Explicit Moduli of Superelliptic Curves with Level Structure

In this article we give an explicit construction of the moduli space of trigonal superelliptic curves with level 3 structure. The construction is given in terms of point sets on the projective line and leads to a closed formula for the number of connected (and irreducible) components of the moduli space. The results of the article generalise the description of the moduli space of hyperelliptic curves with level 2 structure, due to Dolgachev and Ortland, Runge and Tsuyumine.


Introduction
As Mumford describes in [Mum84,§2], 2-torsion divisors on a hyperelliptic curve correspond precisely to degree zero linear combinations of ramification points.Hence, if one takes distinct points P 1 , . . ., P 2g+2 on P 1 and considers C, the unique hyperelliptic curve ramified over these points, then one can hope to explicitly describe symplectic bases for the 2-torsion Jacobian Jac(c) [2] in terms of P 1 , . . ., P 2g+2 .Indeed, after choosing an ordering for the branch points, it turns out that there is a natural way to obtain a (full) symplectic level 2 structure on C from combinations of P 1 , . . ., P 2g+2 .One can then obtain any level 2 structure on C via the symplectic group Sp(2g, F 2 ).On top of this, since each choice of ordering will give rise to a different symplectic level 2, this construction naturally defines an embedding of the symmetric group S 2g+2 into the symplectic group Sp(2g, F 2 ).
Using this construction, Dolgachev and Ortland [DO88] considered Hyp g [2], the moduli space of hyperelliptic curves with level 2 structure.They showed that each irreducible component of Hyp g [2] is isomorphic to M 0,2g+2 , the moduli space of smooth rational curves with 2g + 2 distinct markings, and that the irreducible components of Hyp g [2] are indexed by the set of cosets C := Sp(2g, F 2 )/S 2g+2 .Dolgachev and Ortland also posed the question: What are the connect components of Hyp g [2]?This question was answered independently by Tsuyumine [Tsu91] and Runge [Run97], who showed that the irreducible components are also the connected components.Thus, if we let X c be a variety isomorphic to M 0,2g+2 and indexed by the coset c ∈ C , then there is an isomorphism Dept. of Electronics, Mathematics and Natural Sciences, University of Gävle Dept. of Mathematics, Stockholm University E-mail addresses: olof.bergvall@hig.se,oliver.leigh@math.su.se.
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 investigating cyclic covers of P 1 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 such that 0 ≤ k ≤ 2 and g + 2 = 3n − k.The moduli space Sup 3 g [3] of trigonal superelliptic curves with level 3 structure has connected components.Each connected component is irreducible and is isomorphic to the moduli space M 0,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 formulae 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).

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 → P 1 such that C is a smooth (connected) curve of genus g > 1.
Two hyperelliptic curves π : C → P 1 and π ′ : C ′ → P 1 are isomorphic if there are isomorphisms α : C → C ′ and β : 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 → P 1 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 Hyp g 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 Hyp g ֒→ M g .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 Hyp g instead of the moduli stack of hyperelliptic curves Hyp g .A key difference between the two is that Hyp g 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 Hyp g as a moduli space of maps, this moduli space can also naturally be viewed as the Hurwitz space Hur g (2).

(Affine models for hyperelliptic curves).
It is a well known result that a genus g hyperelliptic curve π : C → P 1 has a non-unique affine model of the form 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 The change of coordinates is then defined by (t, s) → 1 v , u v g+1 .The equivalence relation between two isomorphic affine models is described via 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 → P 1 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 A result of this observation is that one can construct a unique hyperelliptic curve by choosing a set of 2g + 2 points in P 1 .Taking into account the equivalence described in 2.7 we arrive at the following theorem.
Convention 2.11 (Notation for choice of hyperelliptic curve).For the remainder of this section we will assume that π : C → P 1 is a hyperelliptic curve of genus g.Moreover, given the equivalence of affine models described in 2.7, we will make the assumption that ∞ ∈ P 1 is a branch point of π and the affine model is We will denote the ramification points as 2.12 (Natural principal divisors).Recalling convention 2.11 we observe that there are natural principal divisors on C given by: (i) Horizontal: Here a ∈ C * \ {a 1 , . . ., a 2g+1 } and P a ∈ P 1 is the corresponding divisor.The divisor π * P a consists of 2 distinct points.

