Explicit Moduli of Superelliptic Curves with Level Structure

arXiv:2107.00988v1 [math.AG] 2 Jul 2021 ymti group symmetric ogce n rln lopsdtequestion: the posed of also Ortland and Dolgachev opnnsof components yeelpi uv aie vrteepit,te n can one then the for points, bases symplectic these scribe over ramified curve hyperelliptic ieetsmlci level symplectic different group ec,i n ae itntpoints distinct takes one combinati if linear zero Hence, degree to precisely correspond curve mohrtoa uvswith curves rational smooth h coset the ouisaeo yeelpi uvswt level with curves hyperelliptic of space moduli opnns hs fw let we if Thus, components. [ Runge reuil opnn of component irreducible of aua a ooti fl)smlci level symplectic (full) it a points, obtain branch to the way for natural ordering an choosing after Indeed, P Hyp sn hscntuto,DlahvadOtad[ Ortland and Dolgachev construction, this Using -aladdresses University Un E-mail Stockholm Sciences, Mathematics, Natural of and Mathematics Dept. Electronics, of Dept. sMmoddsrbsi [ in describes Mumford As 1 XLCTMDL FSPRLITCCRE WITH CURVES SUPERELLIPTIC OF MODULI EXPLICIT P , . . . , Sp(2 g Run97 [2] Abstract. S ujc lsicto:1D2 42,1H0 44,14H 14H45, 14H10, 14D23, 14D22, Classification: Subject MSC th Hurwitz spaces, Moduli curves, Superelliptic Keywords: Dolgachev to the due Tsuyumine. of structure, and description 2 clo level the the a with generalise of to curves article components leads perelliptic the and irreducible) of line (and results projective connected The the of on number sets the point Th of structure. terms 3 level in with curves superelliptic trigonal of c ? g, ∈ 2 g hsqeto a nwrdidpnetyb symn [ Tsuyumine by independently answered was question This +2 F C ,wosoe htteirdcbecmoet r loteco the also are components irreducible the that showed who ], 2 hnteei nisomorphism an is there then , Hyp ) n a hnoti n ee tutr on structure 2 level any obtain then can One . ntpo hs ic ahcoc fodrn ilgv iet rise give will ordering of choice each since this, of top On . : S [email protected] [email protected] [email protected], 2 nti ril egv nepii osrcino h modul the of construction explicit an give we article this In g g +2 [2] LFBRVL N LVRLEIGH OLIVER AND BERGVALL OLOF notesmlci group symplectic the into r nee ytesto cosets of set the by indexed are 2 hscntuto aual ensa medn fthe of embedding an defines naturally construction this , Hyp EE STRUCTURE LEVEL X 2 2 c g Mum84 trinJacobian -torsion g [2] eavreyioopi to isomorphic variety a be 1. Hyp 2 + P Introduction sioopi to isomorphic is 1 3 P , . . . , [2] itntmrig,adta h irreducible the that and markings, distinct §2], , = ∼ 1 c G 2 ∈ g C +2 2 trindvsr nahyperelliptic a on divisors -torsion 2 2 X on tutr on structure Jac( tutr.Te hwdta each that showed They structure. c htaetecnetcomponents connect the are What . Sp(2 P DO88 M c 1 )[2] n considers and 0 g, , osrcini given is construction e 2 n frmfiainpoints. ramification of ons g F n rln,Runge Ortland, and ouisaeo hy- of space moduli ntrsof terms in considered ] +2 eory C M 2 un u htteei a is there that out turns ) h ouisaeof space moduli the , . 0 oet xlctyde- explicitly to hope e oml for formula sed C =Sp(2 := , C 2 . ouispace. moduli 51 g vriyo Gävle of iversity rmcombinations from +2 i h symplectic the via n nee by indexed and P space i C g, 1 Hyp h unique the , P , . . . , Tsu91 F 2 ) g nnected /S [2] and ] 2 2 the , g g a o +2 +2 . . 2 SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE

It is natural to ask which parts of the above discussion extend to other types of ramiﬁed covers. In the present article we proceed in this direction by investi- gating cyclic covers of P1 called superelliptic curves. We pay special attention to the simplest non-hyperelliptic case, namely trigonal curves. We ﬁnd (somewhat surprisingly) an almost perfect analogy between the case of cyclic trigonal curves with symplectic level 3 structure and hyperelliptic curves with level 2 structure. Our main result is the following. Theorem 1.1. Let g be a positive integer and take the unique integers n and k 3 such that 0 ≤ k ≤ 2 and g +2=3n − k. The moduli space Supg[3] of trigonal superelliptic curves with level 3 structure has

1 1 |Sp(2g, F )| + 3  2  − (3i + k)!(g +2 − (3i + k))! 2(i!) 0≤i<⌈ g+2 2k ⌉ {i∈Z : i= g +1}  X 6 X2  connected components. Each connected component is irreducible and is isomorphic to the moduli space M0,g+2 of smooth rational curves with g +2 marked points. It is plausible that the above theorem can be used to pave a way towards the conjectures of Bergström-van der Geer [Bv20, Sec. 12] and the present work may be useful in the moduli theory related to 2-spin Hurwitz numbers via [Lei18] and [Lei20]. Other possible directions for future work include cohomological computations, see e.g. [BB20] and [Ber19], and investigations of special kinds of superelliptic curves, e.g. Picard curves, Belyi curves or plane curves. Relevant previous work include Accola’s characterization of cyclic trigonal curves in terms of vanishing properties of theta functions [Acc84], Kontogeorgis’ study of automorphism groups of rational function fields [Kon99], Kopeliovich’s computations of Thomae formu- lae for cyclic covers of the projective line [Kop10] and Wangyu’s characterization of cyclic covers of the projective line with prime gonality [Wan15]. For further references, and a survey of the field, see [MS19]. The paper is structured as follows. In Sections 2 and 3 we review the necessary background on hyperelliptic and superelliptic curves. We also rephrase some of the classical material into a setup which is mutually compatible and generalizable. In Section 4 we study divisors on superelliptic curves. In particular, we study divisors generated by ramification points and how the Weil pairing behaves on pairs of such divisors. Finally, in Section 5 we specialize to the case of trigonal curves. Among other results, we obtain an explicit description of the moduli space of superelliptic trigonal curves with level 3 structure and an explicit formula for the number of connected (and irreducible) components of this space (see Theorem 1.1 above).

