Deciding superellipticity and computing the Weierstrass normal form

T. Shaska

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Deciding superellipticity and computing the Weierstrass normal form

T. Shaska

arXiv:2609.00672v1Sep 1, 2026math.AGcs.SCmath.NT

TL;DR

The paper asks how to decide which superelliptic levels an arbitrary plane model admits and how to compute the corresponding normal form. It searches finitely through Weierstrass-point data, handles exceptional hyperelliptic levels separately, and applies Kummer tests. The algorithm decides geometric membership and returns a model and birational transformation, with stated field-of-definition and separability boundaries.

Problem
Given an arbitrary absolutely irreducible plane model, the paper addresses deciding for which n the curve lies in Sg,n and computing a birational transformation to yn = h(x).
Method
The algorithm searches finitely through Weierstrass-point divisors, applies Kummer criteria, verifies outputs by polynomial division, and treats the two m = 2 hyperelliptic levels separately.
Results
Theorem 5.10 states that the algorithm outputs exactly the levels in Sg,n and returns vn = h(u) and the birational transformation over kn.
Takeaways & Limitations
The procedure provides a decision method and normal-form construction for superelliptic levels, while distinguishing geometric membership from models available over the input field.
Takeaways & Limitations
Some cyclic-cover levels have no separable model, and the field-of-definition run may return only a subset of the geometric answer unless additional field obstructions vanish.

Abstract

from arXiv · show

Let $ \mathcal{S}_{g,n} \subset \mathcal{M}_g $ be the locus of curves of genus $ g \geq 2 $ admitting a model $ y^n = h(x) $ with $ h $ separable; such curves $C$ have a cyclic group $ C_n \leq \operatorname{Aut}(C) $ of order $ n $ with $ C/C_n \cong \mathbb{P}^1 $. % We give an algorithm which, given an absolutely irreducible plane model $ F(x,y) = 0 $ of a curve $ C $ over a field $ k_0 $ of characteristic zero, decides for which $ n $ the curve lies in $ \mathcal{S}_{g,n} $ and returns a model $ y^n = h(x) $ together with the birational transformation to it.

1. Introduction

The paper develops an algorithm to decide which cyclic-cover levels a curve from an arbitrary plane model admits and to compute corresponding normal forms. Its key strategy uses special Weierstrass points, while addressing obstacles from nonseparable models and field-of-definition issues.

Motivation: The central problem is deciding, from an absolutely irreducible plane model, for which n the curve lies in Sg,n and computing a birational transformation to yn = h(x).
Motivation: For n ≥ 3, a cyclic function of degree n has a single pole of order n at a totally ramified point, so n must be a non-gap there.
Obstacles: Subfield algorithms do not directly settle the problem because computation over k0 differs from geometric membership over its algebraic closure, genus-zero subfields may be nonsplit conics, and separable models may not exist.
Method: When a separable model has degree m ≥ 3, n ≤ g + 2 and points above roots of h are Weierstrass points; the m = 2 case instead yields hyperelliptic levels n ∈ {2g + 1, 2g + 2}.
Method: The resulting search is finite because it examines finitely many Weierstrass-point divisors and tests each candidate using a Kummer criterion, followed by polynomial-division verification.
Organization: The paper organizes these ingredients into results on cyclic covers, Weierstrass points, normal forms, the membership algorithm, and descent to the field of definition.

2. Cyclic covers and superelliptic curves

The section defines cyclic covers, superelliptic levels, and branch data, then develops the differential and Hurwitz-locus structure used to classify the loci Sg,n.

