ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 24 (2024) #P3.02
https://doi.org/10.26493/1855-3974.2570.5e8
(Also available at http://amc-journal.eu)
On p-gonal fields of definition*
Ruben A. Hidalgo †
Departamento de Matemática y Estadı́stica, Universidad de La Frontera, Temuco, Chile
Received 24 February 2021, accepted 28 November 2023, published online 11 June 2024
Abstract
Let S be a closed Riemann surface of genus g ≥ 2 and φ be a conformal automorphism
of S of prime order p such that S/⟨φ⟩ has genus zero. Let K ≤ C be a field of definition of
S. We prove the existence of a field extension F of K, of degree at most 2(p−1), for which
S is definable by a curve of the form yp = F (x) ∈ F[x], in which case φ corresponds to
(x, y) 7→ (x, e2πi/py). If, moreover, φ is also definable over K, then F can be chosen to
be at most a quadratic extension of K. For p = 2, that is when S is hyperelliptic and φ
is its hyperelliptic involution, this fact is due to Mestre (for even genus) and Huggins and
Lercier-Ritzenthaler-Sijslingit in the case that Aut(S)/⟨φ⟩ is non-trivial.
Keywords: Riemann surfaces, p-gonal curves, automorphisms.
Math. Subj. Class. (2020): 30F10, 30F20, 14H37, 14H55
1 Introduction
In [23], H. A. Schwarz proved that the group Aut(S) of conformal automorphisms of a
closed Riemann surface S of genus g ≥ 2 is finite. Later, in [17], A. Hurwitz obtained the
upper bound |Aut(S)| ≤ 84(g − 1) (this is known as the Hurwitz’s bound).
Let p ≥ 2 be a prime integer. We say that a closed Riemann surface S is cyclic p-
gonal if there exists some φ ∈ Aut(S) of order p such that the quotient orbifold S/⟨φ⟩
has genus zero. In this case, φ is called a p-gonal automorphism and the cyclic group
⟨φ⟩ a p-gonal group of S. The case p = 2 corresponds to S being hyperelliptic and φ
its (unique) hyperelliptic involution. The case p = 3 was studied by R. D. M. Accola in
[1]. In [10], G. González-Diez proved that p-gonal groups are unique up to conjugation in
Aut(S). In [13], it was observed that, if p ≥ 5n − 7, where n ≥ 3 is the number of fixed
*The author would like to express their gratitude to both referees for their valuable feedback, suggestions, and
corrections that have significantly improved the paper.
†Partially supported by projects Fondecyt 1230001 and 1220261.
E-mail address: ruben.hidalgo@ufrontera.cl (Ruben A. Hidalgo)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
2 Ars Math. Contemp. 24 (2024) #P3.02
points of φ, then ⟨φ⟩ is the unique p-group in Aut(S). Results concerning automorphisms
of p-gonal Riemann surfaces can be found, for instance, in [2, 3, 4, 5, 11, 27].
As a consequence of the Riemann-Roch theorem, a closed Riemann surface S can be
described by an (either affine or projective) irreducible complex algebraic curve, i.e., after
desingularization (if it is non-smooth) and filling at some punctures in the affine case, it
carries a Riemann surface structure which is biholomorphic to that of S (see Remark 1.1
for the case of cyclic p-gonal surfaces). A subfield K of the field C of complex numbers is
called a field of definition of S (or that S is definable over K) if there is an irreducible al-
gebraic curve representing S, which is defined as the common zeroes of some polynomials
with coefficients in K. The intersection of all the fields of definition of S is called the field
of moduli of S. In general, it is not a field of definition (see Section 3).
If we are given a (finite) group G < Aut(S) and the geometrical structure of the
quotient orbifold S/G, then it is not a simple task to find an algebraic curve for S reflecting
the action of G. A family of surfaces for which algebraic models are well known is the
case of cyclic p-gonal Riemann surfaces, which we proceed to recall below.
