of notation, we still denote by ${\mathfrak {p}}$ the corresponding primes of F and of the reflex field. We write ${\mathbb {A}}_f$ for the finite adeles of $\mathbb {Q}$. Let $K\subset G({\mathbb {A}}_f)$ be an open compact subgroup, and denote by $\mathrm{Sh}(G,h)$ the associated Shimura variety of level K. Assume p is unramified in E, and $K=K_pK^p\subset G({\mathbb {Q}}_p)\times G({\mathbb {A}}^p_f)$ with $K_p$ hyperspecial.⁵ Note that this assumption holds for any prime p sufficiently large. Then, we denote by ${\mathcal {S}}$ the canonical integral model of $\mathrm{Sh}(G,h)$ at ${\mathfrak {p}}$, and write ${\mathcal {S}}_{\mathfrak {p}}$ for the $\bmod$ ${\mathfrak {p}}$ reduction of ${\mathcal {S}}$, and ${\mathcal {S}}_{\mathfrak {p}}(\nu _b)$ for its basic locus.

1.2 Main theorem

Theorem 8.1

Assume that the signature of the unitary group $U(V,\psi )$ is $(1,n-1)$ or $(n-1,1)$ at one real place of F and is (0, n) or (n, 0) at any other real place. If ${\mathfrak {p}}\mid p$ is inert in E / F, we further assume that n is even. Then for any such prime $\mathfrak p$, the number of (geometrically) irreducible components of ${\mathcal S}_{\mathfrak {p}}(\nu _b)$ grows to infinity with p.

Remark 8.2

When n is odd or when the signature has another form, the statement of Theorem 8.1 does not hold in general. For example, when the center $Z_G$ of G is connected, Xiao and Zhu show that if the dimension of the basic locus is half the dimension of a Hodge type Shimura variety, then the number of its irreducible components is the same for all unramified primes [39, Lemma 1.1.3, Theorem 1.1.4 (1), Remark 1.1.5 (2), Proposition 7.4.2]. This dimension requirement is satisfied for any unitary Shimura variety with the signature as in Theorem 8.1 and n odd at inert prime ${\mathfrak {p}}$. For more examples, see [39, Remark 4.2.11].

Related material is in [40].

Corollary 8.3

The statement of Theorem 8.1 holds true for any connected component of a product of finitely many unitary Shimura varieties satisfying the assumptions of Theorem 8.1.

Definition 8.4

The group of self-isogenies and certain open compact subgroups Fix a point x in ${\mathcal S}_{\mathfrak {p}}(\nu _b)$, and let $A_x$ denote the associated abelian variety, endowed with the additional PEL-structure. We write $I=I_x$ for the group of quasi-isogenies of $A_x$ which are compatible with the E-action and the ${\mathbb {Q}}$-polarization. Then I is an inner form of the algebraic reductive group G. Furthermore, $I({\mathbb {Q}}_\ell )=G({\mathbb {Q}}_\ell )$ at any finite prime $\ell \not = p$, and $I({\mathbb {R}})/{\mathbb {R}}^*$ is compact (due to the positivity of the Rosati involution).

In the following, we construct an open compact subgroup $C_p$ of $I({\mathbb {Q}}_p)$ such that the mass of $C_pK^p$ gives a lower bound of the number of irreducible components of ${\mathcal {S}}_{\mathfrak {p}}(\nu _b)$.

• (1) Assume ${\mathfrak {p}}$ is split in E / F. Then, $I({\mathbb {Q}}_p)$ decomposes as $I({\mathbb {Q}}_p)={\mathbb {Q}}_p^\times \prod _{v|p}I_v$,⁶ where v runs through all places of F above p. Furthermore, $I_v\cong {{\,\mathrm{GL}\,}}_{n}(F_v)$ for $v\ne {\mathfrak {p}}$, and $I_{\mathfrak {p}}$ is isomorphic to $D^\times _{F_{\mathfrak {p}},n}$ the division algebra over $F_{\mathfrak {p}}$ with invariant 1 / n. We define $C_p$ as the maximal compact subgroup of $I({\mathbb {Q}}_p)$ given by

  $$\begin{aligned} C_p={\mathbb {Z}}_p^\times {\mathcal {O}}^\times _{D_{F_{\mathfrak {p}},n}}\prod _{v|p, v\ne {\mathfrak {p}}}{{\,\mathrm{GL}\,}}_n({\mathcal {O}}_{F_v}), \end{aligned}$$

  where ${\mathcal {O}}^\times _{D_{F_{\mathfrak {p}},n}}\subset D^\times _{F_{\mathfrak {p}},n}$ is the subgroup of elements with norm in ${\mathcal {O}}_{F_{\mathfrak {p}}}^\times$. (For more details, see for example [14, Chapter II.1]).