2. Background on Hyperelliptic Curves 2.1 (Overview of section). In this section we will recall facts about hyperelliptic curves and give an overview of an explicit construction of the moduli space of hyperelliptic curves with level 2 structure. This construction, which is based upon ﬁnite subsets of the projective line, can be mainly attributed to Mumford [Mum84] and Dolgachev-Ortland [DO88]. We will present this material in a way that can be readily generalised to superelliptic curves. Deﬁnition 2.2 (Hyperelliptic curves). A hyperelliptic curve is a degree 2 morphism π : C → P1 such that C is a smooth (connected) curve of genus g> 1. SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE 3

Two hyperelliptic curves π : C → P1 and π′ : C′ → P1 are isomorphic if there are isomorphisms α : C → C′ and β : P1 → P1 such that π ◦ α = β ◦ π′. Remark 2.3 (Equivalent defintion of hyperelliptic curve). A much more standard definition of a hyperelliptic curve is a curve C with genus g> 1 such that a degree 2 morphism π : C → P1 exists. However, it is well known that any such degree 2 morphism is unique up to isomorphism, so the two definitions are equivalent (see for example [Har77, IV Prop 5.3]). We use the definition of hyperelliptic curves from definition 2.2 since it is more in-line with the definition of superelliptic curves which will be given in definition 3.2. 2.4 (Moduli space of hyperelliptic curves). Definition 2.2 extends naturally to give a moduli functor. We denote by Hypg the (coarse) moduli space parametrising hyperelliptic curves. Moreover, by the uniqueness of the hyperelliptic morphism .pointed out in remark 2.3, we have that there exists a natural inclusion Hypg ֒→ Mg The image of this inclusion is called the hyperelliptic locus. Remark 2.5. In this article the main focus is on the coarse moduli scheme of hyperelliptic curves Hypg instead of the moduli stack of hyperelliptic curves Hypg. A key difference between the two is that Hypg keeps track of the automorphisms of the hyperelliptic curves. The stack theoretic viewpoint for this space and the analogous superelliptic case (appearing in section 3) was considered in [AV04].

Remark 2.6. Since we have defined Hypg as a moduli space of maps, this moduli space can also naturally be viewed as the Hurwitz space Hurg(2). 2.7 (Affine models for hyperelliptic curves). It is a well known result that a genus g hyperelliptic curve π : C → P1 has a non-unique affine model of the form s2 − f(t)=0 where f ∈ C[t] has unique roots and is either of degree 2g +1 or 2g +2. (see for example [Mum84, p. 3.28]). Moreover, if C◦ is the smooth locus of the above affine variety, then C is the the smooth completion of C◦ and π is the associated projection map arising from the projection to the t coordinate. A second chart for C can be defined by the equation 0= up −h(v) where h ∈ C[v] 2g+2 1 is the unique polynomial with h(v)= v f( v ). The change of coordinates is then 1 u defined by (t,s) 7→ v , vg+1 . The equivalence relation between two isomorphic affine models is described via at + b a b s2 − f(t) ∼ s2 − (ct + d)2g+2f for ∈ GL(2, C). ct + d c d Note that if s2 − h(t) is the resulting polynomial on the right hand side, f and h can have degrees differing by 1 depending on whether a root was moved to or away from infinity. 2.8 (Construction via choosing ramification points). Consider the relationship between the affine model and a hyperelliptic curve π : C → P1 described in 2.7. An immediate observation from this relationship is that the ramification points of π occur at the roots of f(t) for deg(f)=2g +2 and also at π−1(∞) if deg(f)=2g +1. A result of this observation is that one can construct a unique hyperelliptic curve by choosing a set of 2g +2 points in P1. Taking into account the equivalence described in 2.7 we arrive at the following theorem. 4 SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE

Theorem 2.9 (Scheme structure of Hypg, [Fis56, LK79]). There is an isomorphism of schemes

M0,2g+2/S2g+2 −→ Hypg. Remark 2.10. Versions of the isomorphism from theorem 2.9 also exist for compact moduli moduli spaces e.g. [HM98, Ex. 6.25] and [AL02, Cor. 2.5]. Convention 2.11 (Notation for choice of hyperelliptic curve). For the remainder of this section we will assume that π : C → P1 is a hyperelliptic curve of genus g. Moreover, given the equivalence of affine models described in 2.7, we will make the assumption that ∞ ∈ P1 is a branch point of π and the affine model is 2g+1 2 0= s − (t − ai). i=1 Y −1 −1 We will denote the ramification points as Qi := π (ai) and Q∞ := π (∞). 2.12 (Natural principal divisors). Recalling convention 2.11 we observe that there are natural principal divisors on C given by: u 2g+1 (i) Horizontal: (s)= vg+1 = i=1 (Qi − Q∞). (ii) Vertical ramified: (t − a )= 1−aiu =2 Q − 2 Q . i P u i ∞ (iii) Vertical unramified: (t − a)= 1−au = π∗P − 2 Q . u a ∞ ∗ 1 Here a ∈ C \{a1,...,a2g+1} and Pa ∈ P is the corresponding divisor. The divisor ∗ π Pa consists of 2 distinct points. 2.13 (2-torsion in the Jacobian of a hyperelliptic curve). A key focus of this section is related to the study of 2-torsion within the Jacobian of hyperelliptic curves. In this direction, an immediate observation from 2.12 is that for i ∈ {1,..., 2g +1} we have 2 · (Qi − Q∞) ∼ 0.

This shows that each Di := [Qi − Q∞] is a natural element of Jac(C)[2]. We can also now consider the natural F2-vector subspace spanned by the classes

∆ := F2-Span D1,...,D2g+1 ⊆ Jac(C)[2]. On top of this, the horizontal principaln divisoro from 2.12 gives the relationship 2g+1 0 = i=1 Di. This shows that ∆ is spanned by any choice of 2g classes from D1,...,D2g+1. Indeed, it was shown by Mumford that any of these choices gives a P ∼ 2g basis for ∆ since ∆ = F2 . Moreover, combining this result with a basis constructed by Dolgachev and Ortland gives the following key result. Proposition 2.14 (A symplectic basis for Jac(C)[2], [Mum84, IIIa Lem. 2.5] & ∼ ∼ 2g [DO88, §3 Lem. 2]). There are F2-linear isomorphisms Jac(C)[2] = ∆ = F2 . The basis (A1,...,Ag,B1,...,Bg) of Jac(C)[2] deﬁned by

Ai := D2i−1 + D2i and Bi := D2i + ··· + D2g+1, is symplectic with respect to the Weil Pairing. Moreover, the linear map taking this 2g basis to the standard symplectic basis for F2 is an isometry. Remark 2.15. The methods used in [DO88, §3 Lem. 2] to prove Proposition 2.14 are more analytic in nature than the methods employed in this article. Indeed, the authors of [DO88] consider π : C → P1 as a two-sheeted covering and consider SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE 5 a symplectic basis of cycles in H1(C, Z) corresponding to the basis elements of Proposition 2.14. To be more precise, Ai corresponds to a path which goes from Q2i−1 along one sheet to Q2i and then returns along the other sheet, while Bi corresponds to a path which goes from Q2i along one sheet to Q2g+1 and then returns along the other sheet. The symplectic basis of cycles in H1(C, Z) is then combined with a 0 normalised basis of H (C, ΩC ) to construct a branch point period matrix. 2.16 (Inclusion of the symmetric group into the symplectic group). The symmetric group S2g+2 permutes the points Q1,...,Q2g+1,Q∞ and thus deﬁnes an action on the set D := {D1,...,D2g}. Since any permutation of Q1,...,Q2g+1,Q∞ will give the same hyperelliptic curve, the action of S2g+2 on D extends to give a change of symplectic basis. This deﬁnes a group homomorphism S2g+2 → Sp(2g, F2) which is in fact an embedding (see [Ber16, p. 60] for details and further references).