Let S be a p-gonal Riemann surface, φ ∈ Aut(S) be a p-gonal automorphism and
π : S → Ĉ be a regular branched cover with ⟨φ⟩ its deck group. Let {a1, . . . , am} ⊂ Ĉ =
C ∪ {∞} be the set of branch values of π. If aj ̸= ∞, for every j = 1, . . . ,m, then there
exist integers n1, ..., nm ∈ {1, ..., p−1}, n1+ · · ·+nm ≡ 0 mod p, such that S is defined
by the affine, irreducible and smooth p-gonal curve with equation
E : yp = F (x) =
m∏
j=1
(x− aj)nj ∈ C[x]. (1.1)
If one of the branch values is equal to ∞, say am = ∞, then in (1.1) we delete the
corresponding factor (x − am)nm and assume n1 + · · · + nm−1 ̸≡ 0 mod p. In this
affine algebraic model, π(x, y) = x and φ(x, y) = (x, ωpy), where ωp = e2πi/p. In the
hyperelliptic case, i.e., p = 2, in the above one has m ∈ {2g + 1, 2g + 2} and nj = 1.
Remark 1.1. The affine curve (1.1) is smooth at those points (x, y), where y ̸= 0. At
a point (aj , 0), the curve is smooth exactly when nj = 1 (anyway, if nj > 1, it has a
neighborhood homeomorphic to a disc). An irreducible projective algebraic curve defining
S is obtained from the above affine one as
Ê : ypzn1+···+nm−p =
m∏
j=1
(x− ajz)nj . (1.2)
As in the affine model, the projective curve Ê is smooth at the points [x : y : 1], where
y ̸= 0. At the points [aj : 0 : 1] it is smooth if and only if nj = 1 (again, in the other cases
there is a neighborhood homeomorphic to a disc). The curve is also non-smooth at the point
[0 : 1 : 0]. After normalization of the curve, one obtains a closed Riemann surface which
is biholomorphic to S. In this case, π([x : y : z]) = x/z and φ([x : y : z]) = [x : ωpy : z].
If F is a subfield of C such that in (1.1) we have F (x) ∈ F[x], then we say that F is
a p-gonal field of definition of S (and that S is cyclically p-gonally defined over F). Note
that there are infinitely many different p-gonal fields of definition for S (for instance, if T
is a Möbius transformation, then we may replace the values aj by T (aj)).
Given a field of definition of a p-gonal Riemann surface S, it is not clear at first sight if
it is a p-gonal field of definition. Also, it might be that a minimal p-gonal field of definition
R. A. Hidalgo: On p-gonal fields of definition 3
is not a minimal field of definition (see the exceptional case (m, p) = (4, 3) in Section 4.1).
This paper aims to provide an argument to show that, given any field of definition K of S,
there is a p-gonal field of definition F which is an extension of degree at most 2(p−1) over
K.
If φ is an automorphism of S, then we say that S and φ are simultaneously defined
over K if there is an algebraic curve model of S, defined over K, such that φ is given by a
rational map on it with coefficients in K.
Theorem 1.2. Let S be a cyclic p-gonal Riemann surface of genus g ≥ 2, with a p-gonal
automorphism φ, and let K be a field of definition of S. Then
(1) There is p-gonal field of definition of S, this being an extension of degree at most
2(p− 1) of K (which is also a field of definition of φ).
(2) If both S and φ are simultaneously defined over K, then there is a p-gonal field of
definition of S, this being an extension of degree at most two of K.
(3) If in Equation (1.1) n1 = · · · = nm, then there is a p-gonal field of definition of S,
this being an extension of degree at most two of K.
Remark 1.3. Theorem 1.2 is still valid if we change C to any algebraically closed field,
where in positive characteristic we need to assume that p is different from the characteristic.
Remark 1.4. For each integer n ≥ 2, not necessarily prime, the definition of cyclic n-
gonal Riemann surface S, n-gonal automorphism φ and n-gonal group ⟨φ⟩ is the same as
for the prime situation. In the particular case that every fixed point of a non-trivial power
φk is also a fixed point of φ, the definition of an n-gonal curve is the same as in (1.1),
but replacing p by n and assuming each the exponent nj to be relatively prime to n. In
this case, under the assumption that S has a unique n-gonal group ⟨φ⟩ (this is the situation
for generalized superelliptic Riemann surfaces [15]), then the arguments of the proof of
Theorem 1.2 allows us to obtain that: if K is a field of definition of S, then there is an n-
gonal field of definition of S, this being an extension of degree at most 2ϕ(n) of K, where
ϕ(n) is the ϕ-Euler function.