(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 [2].We can also now consider the natural F 2 -vector subspace spanned by the classes On top of this, the horizontal principal divisor from 2.12 gives the relationship 0 = 2g+1 i=1 D i .This shows that ∆ is spanned by any choice of 2g classes from D 1 , . . ., D 2g+1 .Indeed, it was shown by Mumford that any of these choices gives a basis for ∆ since ∆ ∼ = F 2g 2 .Moreover, combining this result with a basis constructed by Dolgachev and Ortland gives the following key result.

Proposition 2.14 (A symplectic basis for
2 .The basis (A 1 , . . ., A g , B 1 , . . ., B g ) of Jac(C)[2] defined by is symplectic with respect to the Weil Pairing.Moreover, the linear map taking this basis to the standard symplectic basis for F 2g 2 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 → P 1 as a two-sheeted covering and consider a symplectic basis of cycles in H 1 (C, Z) corresponding to the basis elements of Proposition 2.14.
To be more precise, A i corresponds to a path which goes from Q 2i−1 along one sheet to Q 2i and then returns along the other sheet, while B i corresponds to a path which goes from Q 2i along one sheet to Q 2g+1 and then returns along the other sheet.The symplectic basis of cycles in H 1 (C, Z) is then combined with a normalised basis of H 0 (C, Ω C ) to construct a branch point period matrix.
2.16 (Inclusion of the symmetric group into the symplectic group).The symmetric group S 2g+2 permutes the points Q 1 , . . ., Q 2g+1 , Q ∞ and thus defines an action on the set D := {D 1 , . . ., D 2g }.Since any permutation of Q 1 , . . ., Q 2g+1 , Q ∞ will give the same hyperelliptic curve, the action of S 2g+2 on D extends to give a change of symplectic basis.This defines a group homomorphism S 2g+2 → Sp(2g, F 2 ) which is in fact an embedding (see [Ber16,p. 60] for details and further references).

(Natural morphism to Hyp g [2]
).We denote the (coarse) moduli space of hyperelliptic curves with level-2 structure by Hyp g [2].This is a moduli space parametrising pairs consisting of a hyperelliptic curve π : C → P 1 and an isometry η : , where F 2g 2 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 Moreover, it is shown in [DO88, §3 Thm.1] that this morphism is an isomorphism onto an irreducible component of Hyp g [2].On top of this, if we consider the morphism that forgets the isometry, Hyp g [2] → Hyp g , then we obtain the following commutative diagram.
In fact, the morphism M 0,2g+2 → Hyp g [2] actually defines an isomorphism to a connected component of Hyp g [2].Using this construction one can describe all of the of connected components of Hyp g [2] and arrive at the following theorem.
Consider the inclusion of the symmetric group in the symplectic group described in 2.16 and denote the quotient set Sp(2g, F 2 )/S 2g+2 by C .Then, if X c denotes a copy of M 0,2g+2 indexed by c ∈ C , there is an isomorphism of schemes 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.