2.17 (Natural morphism to Hypg[2]). We denote the (coarse) moduli space of hyperelliptic curves with level-2 structure by Hypg[2]. This is a moduli space parametrising pairs consisting of a hyperelliptic curve π : C → P1 and an isometry 2g 2g η : F2 → Jac(C)[2], where F2 uses the standard symplectic form and Jac(C)[2] uses the Weil pairing. Now, Proposition 2.14 (which extends easily to families) naturally gives a morphism

M0,2g+2 → Hypg[2]. Moreover, it is shown in [DO88, §3 Thm. 1] that this morphism is an isomorphism onto an irreducible component of Hypg[2]. On top of this, if we consider the morphism that forgets the isometry, Hypg[2] → Hypg, then we obtain the following commutative diagram.

M0,2g+2 Hypg[2]

=∼ M0,2g+2/S2g+2 Hypg

In fact, the morphism M0,2g+2 → Hypg[2] actually deﬁnes an isomorphism to a connected component of Hypg[2]. Using this construction one can describe all of the of connected components of Hypg[2] and arrive at the following theorem.

Theorem 2.18 (Decomposition of Hypg[2], [Tsu91, Thm. 2] & [Run97, Thm. 4.1]). Consider the inclusion of the symmetric group in the symplectic group described in 2.16 and denote the quotient set Sp(2g, F2)/S2g+2 by C . Then, if Xc denotes a copy of M0,2g+2 indexed by c ∈ C , there is an isomorphism of schemes =∼ Hypg[2] −→ Xc, c∈C G so Hypg[2] is smooth and the number of connected components of Hypg[2] is given by |Sp(2g, F2)|/|S2g+2|. Remark 2.19. The result of Tsuyumine from [Tsu91, Thm. 2] is actually slightly diﬀerent from the one stated in theorem 2.18. He shows the more general result that Hypg[n] has |Sp(2g, F2)|/|S2g+2| connected components whenever n is divisible by 2. However, components are only explicitly computed in the case n =2. 6 SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE

Remark 2.20. Modulo minor mistakes (pointed out in [Run97]), Dolgachev-Ortland [DO88, §3] presented a weaker version of the above theorem showing that the irre- C ducible components of Hypg[2] are indexed by and are isomorphic to M0,2g+2. Remark 2.21. Historically, a key purpose of level structures is to rigidify moduli problems. That is, to eliminate the automorphism groups of the objects being parametrised. However, one observation from the constructions in this section is that level-2 structures do not rigidify hyperelliptic curves. To see this, we note that the symplectic basis given in Proposition 2.14 is invariant under the hyperelliptic involution.

3. Background on Superelliptic Curves 3.1. In this section we will take the results from section 2 as inspiration and consider a generalisation of hyperelliptic curves. To be precise, we will consider cyclic morphisms of degree p where p is a prime. We note, however, that many of these concepts can also be extended to the non-prime case. We will also assume that g> 0. Definition 3.2 (Superelliptic curves). A superelliptic curve is a degree p morphism π : C → P1 such that C is a smooth (connected) curve and the Galois group of 1 1 π is cyclic. Two superelliptic curves π1 : C1 → P and π2 : C2 → P are said to 1 1 be equivalent if there are isomorphisms ψ : C1 → C2 and ϕ : P → P such that π2 ◦ ψ = ϕ ◦ π1. 3.3 (Moduli space of superelliptic curves). Definition 3.2 extends naturally to give p a moduli functor. We denote by Supg the (coarse) moduli space of superelliptic curves of degree p and genus g. Here are some well known relationships with other moduli spaces: (i) For d = 2 and g > 1, definition 3.2 matches with that of hyperelliptic ∼ 2 curves and gives rise to an isomorphism Hypg = Supg. This is because a hyperelliptic curve uniquely determines a morphism π : C → P1 of degree 2 and because the Galois group of such a morphism is automatically cyclic. p (ii) There is a natural morphism Supg → Mg to the (coarse) moduli space of smooth genus g curves. In general, this morphism is not an immersion. However, it is an immersion in special cases such as for d =3 when g> 4. (iii) For g> 2, a curve cannot be both hyperelliptic and degree 3 superelliptic, 2 3 meaning that the images of Supg and Supg are disjoint in Mg. 3.4 (Ramification of superelliptic curves). Degree p superelliptic curves have the property of being totally ramified at each ramification point. In other words, each branch point has a unique preimage. Hence, by the Riemann-Hurwitz formula the number of ramification points is 2g m = +2. p − 1 This places a strong condition on the possible choices for g and p. An equivalent condition on g is obtained by specifying m and p to give (p − 1)(m − 2) g = . 2 We note that when p = 2 or p =3, superelliptic curves exist for all g. Indeed, for d =2 we recover m =2g +2 as in Section 2 and when d =3 we obtain m = g +2. SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE 7

3.5 (Affine model for superelliptic curves). It is a well know result (see for example [MS19, Sec. 5]) that a superelliptic curve π : C → P1 has a non-unique affine model given by 0= sp − f(t) where f ∈ C[t] has roots with orders less than p. In particular, if C◦ is the smooth locus of the above affine variety, then C is the the smooth completion of C◦ and π is the associated projection map arising from the projection to the t coordinate. We can construct an affine cover of C by considering a second chart. First, define an integer via the ceiling function n =: ⌈deg(f)/p⌉ and also κ := pn − deg(f). We now define the second chart by 0= up − vκh(v) deg(f) 1 where h ∈ C[v] is the unique polynomial with h(v) = v f( v ). The change of 1 u coordinates is then defined by (t,s) 7→ v , vn . 3.6 (Superelliptic plane curves). An immediate observation from the two chart cover constructed in 3.5 is that the (singular) affine models for super elliptic curves are naturally embedded in the weighted projective plane P(1, 1,n). Furthermore, in the special case where f has no multiple roots and κ = 0 or κ =1, we have that the smooth superelliptic curve itself is a sub-variety of P(1, 1,n). Lemma 3.7 (Equivalent affine models for superelliptic curves). The equivalence relation of superelliptic curves from definition 3.2 is extended to affine models by the relations 2 2 2g+2 at+b a b (i) s − f(t) ∼ s − (ct + d) f ct+d , for c d ∈ GL(2, C); (ii) sp − (t − a )kj ∼ sp − (t − a )ζ·aj , for ζ ∈ F∗ and where ζ · a is j j j j p j multiplication in Fp. Q Q Proof. Property (i) is straight-forward and also arises in the hyperelliptic case, hence we consider only property (ii). For this we define η : C → P1 and η′ : C′ → P1 p kj p ζ·aj to be the superelliptic curves defined by s − j (t − aj ) and s − j (t − aj ) where p divides the sum k . Also, since k ∈ F∗ for each j as well as ζ ∈ F∗, we j j iQ p Q p can assume that k1 =1. We begin by consideringP the monodromies on C arising from the ramification points. Let x ∈ P1 be distinct from the branch points and consider m different loops on P1 starting and finishing at x. The constraint for the loops is that each contains exactly one branch point and different loops contain different branch points. The monodromy arising from the preimages of the loop around ai can be described by considering the equation sp = r while considering the preimage of the unit circle and identifying x with 1. In this case, the preimage of the unit circle is τ πi τ 2πi e 2 ,e p t ∈ [0,p] . n o The points in the preimage of 1 are the p-th roots of unity and we label them by 2πi their power of the primitive root of unity e p . Then the associated permutation from the monodromy representation is the p-cycle (1, 2,..., (p − 1)). Similarly, for the other branch points we consider the equations sp = rkj . In this case, the associated permutation from the monodromy representation is the p-cycle