• (2) Assume ${\mathfrak {p}}$ is inert in E / F. We generalize the discussion in [36, §2], where Vollaard treats the case $F={\mathbb {Q}}$. Here, we follow [23] and recall that n is even in this case.

  Let $V^\bullet$ be the n-dimensional E / F-Hermitian space such that

  • $V^\bullet \otimes _F {\mathbb {A}}_{F,f}^{\mathfrak {p}}\cong V\otimes _F {\mathbb {A}}_{F,f}^{\mathfrak {p}}$ as Hermitian spaces; (we fix such an isomorphism);

  • $V^\bullet$ has signature (n, 0) or (0, n) at all archimedean places (more precisely, we only change the signature by 1 at the one indefinite place of V),

  • $V^\bullet \otimes _F F_{\mathfrak {p}}$ is a ramified Hermitian space over $E_{\mathfrak {p}}/F_{\mathfrak {p}}$.

  Fix an ${\mathcal {O}}_E\otimes {\mathbb {Z}}_p$-lattice $\Lambda ^\bullet _p\subset V^\bullet \otimes _{{\mathbb {Q}}}{\mathbb {Q}}_p$, such that the dual $\Lambda ^{\bullet , \vee }_p$ of $\Lambda ^\bullet _p$ with respect to the Hermitian form satisfies $\Lambda ^\bullet _p\subset \Lambda ^{\bullet ,\vee }_p$ and $\Lambda ^{\bullet ,\vee }_p/\Lambda ^\bullet _p\simeq {\mathcal {O}}_E/{\mathfrak {p}}$ (such lattice exists due to the above assumption on $V^\bullet$). Then I is the unitary similitude group of $V^\bullet$⁷ and we define $C_p\subset I({\mathbb {Q}}_p)$ to be the stabilizer group of $\Lambda ^\bullet _p$.

Proposition 8.5

The number of irreducible components of ${\mathcal S}(\nu _b)_{\mathfrak p}$ is bounded below by the mass of $C_pK^p$,

$$\begin{aligned} m(C_pK^p):=\# I({\mathbb {Q}})\backslash I({\mathbb {A}}_f) /C_pK^p. \end{aligned}$$

Proof

By Rapoport and Zink’s p-adic uniformization theorem [30, Theorem 6.30], to prove the statement it suffices to show that the irreducible components (of a subset) of the Rapoport–Zink space RZ are indexed by $I({\mathbb {Q}}_p)/C_p$.

Assume ${\mathfrak {p}}$ is split in E / F. By Theorem 4.6, there are n possible Newton polygons on ${\mathcal {S}}_{\mathfrak {p}}$ and hence by [13, Theorem 1.1], the basic locus is 0-dimensional. Note that the group $C_p$ is the stabilizer of the Dieudonné lattice of $A_x$. Hence $I({\mathbb {Q}}_p)$ acts on RZ with stabilizer $C_p$ and the number of irreducible components is bounded below by $I({\mathbb {Q}}_p)/C_p$.

Assume ${\mathfrak {p}}$ is inert in E / F. For $F={\mathbb {Q}}$, the statement is [37, Theorem 5.2 (2), Prop. 6.3]. For F totally split at p, the statement is proved in [23]. Here, we sketch a proof that generalizes [23, 37]. The proof is in three steps.