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 → P 1 such that C is a smooth (connected) curve and the Galois group of π is cyclic.Two superelliptic curves π 1 : C 1 → P 1 and π 2 : C 2 → P 1 are said to be equivalent if there are isomorphisms ψ : C 1 → C 2 and ϕ : 3.3 (Moduli space of superelliptic curves).Definition 3.2 extends naturally to give a moduli functor.We denote by Sup p g 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 curves and gives rise to an isomorphism Hyp g ∼ = Sup 2 g .This is because a hyperelliptic curve uniquely determines a morphism π : C → P 1 of degree 2 and because the Galois group of such a morphism is automatically cyclic.(ii) There is a natural morphism Sup p g → M g 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, meaning that the images of Sup 2 g and Sup 3 g are disjoint in M g .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 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 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.
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 → P 1 has a non-unique affine model given by 0 = s p − 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 The change of coordinates is then defined by (t, s) → 1 v , u v n .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 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 → P 1 and η ′ : C ′ → P 1 to be the superelliptic curves defined by s p − j (t − a j ) kj and s p − j (t − a j ) ζ•aj where p divides the sum j k j .Also, since k i ∈ F * p for each j as well as ζ ∈ F * p , we can assume that k 1 = 1.
We begin by considering the monodromies on C arising from the ramification points.Let x ∈ P 1 be distinct from the branch points and consider m different loops on P 1 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 a i can be described by considering the equation s p = r while considering the preimage of the unit circle and identifying x with 1.In this case, the preimage of the unit circle is The points in the preimage of 1 are the p-th roots of unity and we label them by their power of the primitive root of unity e Similarly, for the other branch points we consider the equations s p = r kj .In this case, the associated permutation from the monodromy representation is the p-cycle t ∈ [0, p] and we again label the points in the preimage of 1 by their power of e 2πi p .In this way, the ramification monodromy on C ′ is described by the permutations Now, define σ to be the inverse of the permutation defined by n → ζ • n.This gives 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 Hyp g ∼ = M 0,2g+2 /S 2g+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 s p = 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.
3.9 (Notation for indexing components of Sup p g ).We will index the possible choices of affine model s p = 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 ∞ ∈ P 1 then this allows us to express the affine model in the form An initial observation from this is that the vector (m 1 , . . ., m p−1 ) has the properties: Moreover, the equivalence relation from Lemma 3.7 shows that for ζ ∈ F * p , we have that (m 1 , . . ., m p−1 ) and (m ζ•1 , . . ., m ζ•(p−1) ) give equivalent superelliptic curves.On top of this, there always exists a unique choice of ζ ∈ F * p such that the first entry is m 1 = max {m 1 , . . ., m p−1 }.This leads us to define the following indexing set k m k ≡ 0 (mod p), and 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 S 2g+2 quotient.
However, things are more complicated in the superelliptic case.To begin with, for each m ∈ M the group 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 F * p on the (p − 1)-tuples (m 1 , . . ., m p−1 ) which is defined by Moreover, this action preserves properties (i) and (ii) from 3.8.Hence we consider the subgroup of F * p which stabilises a given m ∈ M and denoted it by Both S m and Stab p (m) are naturally subgroups of S m .Moreover, both of these inclusions arise from the fact that for any given m ∈ M with m = (m 1 , . . ., m p−1 ), we can uniquely describe each n ∈ {1, . . ., m} by a pair (i, k) where i ∈ {1, . . ., p−1} and 0 < k ≤ m i .Explicitly, this relationship is described by Using this relationship can now write: (i) The inclusion δ : S m ֒→ S m is defined by (ii) The inclusion γ : Stab p (m) ֒→ S m is defined by We observe that as subgroups of S m , we have that the intersection S m ∩Stab p (m) is trivial and that ζ • σ = σ • ζ for all ζ ∈ Stab p (m) and σ ∈ S m .Hence, we also have: The group A m describes the equivalent ways that m ∈ M can give rise to a superelliptic curve.This results in Proposition 3.11.
Proposition 3.11 (Decomposition of Sup p g into connected components).The set M defined in 3.9 indexes the connected components of Sup p g to give For each m = (m 1 , . . ., m p−1 ) ∈ M there is an isomorphism M 0,m /A m ∼ = Sup p g,m where A m is the group defined in 3.10.The A m -quotient map composed with this isomorphism gives a morphism which is defined by mapping the equivalence class [a 1 : b 1 ], . . ., [a m : b m ] ∈ M 0,m to the equivalence class of superelliptic curves with the representative Remark 3.12 (Different equivalence conditions).If we changed the equivalence relation between superelliptic curves from definition 3.2 to be the stricter condition: π : C → P 1 and π ′ : C ′ → P 1 are equivalent if there is an isomorphisms α : Then much of this section would still apply except with different equivalence conditions.In this case, the resulting construction in Proposition 3.11 would have been carried out using the configuration space Conf m (P 1 ) in place of M 0,m .Moreover, this construction would result in a natural sub-moduli-space of the moduli space of maps M g (P 1 , p).The resulting sub-moduli-space has links to (p − 1)-spin Hurwitz theory and was studied by the second author in [Lei18] and [Lei20].
Remark 3.13 (Codimension in M g ).An immediate observation from Corollary 3.11 is that superelliptic curves are rare among genus g curves.The image of Sup p g → M g has codimension 3g − 3 − (m − 3) = 3g − 1 − 2g p−1 which we note has a maximum value of 3g − 2 that occurs when 2g = p − 1.