2 An application to hyperelliptic Riemann surfaces
Let S be a hyperelliptic Riemann surface (i.e., p = 2) with hyperelliptic involution φ and let
K be a field of definition of S. As φ is unique, one may consider the group Autred(S) :=
Aut(S)/⟨φ⟩, called the reduced group of automorphisms of S.
For even genus, in [22], J-F. Mestre proved that S is also hyperelliptically definable
over K. If the genus is odd, then the previous fact is in general false; as can be seen from
examples in [8, 9, 20, 21]. In [16], B. Huggins proved that if Autred(S) is neither trivial nor
cyclic, then S is also hyperelliptically definable over K. In [21], R. Lercier, C. Ritzenthaler
and J. Sijslingit proved that S can be hyperelliptically defined over a quadratic extension
of K if the reduced group is a non-trivial cyclic group. Our theorem asserts that this fact is
still valid even if the reduced group is trivial.
Corollary 2.1. If K is a field of definition of a hyperelliptic Riemann surface, then it is
hyperelliptically definable over an extension of degree at most two of K.
4 Ars Math. Contemp. 24 (2024) #P3.02
3 An application to fields of moduli
Let S be a closed Riemann surface and let C be an irreducible algebraic curve representing
it. The field of moduli MS of S is the fixed field of the group ΓC = {σ ∈ Aut(C/Q) :
Cσ ∼= C}; this field does not depend on the choice of the algebraic model C. In [18],
S. Koizumi proved that MS coincides with the intersection of all fields of definition of S,
but in general it might not be a field of definition [6, 7, 12, 16, 19]. If Aut(S) is trivial (the
generic situation for g ≥ 3), then Weil’s descent theorem [25] asserts that MS is a field
of definition of S. In [26], J. Wolfart proved that if S/Aut(S) is the Riemann sphere with
exactly 3 cone points (i.e., S is quasiplatonic), then MS is also a field of definition of S.
In a more general setting, if S/Aut(S) has genus zero, then it is known that S is definable
over an extension of degree at most two of MS (see [14] for a more general statement).
Now, let S be a p-gonal Riemann surface of genus g ≥ 2 and let G = ⟨φ⟩ < Aut(S)
be a p-gonal group. As previously noted, S is either definable over MS or over a suitable
quadratic extension of it (but it might not be cyclically p-gonally definable over such a
minimal field of definition). In the case that G is not a unique p-gonal subgroup, in [28],
A. Wootton noted that S can be cyclically p-gonally defined over an extension of degree at
most 2 of its field of moduli. In the case that G is the unique p-gonal subgroup, the quotient
group Aut(S)/G is called the reduced group of S. In [19], A. Kontogeorgis proved that if
the reduced group is neither trivial nor a cyclic group, then S can always be defined over
its field of moduli. So, a direct consequence of Theorem 1.2 is the following.
Corollary 3.1. Let S be a cyclic p-gonal Riemann surface with a p-gonal group G = ⟨φ⟩.
(1) If G is not a normal subgroup of Aut(S), then S is cyclically p-gonally definable
over an extension of degree at most two of MS .
(2) If G is a normal subgroup of Aut(S) and Aut(S)/G is different from the trivial
group or a cyclic group, then S is cyclically p-gonally definable over an extension of
degree at most 2(p − 1) of MS . Moreover, if φ also is defined over MS , then the
extension can be chosen to be of degree at most two.
(3) If G = Aut(S), then S is cyclically p-gonally definable over an extension of degree
at most 4(p− 1) of its field of moduli. Moreover, if φ also is defined over MS , then
the extension can be chosen of degree at most 4.
As every hyperelliptic Riemann surface is definable over an extension of degree at most
two of its field of moduli, Corollary 2.1 asserts the following.
Corollary 3.2. Every hyperelliptic Riemann surface is hyperelliptically definable over an
extension of degree at most 4 of its field of moduli. Moreover, if either (i) the genus is
even or (ii) the genus is odd and the reduced group is not trivial, then the hyperelliptic
Riemann surface is hyperelliptically defined over an extension of degree at most 2 of its
field of moduli.