(ki, 2 · ki,..., (p − 1) · ki). 8 SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE

′ Now we apply the same process on C beginning with ζ · k1 = ζ. In this case, the preimage of the unit circle is

τ πi τζ 2πi e 2 ,e p t ∈ [0,p]

n o 2πi p and we again label the points in the preimage of 1 by their power of e . In this way, the ramiﬁcation monodromy on C′ is described by the permutations

(ζ · ki, 2 · ζ · ki,..., (p − 1) · ζ · ki). Now, define σ to be the inverse of the permutation defined by n 7→ ζ · n. This gives (ζ · ki, 2 · ζ · ki,..., (p − 1) · ζ · ki) σ = (ki, 2 · ki,..., (p − 1) · ki) and shows that the η and η′ have the same monodromy data (only a different choice of labelling). The uniqueness part of the Riemann existence theorem now shows that η and η′ must be equivalent. Remark 3.8 (Constructing superelliptic curves from points). A key aspect of the results from section 2 was the observation (described in 2.8) that there is a unique hyperelliptic curve associated to a given choice of branch points. Among other ∼ things, this observation leads to the isomorphism Hypg = M0,2g+2/S2g+2 from theorem 2.9. However, in the superelliptic case, things are more complicated. The affine model for superelliptic curves (described in 3.5) is of the form sp = f(t) and suggests that we must also take into consideration the root-orders of f(t). Indeed, this is confirmed by Lemma 3.7. p 3.9 (Notation for indexing components of Supg). We will index the possible choices of affine model sp = f(t) by indexing the possible combinations for the root-orders of f(t). In this light, it makes sense to group the ramification points with same root orders. If we assume that the representative of the superelliptic curve is not branched at ∞ ∈ P1 then this allows us to express the affine model in the form

p−1 mk p k 0= s − (t − αk,i) . k i=1 Y=1 Y An initial observation from this is that the vector (m1,...,mp−1) has the properties: p−1 p−1

(i) mk = m and (ii) kmk ≡ 0 (mod p). k k X=1 X=1 ∗ Moreover, the equivalence relation from Lemma 3.7 shows that for ζ ∈ Fp, we have that (m1,...,mp−1) and (mζ·1,...,mζ·(p−1)) give equivalent superelliptic curves. ∗ On top of this, there always exists a unique choice of ζ ∈ Fp such that the ﬁrst entry is m1 = max {m1,...,mp−1}. This leads us to deﬁne the following indexing set p−1 • mk = m,  k=1  p−1 pP−1 M = (m1,...,mp−1) ∈ (Z≥0) • km ≡ 0 (mod p), and  .  k   k=1 

• mP1 = max {mi}.         SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE 9

3.10 (Group describing equivalent affine models). As was the case for hyperelliptic curve, we have to account for the inherent labelling of ramification points that an affine model gives. In the hyperelliptic case, all the ramification points were the same type, so this choice was accounted for by taking the S2g+2 quotient. However, things are more complicated in the superelliptic case. To begin with, for each m ∈ M the group

Sm := Sm1 ⊕···⊕ Smp−1 accounts for permutations of the ramification points within the groupings that correspond to m. Another complication is that each m ∈ M may define equivalent superelliptic curves in multiple ways. This arises from the equivalence relations described in ∗ Lemma 3.7 part (iii). The lemma shows that there is a natural action of Fp on the (p − 1)-tuples (m1,...,mp−1) which is defined by

ζ · (m1,...,mp−1) := (mζ·1,...,mζ·(p−1)). Moreover, this action preserves properties (i) and (ii) from 3.8. Hence we consider ∗ the subgroup of Fp which stabilises a given m ∈ M and denoted it by ∗ Stabp(m) := ζ ∈ Fp ζ · m = m .

Both Sm and Stab (m) are naturally subgroups of S . Moreover, both of these p m inclusions arise from the fact that for any given m ∈ M with m = (m1,...,mp−1), we can uniquely describe each n ∈{1,...,m} by a pair (i, k) where i ∈{1,...,p−1} and 0 < k ≤ mi. Explicitly, this relationship is described by

n = k + mj . 0

δ(σ1,...,σp−1) k + mj := σj (k) + mj . 0

γ(ζ) k + mj := k + mj . 0

Am := Sm · Stabp(m) =∼ Sm ⊕ Stabp(m). The group Am describes the equivalent ways that m ∈ M can give rise to a superelliptic curve. This results in Proposition 3.11. p Proposition 3.11 (Decomposition of Supg into connected components). The set p M deﬁned in 3.9 indexes the connected components of Supg to give

p p Supg = Supg,m. m∈M G 10 SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE

∼ p For each m = (m1,...,mp−1) ∈ M there is an isomorphism M0,m/Am = Supg,m where Am is the group deﬁned in 3.10. The Am-quotient map composed with this isomorphism gives a morphism p φm : M0,m −→ Supg,m which is deﬁned by mapping the equivalence class [a1 : b1],..., [am : bm] ∈ M0,m to the equivalence class of superelliptic curves with the representative p−1 mi p k 0= s − (aϕ(k,i)t − bϕ(k,i)) , k i=1 Y=1 Y where ϕ(k,i) := k + 0

Remark 3.13 (Codimension in Mg). An immediate observation from Corollary 3.11 p is that superelliptic curves are rare among genus g curves. The image of Supg → Mg 2g has codimension 3g − 3 − (m − 3)= 3g − 1 − p−1 which we note has a maximum value of 3g − 2 that occurs when 2g = p − 1.

4. Divisors on superelliptic curves 4.1 (Notation for this section). In this section we will consider a fixed superelliptic curve π : C → P1 with g > 0 and a choice of affine model with the same notation given in 3.5. Namely, π will be given by the equations sp = f(t) and up = vκh(v) where f,h ∈ C[t] such that for n =: ⌈deg(f)/p⌉, κ := pn − deg(f) we have that h κ 1 is the unique polynomial with v h(v)= f( v ). The change of coordinates between 1 u the two charts is then defined by (t,s) 7→ v , vn . Moreover, in this section, we will assume that these coordinates have been chosen such that π is has a branch point at ∞ ∈ P1. This condition is equivalent to ∗ requiring that p ∤ deg(f) or equivalently requiring that κ ∈ Fp. Lastly, we will assume an ordering for the branch points in P1 \ {∞}. With this ki ordering assumed we will write f(t)= i(t − ai) where we have also invoked the equivalence described in Lemma 3.7 to make this a monic polynomial. Q 4.2 (Points at infinity and branch points). We will follow the standard convention −1 and refer to the preimage π (∞) as the point at infinity and denote it by Q∞. In the notation from 4.1 this corresponds to the point defined by v = u = 0 in the second chart. SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE 11