Divisors on superelliptic curves
4.1 (Notation for this section).In this section we will consider a fixed superelliptic curve π : C → P 1 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 The change of coordinates between the two charts is then defined by (t, s) → 1 v , u v n .Moreover, in this section, we will assume that these coordinates have been chosen such that π is has a branch point at ∞ ∈ P 1 .This condition is equivalent to requiring that p ∤ deg(f ) or equivalently requiring that κ ∈ F * p .Lastly, we will assume an ordering for the branch points in P 1 \ {∞}.With this ordering assumed we will write f (t) = i (t − a i ) ki where we have also invoked the equivalence described in Lemma 3.7 to make this a monic polynomial.

(Points at infinity and branch points
).We will follow the standard convention and refer to the preimage π −1 (∞) 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.

(Natural principal divisors and the superelliptic divisor class).
Recall from 4.1 that f (t) = i (t−a i ) ki and observe that pn = κ+ i k i .Then we have the natural principal divisors given by: (i) Horizontal: Here a ∈ C * \ {a 1 , . . ., a m−1 } and P a ∈ P 1 is the corresponding divisor.The divisor π * P a 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.

(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 is an element of Jac(C)[p] (c.f. the construction in Section 2).We can now consider the F p -vector subspace of Jac(C)[p] spanned by the classes and denote it by We also have the relationship which shows that D 1 , . . ., D m−2 is a F 3 -spanning set for ∆.In fact, by the following theorem, this turns out to be a basis for ∆.
Remark 4.7.An immediate observation from the construction in 4.5 is that for p > 2 we will not get all of Jac(C)[p] ∼ = F 2g p .Indeed, when 2g = p − 1, the codimension of ∆ in Jac(C)[p] is 2g − 1.

(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 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 D i is the divisor class defined in 4.5 then w(D i , D j ) = −1 for i = j.
Proof.We will use the notation from 4.1.Namely that π : C → P 1 is given by the polynomials s p = f (t) and u = v κ g(v).Furthermore, using Lemma 3.7, we may assume that κ = 1 and that f (t) = i (t − a i ) ki such that a i ∈ C * are non-zero and distinct.
From the definition in 4.5, we have that ∆ is generated by classes with representatives of the form Moreover, since the Weil pairing is alternating we have that w( Hence, we assume that i = j.Since E i and E j do not have disjoint support we need to find a divisor F j in the class E j such that F j has support disjoint from E i .The construction of the divisor F j is a key aspect of this proof. To construct F j we consider the rational function (t n−1 −s)/t n ∈ k(C) * and note that it corresponds to u − v in the other chart (where C is defined by the equation u p − v g(v) = 0).To determine the divisor (u − v), we consider the isomorphism and recall that g(v) = i (1−a i v) ki .Then we consider the two unique factorisations which are equivalent under the change of variables u → 1 t .We then denote by G α the divisors on C given by the points (t, s) = (b α , b α ).Then the principal divisor (u − v) is given by where S β is the divisor of the p disjoint points π −1 (0).Now we can define F j by and define the rational function ψ ∈ C(C) * by ψ = (t − a j )(t n−1 − s) p /t pn , while defining the rational function ϕ ∈ C(C) * by ϕ = (t − a i ).We note that by construction we have the properties required to evaluate the Weil pairing.Namely, we have that (i) We will evaluate the numerator and denominator of w([E i ], [F j ]) individually beginning with ψ(E i ).First note that in the second chart we have that the rational function ψ is given by (−1) p (1 − a j v)(u − v) p /v and recall that C is defined in this chart by the equation u p − v g(v) = 0. Now the binomial theorem shows that the rational function is given by In this chart we have that Q i is given by (v, u) = ( 1 ai , 0) and Q ∞ is given by (v, u) = (0, 0), hence we can evaluate ψ(E i ) as: Now we will evaluate ϕ at F j after recalling that Q j corresponds to (t, s) = (a j , 0), G α corresponds to (t, s) = (b α , b α ) and S β corresponds to pairs with t = 0. Hence we have Now considering equation (4.1) in the t coordinate and noting that f (a i ) = 0 and deg(f ) = pn − 1 we have the equations Finally, combining equations (4.2) and (4.3), we have 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, conversely, 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.