Cyclic covers: A cyclic cover of degree n is determined by an order-n automorphism with genus-zero quotient, while the level spectrum records all such degrees.The spectrum is finite and is an invariant of the curve rather than a chosen model.
Branch data: Branch data records the local monodromies and determines ramification through d_j = gcd(n,l_j) and e_j = n/d_j, including possible branching at infinity.Isomorphism classes are preserved under permuting branch points, Möbius transformations, and multiplying all monodromies by a unit modulo n.
Superelliptic curves: A curve is superelliptic of level n precisely when it has a separable model y^n = h(x), with every finite branch point totally ramified.This convention differs from literature definitions based on normality or centrality in the automorphism group.
Differentials at ramified points: The vanishing sequence at a totally ramified point is computed from explicit holomorphic differentials, whose indexed family forms a basis of H0(C, Ω1).These sequences identify special points and provide the data used to restrict the search for separable models.
Hurwitz loci: For fixed genus g and level n, the genus equation has at most two solutions for the degree m, and any two differ by one with n dividing m + 1.Consequently, the corresponding separable branch data agree up to the allowed Möbius transformation.
The loci Sg,n: The loci satisfy Sg,n = ∅ for n > 2g + 2, while levels above g + 2 occur only at 2g + 1 and 2g + 2 as single hyperelliptic points.For 3 ≤ n ≤ g + 2, nonempty loci have m ≥ 3; when 3 divides g + 2, Sg,g+2 contains y^(g+2) = x^3 − x.
The loci Sg,n: Distinct levels can intersect, as shown by the genus-three Picard curve lying in S3,3 ∩ S3,4.In genus five, no non-hyperelliptic curve has a separable model under this convention, although many cyclic-cover levels are realised.

3. Bounds on the level and Weierstrass points

The paper bounds separable superelliptic levels and identifies branch points through Weierstrass theory, with hyperelliptic models as the exceptional case. These results make the search for admissible levels finite and expose limits of earlier bounds.

Level bounds: For separable models with deg h=m≥3, 2g≥(m−1)(n−2) and n≤g+2; when m=2, C is hyperelliptic with n∈{2g+1,2g+2}.The bound n≤2g+2 is attained by y^(2g+2)=x^2−1.
Level bounds: If C is nonhyperelliptic, every separable level satisfies n≤g+2, attained by y^n=x^3−x when 3 divides n and n≥6.
Scope and prior bounds: The separability hypothesis sharpens the general automorphism bound 4g+2 to 2g+2, whereas without separability examples attain Wiman’s bound.
Scope and prior bounds: Earlier inequalities assuming m>n fail outside that range, including the genus-four example y^5=x(x^2−1), where n=g+1.
Scope and prior bounds: A cyclic cover of degree ten can exist without a separable level-ten model: the genus-four curve y^5=x^3−x lies in S_4,5 but not S_4,10.
Weierstrass points: For m≥3, points above roots of h are Weierstrass points; for m=2, their vanishing sequence is generic and the degree-two map is instead y.
Weierstrass points: A separable model with m≥3 yields two distinct Weierstrass points P,P′ satisfying nP∼nP′, while m=2 yields only the hyperelliptic alternatives n=2g+1 or 2g+2.

4. The superelliptic normal form

The normal-form construction uses Kummer theory to recover a cyclic generator and then reduces local exponents modulo n to obtain a polynomial model. A congruence criterion decides when that model can be made separable, as illustrated by an explicit genus-four example.

4. The superelliptic normal form: Given a cyclic degree-n map t:C→P^1 with cyclic Galois extension k(C)/k(t), Kummer theory constructs a birational model v^n=h(u) with k(u)=k(t).
4.1. The Kummer step.: The eigenspace W_1 is one-dimensional, and any nonzero s∈W_1 satisfies s^n∈k(t) and generates k(C) over k(t).
4.1. The Kummer step.: The construction requires a base field containing ζ_n; it also applies over any characteristic-zero field containing ζ_n where C, t, and the generator τ are defined.
4.1. The Kummer step.: Local monodromy determines ord_a(h̃) modulo n, with exponent zero modulo n at every unbranched place.
4.1. The Kummer step.: Multiplying s by r(t) changes h̃ by an n-th power, so only exponents modulo n matter; negative exponents are removed by a rational substitution.
4.2. Reduction of the exponents.: Writing each exponent as its remainder modulo n and choosing r(t) produces v^n=h(u) with u=t and k(u,v)=k(C).
4.3. When h is separable.: A separable polynomial model exists exactly when some unit c modulo n satisfies c l_j≡1 modulo n for every branch point except at most one.
4.3. When h is separable.: For y^4=x^2(x^3−1), the criterion fails for both units modulo four, so this degree-four cover has no separable model, although the curve is superelliptic of level six.