4.3 (Natural principal divisors and the superelliptic divisor class). Recall from 4.1 ki that f(t)= i(t−ai) and observe that pn = κ+ i ki. Then we have the natural principal divisors given by: Q u P (i) Horizontal: (s)= vn = i ki(Qi − Q∞). (ii) Vertical ramified: (t − a )= 1−aiu = pQ − pQ . i P u i ∞ (iii) Vertical unramified: (t − a)= 1−au = π∗P − pQ . u a ∞ ∗ 1 Here a ∈ C \{a1,...,am−1} and Pa ∈ P is the corresponding divisor. The divisor ∗ π Pa consists of p distinct points. Remark 4.4. In 4.3 and in what follows, we will always mean the divisor on the smooth curve C and not the (potentially) singular affine model. 4.5 (p-torsion in the Jacobian of a superelliptic curve). A key focus of this article is to study p-torsion within the Jacobian of a superelliptic curve (for related, but somewhat orthogonal, investigations of this topic, see Arul’s thesis [Aru20]). An immediate observation from 4.3 is that

p · (Qi − Q∞) ∼ 0 for each i. This shows that [Qi − Q∞] is an element of Jac(C)[p] (c.f. the construction in Section 2). We can now consider the Fp-vector subspace of Jac(C)[p] spanned by the classes Di := [Qi − Q∞] and denote it by

∆ := Fp-Span D1,...,Dm−1 ⊆ Jac(C)[p]. We also have the relationship n o

kiDi = ki[Qi − Q∞]= (s) =0 i i X X which shows that D1,...,Dm−2 is a F3-spanning set for ∆. In fact, by the following theorem, this turns out to be a basis for ∆.

Theorem 4.6 (A Fp-basis for ∆, [PS97, Prop. 6.1] & [Waw21, Thm. 1]). Let ∆ be the subgroup of Jac(C)[p] generated by classes of the form Di = [Qi −Q∞]. Then m−2 ∆ is isomorphic to Fp and D1,...,Dm−2 is a Fp-basis for ∆. Remark 4.7. An immediate observation from the construction in 4.5 is that for ∼ 2g p > 2 we will not get all of Jac(C)[p] = Fp . Indeed, when 2g = p − 1, the codimension of ∆ in Jac(C)[p] is 2g − 1. 4.8 (Weil pairing on superelliptic curves). Let [E] and [E′] be two elements of Jac(C)[p] and let E ∈ [E] and E′ ∈ [E′] be divisors with disjoint support. Let f and g be functions on C such that pE = (f) and pE′ = (g). The Weil pairing of [E] and [E′] is then defined as the quotient f(E′) w([E], [E′]) := ∈ µ g(E) p ′ ′ ′ multP (E ) where f(E ) is defined by f(E ) = P ∈E′ f(P ) and g(E) is analogously defined. The Weil pairing is a symplectic pairing on Jac(C)[p]. For more details and relations to moduli, see e.g. [ACGH85Q , Appx. B] and [MFK94, Sec. 7.2]. 12 SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE

Proposition 4.9. When p is odd the vector space ∆ is an isotropic subspace of Jac(C)[p] with respect to the Weil pairing. In other words, for any two divisor classes D,D′ in Jac(C)[p] we have that w(D,D′)=1. If p is even and Di is the divisor class deﬁned in 4.5 then w(Di,Dj) = −1 for i 6= j. Proof. We will use the notation from 4.1. Namely that π : C → P1 is given by the polynomials sp = f(t) and u = vκg(v). Furthermore, using Lemma 3.7, we may ki ∗ assume that κ =1 and that f(t)= i(t − ai) such that ai ∈ C are non-zero and distinct. From the deﬁnition in 4.5, we haveQ that ∆ is generated by classes with represen- tatives of the form Ei = Qi − Q∞, i =1,...,m − 1.

Moreover, since the Weil pairing is alternating we have that w([Ei], [Ei]) = 1, it will suffice to show that w([Ei], [Ej ]) = 1 for i 6= j. Hence, we assume that i 6= j. Since Ei and Ej do not have disjoint support we need to find a divisor Fj in the class Ej such that Fj has support disjoint from Ei. The construction of the divisor Fj is a key aspect of this proof. n−1 n ∗ To construct Fj we consider the rational function (t −s)/t ∈ k(C) and note that it corresponds to u − v in the other chart (where C is defined by the equation up − vg(v)=0). To determine the divisor (u − v), we consider the isomorphism C[u, v]/(vg(v) − up,u − v) =∼ C[v]/ v(g(v) − vp−1)

ki and recall that g(v)= i(1−aiv) . Then we consider the two unique factorisations

p−1 Q λα p(n−1) λα (4.1) g(v) − v = (1 − bαv) and f(t) − t = (t − bα) . α α Y Y 1 which are equivalent under the change of variables u 7→ t . We then denote by Gα the divisors on C given by the points (t,s) = (bα,bα). Then the principal divisor (u − v) is given by

(u − v)= Q∞ + λαGα − n · Sβ. α β X X −1 where Sβ is the divisor of the p disjoint points π (0). Now we can define Fj by P Fj := Ej + (u − v)= Qj + λαGα − n · Sβ α β X X ∗ n−1 p pn and define the rational function ψ ∈ C(C) by ψ = (t − aj)(t − s) /t , while ∗ defining the rational function ϕ ∈ C(C) by ϕ = (t − ai). We note that by construction we have the properties required to evaluate the Weil pairing. Namely, we have that

(i) Ei ∩ Fj = ∅, (ii) (φ)= p · Ei and (iii) (ψ)= p · Fj .

We will evaluate the numerator and denominator of w([Ei], [Fj ]) individually beginning with ψ(Ei). First note that in the second chart we have that the rational p p function ψ is given by (−1) (1 − ajv)(u − v) /v and recall that C is deﬁned in this SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE 13 chart by the equation up − vg(v)=0. Now the binomial theorem shows that the rational function is given by p p ψ = (1 − a v) (−1)pg(v)+ (−u)p−γvγ−1 . j γ γ=1 ! X 1 In this chart we have that Qi is given by (v,u) = ( , 0) and Q∞ is given by ai (v,u)=(0, 0), hence we can evaluate ψ(Ei) as: 1−p ψ(Qi) (1 − aj /ai)ai ai − aj (4.2) ψ(Ei)= = p = p . ψ(Q∞) (−1) (−ai)

Now we will evaluate ϕ at Fj after recalling that Qj corresponds to (t,s) = (aj , 0), Gα corresponds to (t,s) = (bα,bα) and Sβ corresponds to pairs with t =0. Hence we have

λα λα −n (aj − ai) α(bα − ai) ϕ(Fj )= ϕ(Qj ) · ϕ(Gα) · ϕ(Sβ ) = pn . (−ai) β α Y Y Q Now considering equation (4.1) in the t coordinate and noting that f(ai)=0 and deg(f)= pn − 1 we have the equations

λα (bα − aj ) −(a − a ) (4.3) (a )−p = α and ϕ(F )= i j . i (−a )pn j (a )p Q i i Finally, combining equations (4.2) and (4.3), we have

ψ(Ei) p−1 w([Ei], [Fj ]) = = (−1) ϕ(Fj ) which is −1 when p is even and 1 when p is odd. Corollary 4.10. The vector space ∆ is a maximal isotropic subspace of Jac(C)[3] with respect to the Weil pairing. Proof. Any maximal isotropic subspace of Jac(C)[3] has dimension g and, con- versely, any isotropic subspace of Jac(C)[3] of dimension g is a maximal isotropic subspace. Thus, since ∆ has dimension g the result needs follows immediately from Proposition 4.9.