• (1) Construct a 0-dimensional Shimura variety $\mathrm{Sh}(GU(V^\bullet ))$ of level $C_pK^p$, parametrizing abelian varieties $A^\bullet$ of dimension $[F:{\mathbb {Q}}]n$ with ${\mathcal {O}}_E$-action and polarization $\lambda ^\bullet$ satisfying Kottwitz’s determinant condition. Here we assume that the polarization $\lambda ^\bullet$ admits the same polarization type as in $\mathrm{Sh}(G,h)$ outside ${\mathfrak {p}}$; at ${\mathfrak {p}}$ we assume that $(\mathrm{Ker}\lambda ^\bullet )[p^\infty ]=(\mathrm{Ker}\lambda ^\bullet )[{\mathfrak {p}}]$ and that the latter is a finite flat group scheme of order $\#{\mathcal {O}}_E/{\mathfrak {p}}$. To describe the Kottwitz’s condition, let $\Phi \subset \mathrm{Hom}(E,{\mathbb {C}})$ be the subset of $\tau$ such that the signature $({\mathfrak {f}}(\tau ),{\mathfrak {f}}(\tau ^*))$ of $V^\bullet$ is (n, 0). Note that $\Phi \sqcup \Phi ^*=\mathrm{Hom}(E,{\mathbb {C}})$. Kottwitz’s determinant condition says that the characteristic polynomial of the action $b\in {\mathcal {O}}_E$ on ${{\,\mathrm{Lie}\,}}(A^\bullet )$ is given by $\prod _{\tau \in \Phi }(x-\tau (b))^n$.

• (2) We will construct a Deligne–Lusztig variety DL and, for a fixed $A^\bullet$ as in (1), a family of abelian varieties A in ${\mathcal {S}}_{\mathfrak {p}}(\nu _b)$ together with a universal isogeny from $A^\bullet$ parametrized by $DL^{{{\,\mathrm{perf}\,}}}$, the perfection of DL.

  Let $\tau _0$ denote the unique indefinite real place for U(V) and let f denote the inertia degree of ${\mathfrak {p}}$ in F. Consider the $\sigma$-orbit ${\mathfrak {o}}_{\tau _0}=\{ \tau _0, \sigma \tau _0, \cdots , \sigma ^f\tau _0=\tau _0^*,\sigma \tau _0^*, \cdots , \sigma ^{f-1}\tau _0^*\}$. For $\tau \in {\mathfrak {o}}_{\tau _0}$, define $\widetilde{{\varvec{F}}}^f=F_f\circ \cdots F_1$, where $F_i:H_1^\mathrm{dR}(A^\bullet )_{\sigma ^{i-1}\tau _0}\rightarrow H_1^\mathrm{dR}(A^\bullet )_{\sigma ^i \tau _0}$ is equal to ${\varvec{F}}$ if $\sigma ^{i-1}\tau \in \Phi$, and to ${\varvec{V}}^{-1}$ otherwise.⁸ Then, there exists a submodule $M\subset H^\mathrm{dR}_1(A^\bullet )_{\tau _0}\otimes \bar{{\mathbb {F}}}_p$ of rank $n/2+1$ satisfying the condition

  $$\begin{aligned} \widetilde{{\varvec{F}}}^f((M^{(p^f)}))^\perp \subset M, \end{aligned}$$

  where $^\perp$ is taken with respect to the pairing⁹

  $$\begin{aligned} \langle -,-\rangle ^\bullet : H_1^{\mathrm{dR}}(A^\bullet )_{\tau _0}\times H_1^\mathrm{dR}(A^\bullet )_{\tau _0^*}\rightarrow \bar{{\mathbb {F}}}_p. \end{aligned}$$

  The Deligne–Lusztig variety DL is the moduli of all submodules M satisfying the above conditions.

  To construct A over $DL^{{{\,\mathrm{perf}\,}}}$, we use covariant Dieudonné theory and by a theorem of Gabber (see for instance [20, Theorem D]), we only need realize the Dieudonné module of A as a sub-Dieudonné module of $D(A^\bullet )$.

  We define $D(A)_\tau \subset D(A^\bullet )_\tau$ as follows.

  • (a) For $\tau \notin {\mathfrak {o}}_{\tau _0}$: we define $D(A)_\tau =D(A^\bullet )_\tau$ if $\tau \notin \Phi$, and $D(A)_\tau =pD(A^\bullet )_\tau$ if $\tau \in \Phi$.