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 In this case we have F * 3 ∼ = S 2 and the indexing set M from 3.9 contains ordered pairs (m 1 , m 2 ).Explicitly, the indexing set can be expressed as . Some examples of the indexing set are given in below in table 5.3.We can also examine the group A m describing equivalent affine models from 3.10.There are two cases:  ).Here we recall that the definition of a superelliptic curve with a level-3 structure is a pair where π is a superelliptic curve and η is an isometry from F 2g 3 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 denoted by Sup 3 g [3] and comes with a natural morphism g which forgets the level 3 structure.The group Sp(2g, F 3 ) acts naturally on Sup 3 g [3] by changing basis and has no fixed points.The morphism F is then equivalent to taking the quotient by Sp(2g, F 3 ) which shows that F is étale of degree |Sp(2g, F 3 )|.
For each m ∈ M, we can extend this concept to the sub-moduli space Sup 3 g,m (defined via the decomposition in proposition 3.11) by the following diagram where all squares are cartesian.
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 (D 1 , . . ., D g ) to define a basis for the complimentary isotropic subspace ∆ c ⊂ Jac(C) [3].In particular, for each j ∈ {1, . . ., g} the conditions define 1-dimensional subspaces Λ j ⊂ ∆ c .On top of this, the condition uniquely defines an element E j ∈ Λ j .So, by construction, we have a natural symplectic basis for Jac(C) [3] given by (D 1 , . . ., D g , E 1 , . . ., E g ).