5. Trigonal Superelliptic Curves 5.1 (Overview of this section). The case p = 3 is the natural next step from the hyperelliptic case. In this case, superelliptic curves exist for each g> 0 and we will see that the key decomposition result of the moduli space of hyperelliptic curves with level-2 structure from theorem 2.18 has an analogue in the p =3 case. 5.2 (Properties of the case p =3). As discussed in 3.4, trigonal superelliptic curves exist for every genus g > 0 and the number of ramiﬁcation points is m = g +2. ∗ ∼ In this case we have F3 = S2 and the indexing set M from 3.9 contains ordered pairs (m1,m2). Explicitly, the indexing set can be expressed as r ∈{0, 1, 2} with m =3n − r for some n ∈ Z and M = m − 3i − r, 3i + r . i ∈ Z with 0 ≤ i ≤ (m − 2r)/6.

14 SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE

Some examples of the indexing set are given in below in table 5.3. We can also examine the group Am describing equivalent aﬃne models from 3.10. There are two cases: (i) m = (a,a): This case occurs whenever m is even and in this case we have Stabp(m) = S2 and Am =∼ S2 · (Sa ⊕ Sa) =∼ S2 ⊕ Sa ⊕ Sa. In terms of the explicit description of M given above, this occurs when i = (m − 2r)/6. (ii) m = (a,b) where a>b: In this case we have that Stabp(m) is trivial and hence Am =∼ Sa ⊕ Sb. Table 5.3. Examples of the indexing set M for p = 3. The values of m for which

Sm 6= Sm1 ⊕ Sm2 are shown in bold. g m M g m M 1 3 {(3, 0)} 7 9 {(9, 0), (6, 3)} 2 4 {(2, 2)} 8 10 {(8, 2), (5, 5)} 3 5 {(4, 1)} 9 11 {(10, 1), (7, 4)} 4 6 {(6, 0), (3, 3)} 10 12 {(12, 0), (9, 3), (6, 6)} 5 7 {(5, 2)} 11 13 {(11, 2), (8, 5)} 6 8 {(7, 1), (4, 4)} 12 14 {(13, 1), (10, 4), (7, 7)}

3 5.4 (Level 3 structures and moduli description of Supg[3]). Here we recall that the deﬁnition of a superelliptic curve with a level-3 structure is a pair ∼ 1 2g = π : C → P , η : F3 −→ Jac(C)[3]

2g where π is a superelliptic curve and η is an isometry from F3 with the standard symplectic form to Jac(C)[3] with the Weil pairing. In other words, a level-3 structure is a choice of an ordered symplectic basis for Jac(C)[3] compatible with the Weil paring. The moduli space of p = 3 superelliptic elliptic curves with level 3 structure is 3 denoted by Supg[3] and comes with a natural morphism 3 3 F : Supg[3] −→ Supg 3 which forgets the level 3 structure. The group Sp(2g, F3) acts naturally on Supg[3] by changing basis and has no ﬁxed points. The morphism F is then equivalent to taking the quotient by Sp(2g, F3) which shows that F is étale of degree |Sp(2g, F3)|. 3 For each m ∈ M, we can extend this concept to the sub-moduli space Supg,m (deﬁned via the decomposition in proposition 3.11) by the following diagram where all squares are cartesian.

3 3 Supg,m[3] Supg[3]

Fm F

3 3 Supg,m Supg

5.5 (A natural basis for Jac(C)[3]). Consider a moduli point [P1,...,Pm] ∈ M0,m 1 3 and let [π : C → P ] be the associated point in Supg,m. Recall from section 4 that the divisor classes

Dj := [Qj − Qm] SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE 15 for j ∈{1,...,g} form a basis for a natural subspace ∆ ⊂ Jac(C)[3]. It was further shown in Corollary 4.10 that ∆ is a Lagrangian (a maximal isotropic subspace) and hence that ∆ is a natural choice of polarisation for the pair (Jac(C)[3], w) where w is the Weil pairing. Moreover, using standard techniques, we can use the basis (D1,...,Dg) to define a basis for the complimentary isotropic subspace ∆c ⊂ Jac(C)[3]. In particular, for each j ∈{1,...,g} the conditions c w(Dk, E)=1, k 6= j, E ∈ ∆ c define 1-dimensional subspaces Λj ⊂ ∆ . On top of this, the condition 2π i w(Dj , Ej )= e 3 uniquely defines an element Ej ∈ Λj. So, by construction, we have a natural symplectic basis for Jac(C)[3] given by (D1,...,Dg, E1,...,Eg). 3 5.6 (Natural morphisms to Supg,m[3]). Continuing from the situation in 5.5, we 1 take a moduli point [P1,...,Pm] ∈ M0,m and let [π : C → P ] be the associated 3 point in Supg,m. Using the basis from 5.5, we can construct an isometry 2g =∼ η : F3 −→ Jac(C)[3] 2g by sending the standard basis (e1,...,eg,f1,...,fg) for F3 to the natural basis (D1,...,Dg, E1,...,Eg) for Jac(C)[3]. This then gives rise to a natural morphism 3 ψid : M0,m −→ Supg,m[3] defined by mapping the equivalence class [P1,...,Pm] ∈ M0,m to the equivalence class [π, η]. Here the subscript id refers to the identity in Sp(2g, F3). We can also extend this concept to any A ∈ Sp(2g, F3) by first considering the 3 3 automorphism θA : Supg,m[3] → Supg,m[3] defined by [π, η] 7→ [π,TA ◦ η], where TA : F3 → F3 is the linear change of basis associated to A. The morphism θA ◦ ψid is denoted by 3 ψA : M0,m −→ Supg,m[3]. Combining this concept with the forgetful morphism and the isomorphism from Proposition 3.11 gives the following commutative diagram.

ψA 3 M0,m Supg,m[3]

(5.1) Fm

=∼ 3 M0,m/Am Supg,m

5.7 (Natural group homomorphism deﬁned by ψA). For each m = (m1,m2) ∈ M, 3 P ∈ M0,m and A ∈ Sp(2g, F3), the morphism ψA : M0,m → Supg[3] from 5.5 deﬁnes a group homomorphism which we will denote by

ΨP,A : Am → Sp(2g, F3). This homomorphism can be describe explicitly by initially considering the case 1 A = id. To begin, let [P1,...,Pm] ∈ M0,m and (π : C → P , η) be the superelliptic curve and the level-3 structure which are given by ψid. The homomorphism ΨP,id is constructed by considering the action of Sm on M0,m and showing that this deﬁnes a change of basis for Jac(C)[3]. A straight-forward 16 SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE way of accomplishing this is to consider how the generators of Am deﬁne change of basis matrices for Jac(C)[3].