  • (b) For $\tau \in {\mathfrak {o}}_{\tau _0}$: let $\tilde{M}$ be the preimage of M in $D(A^\bullet )_{\tau _0}$ under the $\bmod$ p map, and $\tilde{M}^*$ the preimage of $M^\perp$ in $D(A^\bullet )_{\tau _0^*}$. We define $D(A)_{\tau _0^*}={\varvec{F}}(\tilde{M})$ and $D(A)_{\tau _0}={\varvec{V}}^{-1}(\tilde{M}^*)$. For $\tau =\sigma ^i\tau _0$, where $1\le i\le f$ (resp. $f+1\le i\le 2f$), we define $D(A)_\tau =\widetilde{{\varvec{V}}}D(A)_{\sigma \tau }$ inductively from $\tau _0^c=\sigma ^f\tau _0$ (resp. $\tau _0=\tau _0^{2f}$), where $\widetilde{{\varvec{V}}}={\varvec{V}}$ if $\tau \notin \Phi$ and $\widetilde{{\varvec{V}}}=p^{-1}{\varvec{V}}$ otherwise.

  Note that by definition the submodule D(A) of $D(A^\bullet )$ is invariant under ${\varvec{F}}$ and ${\varvec{V}}$. Hence, it is a sub-Dieudonné module.¹⁰ Furthemore, the abelian variety A inherits a polarization and additional PEL-structures from those of $A^\bullet$, and satisfies Kottwitz’s determinant condition.

• (3) Show that step (2) constructs an irreducible component of ${\mathcal {S}}_{\mathfrak {p}}(\nu _b)$, or similarly of RZ. Indeed, this can be proven by showing that the image of DL has the correct dimension. On one hand, by the same argument as in the proof of [36, Proposition 2.13] (replacing the p-Frobenius by the q-Frobenius), the dimension of the Deligne–Lusztig variety DL in step (2) is $n/2-1$. On the other hand, by Theorem 4.6, the $\mu$-ordinary Newton polygon $\nu _o({\mathfrak {o}}_{\tau _0})$ has break points

  $$\begin{aligned} (2f,f), \ (2f(n-1), (n-1)f-1), \ \mathrm{and} \ (2fn, fn). \end{aligned}$$

  Hence all possible non-supersingular Newton polygons are in one-to-one correspondence with integer points with abscissa 2ft, for some $t\in {\mathbb {Z}}\cap [1,n/2]$. In particular, by [13, Theorem 1.1] (combined with Theorem 4.6) the basic locus (i.e., the supersingular locus) has codim n / 2, and thus dimension $n/2-1$.

To conclude, we observe that $C_p$ is the stabilizer of this irreducible component under the action of $I({\mathbb {Q}}_p)$ on RZ arising from its natural action on the associated Dieudonné modules.

Proof of Theorem 8.1

By Proposition 8.5, the theorem follows once we provide an asymptotic lower bound for $m(C_pK^p)$ which grows to infinity with p. By [11, Proposition 2.13],¹¹

$$\begin{aligned} m(C_pK^p)= c\cdot \lambda _S \end{aligned}$$

where $\lambda _S=\prod _{p\in S} \lambda _p$, for $\lambda _p$ an explicit local factor at p and S the set of finite places v of ${\mathbb {Q}}$ where $I_v$ is not isomorphic to $G_v$, and $c={2^{-(n[F:{\mathbb {Q}}]+1)}}c'\cdot L(M_I)\cdot \tau (I),$ where

• $M_I$ is a motive of Artin–Tate type attached to I by Gross,

• $L(M_I)$ is the value of the L-function of $M_I$ at 0,

• $\tau (I)$ is the Tamagawa number of I, and

• $c'$ depends only on the non-hyperspecial piece of the level K.

Note that in our case $S=\{p\}$ (see Section 8.4). We claim that c is independent of p. Indeed, the constant $L(M_I)$ only depends on the quasi-split inner form of I over ${\mathbb {Q}}$, which is independent of p; by [18], $\tau (I)$ is independent of p because the center of the neutral connected component of the Langlands dual group of I is independent of p; since K is hyperspecial at p, then $c'$ is independent of p.