5
.6 (Natural morphisms to Sup 3 g,m [3]).Continuing from the situation in 5.5, we take a moduli point [P 1 , . . ., P m ] ∈ M 0,m and let [π : C → P 1 ] be the associated point in Sup 3 g,m .Using the basis from 5.5, we can construct an isometry by sending the standard basis (e 1 , . . ., e g , f 1 , . . ., f g ) for F 2g 3 to the natural basis (D 1 , . . ., D g , E 1 , . . ., E g ) for Jac(C) [3].This then gives rise to a natural morphism We can also extend this concept to any A ∈ Sp(2g, F 3 ) by first considering the automorphism θ A : Sup (5.1) 5.7 (Natural group homomorphism defined by ψ A ).For each m = (m 1 , m 2 ) ∈ M, P ∈ M 0,m and A ∈ Sp(2g, F 3 ), the morphism ψ A : M 0,m → Sup 3 g [3] from 5.5 defines a group homomorphism which we will denote by This homomorphism can be describe explicitly by initially considering the case A = id.To begin, let [P 1 , . . ., P m ] ∈ M 0,m and (π : C → P 1 , η) be the superelliptic curve and the level-3 structure which are given by ψ id .
The homomorphism Ψ P,id is constructed by considering the action of S m on M 0,m and showing that this defines a change of basis for Jac(C) [3].A straight-forward way of accomplishing this is to consider how the generators of A m define change of basis matrices for Jac(C) [3].
First consider the generators of S m = S m1 ⊕ S m2 while considering S m as a subgroup of S m .Recall that S m is generated by the m − 1 transpositions (i, i + 1) for i ∈ {1, . . ., m − 1}.In the case at hand, S m is generated by the collection (S m1 ⊕ S m2 ) ∩ {(i, i + 1)}.We can now consider how these permutations define changes of basis: The change of basis Ψ P,id (σ) is defined on each D j by considering the action of S m on {Q 1 , . . .Q m }.The result is 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 m 2 = 0, m 2 = 1 and m 2 > 3.In these cases we have To see the associated change of basis, we recall from 4.3 that the natural horizontal principal divisor gives: (a) For m 2 = 0: The generator σ := (g, g + 1) exists in the cases m 2 = 0 and m 2 > 2. In these cases we have the same change of basis and relations as those in part (ii) except We now consider the extra generators in the case when Stab p (m) = 0.This case occurs when m 2 ∈ Z >0 and m = ( m 2 , m 2 ) so that Stab p (m) ∼ = S 2 .Let ζ be the generator of Stab p (m) then we have Hence, we have defined a homomorphism Ψ P,id : A m → Sp(2g, F 3 ).Now, consider the automorphism Θ A : Sp(2g, F 3 ) → Sp(2g, F 3 ) which is defined by B → AB.The group homomorphism Ψ P,A is now defined by Ψ P,A := Θ A • Ψ P,id .
Proof.Since for A = id, the morphisms ψ A and Ψ A are defined by post-composing with automorphisms, we are only required to consider the case when A = id.
To consider whether ψ id is injective take two points in P, P ′ ∈ M 0,m which have ψ id (P ) = ψ id (P ′ ).We also let D 1 , . . ., D g and D ′ 1 , . . ., D ′ g be the associated natural bases constructed from divisors in 4.5 and corresponding to the images ψ id (P ) and ψ id (P ′ ).
The combination of the condition ψ id (P ) = ψ id (P ′ ) and fact that the diagram (5.1) commutes shows that P and P ′ must be in the same A m -class.Taking σ ∈ A m to be such that P ′ = σ • P , we observe that this implies D ′ i = Ψ P,id (σ) • D i for all i ∈ {1, . . ., g}.The condition ψ id (P ) = ψ id (P ′ ) is now only true if D i = Ψ P,id (σ)•D i for all i ∈ {1, . . ., g}.Hence we have shown that ψ id is injective only if Ψ P,id is injective for all P ∈ M 0,m .
Lemma 5.9.For A ∈ Sp(2g, F 3 ) the morphism ψ A is an isomorphism onto an irreducible component of Sup 3 g,m [3].Proof.Denote the quotient map by q m : M 0,m → M 0,m /A m .We have that both q m and the morphism F m forgetting the level-3 structure are finite.Hence we have that ψ A is finite as well (see for example [Sta21, Tag 01WJ & Tag 035D]).Thus, since Lemma 5.8 shows that ψ A is injective we have that it is a closed immersion (see for example [Sta21, Tag 03BB]).
Since the quotient map q m and F m are both finite and surjective, we have that M 0,m and Sup 3 g,m [3] are both the same dimension (see for example [Sta21, Tag 01WJ & Tag 0ECG]).Now, let A ⊆ Sup 3 g,m be the irreducible component that contains the image of ψ A , and let ε : M 0,m → A be the associated morphism.Since M 0,m is irreducible, we must have that ε is surjective.
Let B ⊂ Sp(2g, F 3 ) have the following properties: (i) A∈B ψ A is surjective.
(ii) If B ′ ⊂ Sp(2g, F 3 ) has the property that A∈B ′ ψ A is surjective then we have |B| ≤ |B ′ |.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 = A ′ that the images Im ψ A and Im ψ A ′ are disjoint.Now, letting x ∈ Sup 3 g be a geometric point, we have that 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.In particular, Sup 3 g [3] is smooth.
Now we apply the same process on C ′ beginning with ζ • k 1 = ζ.In this case, the preimage of the unit circle is e τ 2πi , e τ ζ p 2πi Theorem 4.6 (A F p -basis for ∆, [PS97, Prop.6.1] & [Waw21, Thm.1]).Let ∆ be the subgroup of Jac(C)[p] generated by classes of the form a): This case occurs whenever m is even and in this case we have Stab p (m) = S 2 and A m ∼ = S 2 • (S a ⊕ S a ) ∼ = S 2 ⊕ S a ⊕ S a .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 Stab p (m) is trivial and hence A m ∼ = S a ⊕ S b .
3 g,m [3] → Sup 3 g,m [3] defined by [π, η] → [π, T A • η], where T A : F 3 → F 3 is the linear change of basis associated to A. The morphism θ A • ψ id is denoted by ψ A : M 0,m −→ Sup 3 g,m [3].Combining this concept with the forgetful morphism and the isomorphism from Proposition 3.11 gives the following commutative diagram.