First consider the generators of Sm = Sm1 ⊕ Sm2 while considering Sm as a subgroup of Sm. Recall that Sm is generated by the m − 1 transpositions (i,i + 1) for i ∈ {1,...,m − 1}. In the case at hand, Sm is generated by the collection

(Sm1 ⊕ Sm2 ) ∩{(i,i + 1)}. We can now consider how these permutations deﬁne changes of basis:

(i) For i∈ / {m − 1,m − 2}: The change of basis ΨP,id(σ) is deﬁned on each Dj by considering the action of Sm on {Q1,...Qm}. The result is

Dj for j∈{ / i,i +1}, Ψ (σ) · D = D for j = i, P,id j  i+1  Di for j = i +1, which is the usual “row-swap ” elementary row-operation. (ii) For σ := (g,g +1)=(m − 2,m − 1): The generator σ := (g,g + 1) exists in the cases m2 =0, m2 =1 and m2 > 3. In these cases we have

Dj for j 6= g, ΨP,id(σ) · Dj = Dm− for j = g. 1 To see the associated change of basis, we recall from 4.3 that the natural horizontal principal divisor gives: m−2 (a) For m2 =0: Dm−1 ∼ 2 i=1 Di. m−2 (b) For m2 =1: Dm−1 ∼ i=1 Di. Pm1 m−2 (c) For m2 ≥ 2: Dm−1 ∼ Di +2 Di. Pi=1 i=m1+1 (iii) For σ := (g +1,g +2)=(m − 1,m): The generator σ := (g,g + 1) exists P P in the cases m2 =0 and m2 > 2. In these cases we have the same change of basis and relations as those in part (ii) except

ΨP,id(σ) · Dj = Dj +2 Dm−1.

We now consider the extra generators in the case when Stabp(m) 6= 0. This m m m ∼ case occurs when 2 ∈ Z>0 and m = ( 2 , 2 ) so that Stabp(m) = S2. Let ζ be the generator of Stabp(m) then we have

m m m Dj+ 2 +2 D 2 for 1 ≤ j < 2 − 1, m m Dm−1 +2 D 2 for j = 2 − 1, ΨP,id(ζ) · Dj =  m 2 D m for j = ,  2 2  m  D m +2 D m for < j ≤ m − 2, j− 2 2 2  where Dm−1 has been calculated above in (ii). Hence, we have defined a homomorphism ΨP,id : Am → Sp(2g, F3). Now, consider the automorphism ΘA : Sp(2g, F3) → Sp(2g, F3) which is defined by B 7→ AB. The group homomorphism ΨP,A is now defined by ΨP,A := ΘA ◦ ΨP,id.

Lemma 5.8. Let A ∈ Sp(2g, F3) and P ∈ M0,m. The morphism ψA from 5.6 and the group homomorphism ΨP,A from 5.7 are injective.

Proof. Since for A 6= id, the morphisms ψA and ΨA are deﬁned by post-composing with automorphisms, we are only required to consider the case when A = id. ′ To consider whether ψid is injective take two points in P, P ∈ M0,m which have ′ ′ ′ ψid(P )= ψid(P ). We also let D1,...,Dg and D1,...,Dg be the associated natural SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE 17 bases constructed from divisors in 4.5 and corresponding to the images ψid(P ) and ′ ψid(P ). ′ The combination of the condition ψid(P ) = ψid(P ) and fact that the diagram ′ (5.1) commutes shows that P and P must be in the same Am-class. Taking σ ∈ Am ′ ′ to be such that P = σ · P , we observe that this implies Di =ΨP,id(σ) · Di for all ′ i ∈{1,...,g}. The condition ψid(P )= ψid(P ) is now only true if Di =ΨP,id(σ)·Di for all i ∈ {1,...,g}. Hence we have shown that ψid is injective only if ΨP,id is injective for all P ∈ M0,m. Let σ ∈ Am be such that ΨP,id(σ) · Di = Di for all i ∈{1,...,g}. The relations given in 4.5 show that ΨP,id(σ)·Dg+1 = Dg+1 as well as ΨP,id(σ)·Dg+2 = Dg+2 =0. −1 Now, let n ∈{1,...,m} be n := σ (m). Then the condition ΨP,id(σ) · Dn = Dn implies Qn − Qm ∼ Qm − Qσ(n) and hence Dn = −Dσ(n). This is not one of relations from 4.5, which are shown to be all the relations in Theorem 4.6, so we must have Dn =0 and hence n = m. So now, for any i ∈{1,...,g}, the condition ΨP,id(σ) · Di = Di implies

Qi − Qm ∼ Qσ(i) − Qm and Di = Dσ(i). Again, the relations from Deﬁnition 4.5 and Theorem 4.6 show that this implies σ(i)= i for any i ∈{1,...,g}. Hence we have that σ = id. Thus, we have shown that ΨP,id has trivial kernel and so is injective.

Lemma 5.9. For A ∈ Sp(2g, F3) the morphism ψA is an isomorphism onto an 3 irreducible component of Supg,m[3].

Proof. Denote the quotient map by qm : M0,m → M0,m/Am. We have that both qm and the morphism Fm forgetting the level-3 structure are finite. Hence we have that ψA is finite as well (see for example [Sta21, Tag 01WJ & Tag 035D]). Thus, since Lemma 5.8 shows that ψA is injective we have that it is a closed immersion (see for example [Sta21, Tag 03BB]). Since the quotient map qm and Fm are both finite and surjective, we have that 3 M0,m and Supg,m[3] are both the same dimension (see for example [Sta21, Tag 3 01WJ & Tag 0ECG]). Now, let A ⊆ Supg,m be the irreducible component that contains the image of ψA, and let ε : M0,m → A be the associated morphism. Since M0,m is irreducible, we must have that ε is surjective. On top of this, both qm and Fm are étale, so by the vanishing of cotangent complexes we have that ε is étale as well. In conclusion, ε is a surjective flat closed immersion and hence ε is an isomorphism (see [Sta21, Tag 04PW]).

3 Corollary 5.10. Let x = [π] ∈ Supg be a geometric point and let A ∈ Sp(2g, F3). ′ If [π, η] and [π, η ] are both in the image Im ψA then there is a unique B ∈ Am such ′ that η = TB ◦ η (where TB is as deﬁned in 5.6). Moreover, we have that −1 Im ψA ∩ Fm (x) = |Am|.

Proof. This follows from Lemma 5.9 since ψA is an isomorphism on an irreducible 3 3 component of Supg,m[3] and the morphism M0,m → Supg,m described by diagram (5.1) is the ﬁxed-point-free quotient of M0,m by Am. ′ Lemma 5.11. Let A, A ∈ Sp(2g, F3). If the images Im ψA and Im ψA′ are not disjoint, then Im ψA = Im ψA′ . 18 SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE

Proof. Suppose that the images Im ψA and Im ψA′ are not disjoint and take a point 3 [π, η] ∈ Supg[3] in the intersection. Then, by the construction of ψA and ψA′ , we ′ ′ must have that η = TA ◦ τ and η = TA′ ◦ τ for points [π, τ], [π, τ ] ∈ Im ψid. ′ Hence by Corollary 5.10 we have that there is a B ∈ Am such that τ = TB ◦ τ. ′ Combining that with the equality η = TA ◦ τ = TA′ ◦ τ we see that TA = TA′ ◦ TB and hence A = A′B. Now, by the deﬁnition of ψA we have that ψA =ΘA′ ◦ ψB and ψA′ =ΘA′ ◦ ψid. The result now follows from the observation that Im ψid = Im ψB . 3 Lemma 5.12. The scheme Supg,m[3] has |Sp(2g, F3)|/|Am| connected components. Each connected component is irreducible and is isomorphic to M0,m. Proof. We begin by claiming that the morphism 3 ψA : M0,m −→ Supg[3] F F A∈Sp(2Gg, 3) A∈Sp(2Gg, 3) 1 ′ 3 is surjective. So see this, take any geometric point [π : C → P , η ] ∈ Supg[3] and 3 consider the point [π] ∈ Supg. We know from diagram (5.1) that the morphism 3 M0,m → Supg is surjective, so let P ∈ M0,m be be any point in the preimage of [π] under this morphism. We now have that ψid(P ) = [π, η] for some isometry 2g ′ −1 ′ η : F3 → Jac(C)[3]. Considering the composition η ◦ η ◦ η = η we observe that ′ −1 2g 2g ′ −1 η ◦η corresponds to symplectic change of basis F3 → F3 and hence η ◦η = TA for some A ∈ Sp(2g, F3). This proves the claimed surjectivity. Let B⊂ Sp(2g, F3) have the following properties:

(i) A∈B ψA is surjective. ′ (ii) If B ⊂ Sp(2g, F3) has the property that A∈B′ ψA is surjective then we haveF |B| ≤ |B′|. F Such a B will always exist but may not be unique. Moreover, combining property (ii) with Lemma 5.11 we must have that for each pair A, A′ ∈B with A 6= A′ that the images Im ψA and Im ψA′ are disjoint. 3 Now, letting x ∈ Supg be a geometric point, we have that −1 −1 Fm (x)= Im ψA ∩ Fm (x) . A∈B G −1 Hence using Corollary 5.10 and the fact that |Fm (x)| = |Sp(2g, F3)| we have −1 |Sp(2g, F3)| = Im ψA ∩ Fm (x) = |B| · |Am|. A∈B X Since |B| is the number of connected components, the desired result now follows. Theorem 1.1 now follows immediately from Lemma 5.12 and the description of the indexing set M from 5.2. We also summarize what we have obtained in alternative way to highlight the analogy with Theorem 2.18. 3 Theorem 5.13 (Connected components of Supg[3]). For m ∈ M, denote the quotient set Sp(2g, Fp)/Am by Cm. Then, if Xc denotes a copy of M0,m for each c ∈ Cm, there is an isomorphism of schemes 3 =∼ Supg[3] −→ Xc. m∈M c∈Cm G G 3 In particular, Supg[3] is smooth. SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE 19

Acknowledgements. The authors thank Jonas Bergstöm and Dan Petersen for useful conversations.

References

[Acc84] Accola, R., On cyclic trigonal Riemann surfaces. I, Trans. Amer. Math. Soc. 283 (1984), no. 2, 423–449. MR 737877 [ACGH85] Arbarello, E., Cornalba, M., Griffiths, P., and Harris, J., Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. MR 770932 [AL02] Avritzer, D. and Lange, H., The moduli spaces of hyperelliptic curves and binary forms, Math. Z. 242 (2002), no. 4, 615–632. MR 1981190 [Aru20] Arul, V., Explicit Division and Torsion Points on Superelliptic Curves and Jacobians, Phd thesis, Massachusetts Institute of Technology, 2020. MR 4158225 [AV04] Alessandro Arsie and Angelo Vistoli, Stacks of cyclic covers of projective spaces, Com- pos. Math. 140 (2004), no. 3, 647–666. MR 2041774 [BB20] Bergström, J. and Bergvall, O., The equivariant Euler characteristic of A3[2], Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 20 (2020), no. 4, 1345–1357. MR 4201184 [Ber16] Bergvall, O., Cohomology of arrangements and moduli spaces, Phd thesis, Stockholms Universitet, 2016. [Ber19] , Equivariant cohomology of moduli spaces of genus three curves with level two structure, Geom. Dedicata 202 (2019), 165–191. MR 4001812 [Bv20] Bergström, J. and van der Geer, G., Picard modular forms and the cohomology of local systems on a Picard modular surface, arXiv:2012.07673 (to appear in Commentarii Mathematici Helvetici), 2020. [DO88] Dolgachev, I. and Ortland, D., Point Sets in Projective Spaces, Astérisque 165 (1988), 1–210. MR 1007155 [Fis56] Fischer, I., The moduli of hyperelliptic curves, Trans. Amer. Math. Soc. 82 (1956), 64–84. MR 77997 [Har77] Hartshorne, R., Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977, Grad- uate Texts in Mathematics, No. 52. MR 0463157 [HM98] Harris, J. and Morrison, I., Moduli of curves, Graduate Texts in Mathematics, vol. 187, Springer-Verlag, New York, 1998. MR 1631825 [Kon99] Kontogeorgis, A., The group of automorphisms of cyclic extensions of rational function fields, J. Algebra 216 (1999), no. 2, 665–706. MR 1692965 [Kop10] Kopeliovich, Y., Thomae formula for general cyclic covers of CP1, Lett. Math. Phys. 94 (2010), no. 3, 313–333. MR 2738563 [Lei18] Leigh, O., The Moduli Space of Stable Maps with Divisible Ramification, arXiv:1812.06933, 2018. [Lei20] , The r-ELSV Formula via Localisation on the Moduli Space of Stable Maps with Divisible Ramification, arXiv:2004.06739, 2020. [LK79] Lønsted, K. and Kleiman, S. L., Basics on families of hyperelliptic curves, Compositio Math. 38 (1979), no. 1, 83–111. MR 523266 [MFK94] Mumford, D., Fogarty, J., and Kirwan, F., Geometric invariant theory, third ed., Ergeb- nisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. MR 1304906 [MS19] Malmendier, A. and Shaska, T., From hyperelliptic to superelliptic curves, Albanian J. Math. 13 (2019), no. 1, 107–200. MR 3978315 [Mum84] Mumford, D., Tata lectures on theta. II, Progress in Mathematics, vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984, Jacobian theta functions and differential equations, With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. MR 742776 [PS97] Poonen, B. and Schaefer, E., Explicit descent for Jacobians of cyclic covers of the projective line, Journal für die Reine und Angewandte Mathematik. [Crelle’s Journal] 488 (1997), 141–188. MR 1465369 [Run97] Runge, B., Level-Two-Structures and Hyperelliptic Curves, Osaka Journal of Mathe- matics 34 (1997), no. 1, 21–51. MR 1438996 20 SUPERELLIPTIC CURVES WITH LEVEL STRUCTURE

[Sta21] The Stacks Project Authors, The stacks project, https://stacks.math.columbia.edu, 2021. [Tsu91] Tsuyumine, S., Thetanullwerte on a Moduli Space of Curves and Hyperelliptic Loci, Mathematische Zeitschrift 207 (1991), no. 1, 539–568. MR 1119956 [Wan15] Wangyu, N., Cyclic covering of the projective line with prime gonality, J. Pure Appl. Algebra 219 (2015), no. 5, 1704–1710. MR 3299702 [Waw21] Wawrów, W., On torsion of superelliptic Jacobians, Journal de Théorie des Nombres de Bordeaux 33 (2021), no. 1, 223–235.