Hence, to conclude, it suffices to prove that the local factor $\lambda _p$ is unbounded as p grows to infinity. In [11, Formula (2.12)], the local factor $\lambda _p$ is explicitely computed as

$$\begin{aligned} \lambda _p=\frac{p^{-N(\overline{G}_p)}\cdot \# \overline{G}_p({\mathbb {F}}_p)}{{p^{-N(\overline{I}_p)}\cdot \# \overline{I}_p({\mathbb {F}}_p)}}, \end{aligned}$$

where, for $H=G,I$, the group $\overline{H}_p$ denotes the maximal reductive quotient of the special fiber of $H_p$, and $N(\overline{H}_p)$ denotes the number of positive roots of $\overline{H}_p$ over $\overline{\mathbb {F}}_p$. Note that the integral structure of $I_p$ is given by $C_p$.

Assume p is split in E / F. Then, G and I only differ at ${\mathfrak {p}}$. More precisely, $G_{{\mathbb {Z}}_p}={\mathbb {G}}_m\prod _{v|p} G_v$ and $G_v\cong I_v$ for $v\ne {\mathfrak {p}}$. At ${\mathfrak {p}}$, the group $C_{\mathfrak {p}}={\mathcal {O}}^\times _{D_{F_{\mathfrak {p}},n}}\subset I_{\mathfrak {p}}$ is Iwahori and $\overline{I}_{\mathfrak {p}}$ modulo ${\mathbb {G}}_m$ is a totally non-split torus of rank $n-1$. More precisely, $\overline{I}_{\mathfrak {p}}$ is the multiplicative group of the degree n extension of ${\mathcal {O}}_F/{\mathfrak {p}}$. Hence, for $q=\# {\mathcal {O}}_F/{\mathfrak {p}}$, we have

$$\begin{aligned} \lambda _p=\frac{q^{-N({{\,\mathrm{GL}\,}}_n)}\# {{\,\mathrm{GL}\,}}_n({\mathbb {F}}_q)}{\# \overline{I}_{\mathfrak {p}}({\mathbb {F}}_q)}=\frac{q^{(1-n)n/2}\prod _{i=1}^n(q^n-q^{i-1})}{q^n-1}=\prod _{i=2}^n(q^{n-i+1}-1). \end{aligned}$$

Assume ${\mathfrak {p}}$ is inert in E / F. In Lemma 8.6 below, we verify that the ${\mathcal {O}}_{E}\otimes {\mathbb {Z}}_p$-lattice $\Lambda _p^\bullet$ is maximal. Hence, the computation in [11, §3, Table 2] applies for n even. For $q=\# {\mathcal {O}}_F/{\mathfrak {p}}$, we have

$$\begin{aligned} \lambda _p=(q^{n}-1)/(q+1). \end{aligned}$$

$\square$

Lemma 8.6

Maintaining the same assumptions as in Theorem 8.1, suppose ${\mathfrak {p}}$ is inert in E / F and n is even. Then the ${\mathcal {O}}_{E}\otimes {\mathbb {Z}}_p$-lattice $\Lambda _p^\bullet$ is maximal. More precisely, let ${\mathbb {H}}$ be the split rank 2 Hermitian space over $E_v$ with a standard basis $\{e,f\}$ and let $\Delta$ be the maximal lattice ${\mathcal {O}}_{E,v}e\oplus {\mathcal {O}}_{E,v}f$.

• (1) if $v\ne {\mathfrak {p}}$, then $\Lambda _v^\bullet \cong \Delta ^n$ as Hermitian lattices.

• (2) if $v={\mathfrak {p}}$, then, as Hermitian lattices, $\Lambda _v^\bullet \cong \Delta ^{n-1}\oplus {\mathcal {O}}_D$, where ${\mathcal {O}}_D$ is the maximal order of the unique quaternion algebra over $F_v$.¹²

Proof

Let $\Lambda$ be a maximal lattice containing $\Lambda ^\bullet _p$. We use integral Witt decomposition for the maximal lattice $\Lambda$ to check that $\Lambda$ itself satisfies the duality condition $\Lambda ^\vee /\Lambda \simeq {\mathcal {O}}_E/{\mathfrak {p}}$. Hence, $\Lambda =\Lambda ^\bullet _p$ and it is maximal. (For definition and results on the integral Witt decomposition, we refer to [32, Chapter 1]. Also, note that since p is unramified in $E/{\mathbb {Q}}$, the inverse of the different ideal ${\mathfrak {d}}^{-1}$ is relatively prime to p.) $\square$