Corollary 5. 10 .
Let x = [π] ∈ Sup 3 g be a geometric point and let A ∈ Sp(2g, F 3 ).If [π, η] and [π, η ′ ] are both in the image Im ψ A then there is a unique B ∈ A m such that η ′ = T B • η (where T B is as defined in 5.6).Moreover, we have thatIm ψ A ∩ F −1 m (x) = |A m |.Proof.This follows from Lemma 5.9 since ψ A is an isomorphism on an irreducible component of Sup 3 g,m [3] and the morphism M 0,m → Sup 3 g,m described by diagram (5.1) is the fixed-point-free quotient of M 0,m by A m .Lemma 5.11.Let A, A ′ ∈ Sp(2g, F 3 ).If the images Im ψ A and Im ψ A ′ are not disjoint, then Im ψ A = Im ψ A ′ .Proof.Suppose that the images Im ψ A and Im ψ A ′ are not disjoint and take a point [π, η] ∈ Sup 3 g [3] in the intersection.Then, by the construction of ψ A and ψ A ′ , we must have that η= T A • τ and η = T A ′ • τ ′ for points [π, τ ], [π, τ ′ ] ∈ Im ψ id .Hence by Corollary 5.10 we have that there is aB ∈ A m such that τ ′ = T B • τ .Combining that with the equality η = T A • τ = T A ′ • τ ′ we see that T A = T A ′ • T B and hence A = A ′ B.Now, by the definition of ψ A we have thatψ A = Θ A ′ • ψ B and ψ A ′ = Θ A ′ • ψ id .The result now follows from the observation that Im ψ id = Im ψ B .Lemma 5.12.The scheme Sup 3 g,m [3] has |Sp(2g, F 3 )|/|A m | connected components.Each connected component is irreducible and is isomorphic to M 0,m .Proof.We begin by claiming that the morphism A∈Sp(2g,F3) ψ A : A∈Sp(2g,F3) M 0,m −→ Sup 3 g [3] Hence using Corollary 5.10 and the fact that|F −1 m (x)| = |Sp(2g, F 3 )| we have |Sp(2g, F 3 )| = A∈B Im ψ A ∩ F −1 m (x) = |B| • |A m |.
Theorem 5.13 (Connected components of Sup 3 g [3]).For m ∈ M, denote the quotient set Sp(2g, F p )/A m by C m .Then, if X c denotes a copy of M 0,m for each c ∈ C m , there is an isomorphism of schemes Sup 3 g [3] smooth and the number of connected components of Hyp g [2] is given by |Sp(2g, F 2 )|/|S 2g+2 |.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 irreducible components of Hyp g [2] are indexed by C and are isomorphic to M 0,2g+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 Hyp g [n] has |Sp(2g, F 2 )|/|S 2g+2 | connected components whenever n is divisible by 2. However, components are only explicitly computed in the case n = 2.
Table 5.3.Examples of the indexing set M for p = 3.The values of m for which S m = S m1 ⊕ S m2 are shown in bold.