5. Deciding superellipticity

The algorithm decides whether a curve belongs to Sg,n by finitely searching Weierstrass-point divisors and testing candidate cyclic functions, including separate treatment of two hyperelliptic levels. It returns separable normal forms and birational transformations over an explicit finite extension, with correctness guaranteed by Theorem 5.10.

Input and field of computation: The input is an absolutely irreducible plane model over k0, while membership is tested geometrically after extending to a field generated by relevant Weierstrass points and roots of unity.The plane model need not be smooth; its smooth projective model is the curve being tested.
Finite candidate search: The search is finite because candidate separable branch divisors are sums of Weierstrass points with prescribed weights, and genus-g curves have at most g^3 − g Weierstrass points.Candidates are restricted to levels n ≤ g + 2 and pairs (n,m) satisfying the genus condition.
Candidate testing and construction: Each candidate is tested by a dimension condition and a Kummer criterion, then verified by polynomial division; the resulting cyclic function is a Möbius transform of the quotient function when the relevant eigenspace has dimension two.The construction uses Riemann–Roch spaces of rational divisors over the computation field.
Algorithm output: Algorithm 5.9 outputs every level admitting a separable model together with u, v, and h satisfying v^n = h(u), with h separable and the generated function field equal to that of C.The output field is the explicit extension kn used to split candidate points pointwise.
Hyperelliptic exception: Levels 2g + 1 and 2g + 2 are handled separately because their separable models have m = 2 and their branch points are not Weierstrass points, making them invisible to the main search.These cases are hyperelliptic and reduce after completing the square to y^n = x^2 − c.
Correctness and field bounds: Theorem 5.10 proves that the algorithm outputs n exactly when [C] belongs to Sg,n, and that the returned model and birational transformation are defined over kn.The uniform field kn need not be minimal, and some levels may be defined over smaller fields.

6. Descent to the field of definition

The descent algorithm decides superelliptic levels over the field of definition and constructs normal forms, usually over k0, with a controlled exceptional extension. Its guarantees cover separable models of degree at least three and degree two, while timings identify the Wronskian determinant as a major cost in higher genus.

Descent assumptions: A quotient group must be Galois-stable and its genus-zero quotient must be a k0-form of P1; over a number field, the quotient becomes P1 after at most one quadratic extension.When the level group is not unique, conjugate groups can yield distinct covers with the same branch data.
Candidate construction and testing: The algorithm enumerates k0-rational candidate branch divisors and tests each candidate using Riemann–Roch constructions, divisor-shape checks, and the Kummer criterion.Candidates are discarded when asserted divisor shapes or dimensions fail; the genuine branch divisor is guaranteed to succeed.
Candidate construction and testing: The five cases of Theorem 6.2 are exhaustive, with case (d′) requiring one base change of degree at most m when the branch divisor lacks a k0-rational point.Case (d′) applies when A = 0, c = 0, d > 1, S has no k0-rational point, and m = d or m ≥ n.
Completeness guarantees: For degree at least three, Theorem 6.4 outputs the level and a normal form, extending constants at most once and only in the specified non-rational-root cases.The extension has degree at most deg h and occurs only when no root of h is k0-rational and either deg h divides n or deg h = n + gcd(n, deg h) with gcd(n, deg h) > 1.
Completeness guarantees: For degree-two models with n in {2g + 1, 2g + 2}, Proposition 6.5 returns a separable quadratic model and transformation over k0 without extending constants.The broader corollary combines the degree-two result with the degree-at-least-three guarantee.
Scope over the field of definition: The run over k0 returns a subset of the geometric answer, becoming equal only when every separable level has a cover and model defined over k0.For odd n, the subgroup obstruction is removed and the quotient conic splits after at most one quadratic extension, but Kummer obstructions may remain.
Implementation and timings: On genus-six examples, the Wronskian determinant consumed 130.8 of 133.9 seconds and 352.6 of 361.5 seconds, making it the dominant reported cost there.The determinant was not the leading cost elsewhere, where norm factorization and level attempts could dominate.