Examples of hyperelliptic Riemann surfaces with a trivial reduced group that cannot be
defined over their field of moduli were provided by C. J. Earle [6, 7] and G. Shimura [24].
The same type of examples, but with a non-trivial cyclic reduced group, were provided by
B. Huggins [16].
R. A. Hidalgo: On p-gonal fields of definition 5
4 Proof of Theorem 1.2
We assume the p-gonal Riemann surface S to be provided by an irreducible curve C, de-
fined over a subfield K of C. If K is the algebraic closure of K inside C, then (in this
algebraic model) the p-gonal automorphism φ is given by a rational map defined over K.
We divide the arguments depending on the uniqueness of the cyclic group ⟨φ⟩.
4.1 The case when ⟨φ⟩ is not unique
The following result, due to A. Wootton, describes those cases where the uniqueness fails.
Theorem 4.1 ([28, A. Wootton]). Let S be a cyclic p-gonal Riemann surface of genus
g ≥ 2 and let m = 2(g+p−1)/(p−1). If (m, p) is different from any the following tuples
(i) (3, 7), (ii) (4, 3), (iii) (4, 5), (iv) (5, 3), (v) (p, p), p ≥ 5, (vi) (2p, p), p ≥ 3,
then S has a unique p-gonal group.
In the same paper, Wootton describes the exceptional cyclic p-gonal Riemann surfaces,
ie., where the p-gonal group is non-unique.
(i) Case (m, p) = (3, 7) corresponds to Klein’s quartic (a non-hyperelliptic Riemann
surface of genus 3) x3y+y3z+z3x = 0, whose group of automorphisms is PGL2(7)
(of order 168). This surface is cyclically 7-gonally defined as y7 = x2(x− z)z4.
(ii) Case (m, p) = (4, 3) corresponds to the genus 2 Riemann surface defined hyperel-
liptically by y2z3 = x(x4−z4), whose group of automorphisms is GL2(3) (of order
48). This surface is cyclically 3-gonally defined as y3z3 = (x2 − z2)(x2 − (15
√
3−
26)z2)2.
(iii) Case (m, p) = (4, 5) corresponds to the genus 4 non-hyperelliptic Riemann surface,
called Bring’s curve, which is the complete intersection of the quadric x1x4+x2x3 =
0 and the cubic x21x3 + x
2
2x1 + x
2
3x4 + x
2
4x2 = 0 in the 3-dimensional complex
projective space. Its group of automorphisms is S5, the symmetric group in five
letters S5. This surface is cyclically 5-gonally defined as y5z5 = (x2−z2)(x2+z2)4.
(iv) Case (m, p) = (5, 3) corresponds to the genus 3 non-hyperelliptic closed Riemann
surface x4+y4+ z4+2i
√
3z2y2 = 0, whose group of automorphisms has order 48.
The quotient of that surface by its group of automorphisms has signature (0; 2, 3, 12).
This surface is cyclically 3-gonally defined as y3z3 = x2(x4 − z4).
(v) Case (m, p) = (p, p), where p ≥ 5, corresponds to the Fermat curve xp+yp+zp = 0,
whose group of automorphisms is Z2p ⋊ S3. This is already in a p-gonal form as
yp = −zp − xp.
(vi) Case (m, p) = (2p, p), where p ≥ 3. There is a 1-dimensional family with group
of automorphisms Z2p ⋊ Z22 (the quotient by that group has signature (0; 2, 2, 2, p)).
Also, there is a surface with group of automorphisms Z2p ⋊D4 (the quotient by that
group has signature (0; 2, 4, 2p). These surfaces are cyclically p-gonally defined as
ypzp = (xp − apzp)(xp − zp/ap) = x2p − (ap + 1/ap)xpzp + z2p.
6 Ars Math. Contemp. 24 (2024) #P3.02
Note that, in all the above exceptional cases, the surface S is cyclically p-gonally de-
fined over an extension of degree at most 2 over the field of moduli. In fact, with only the
exception of case (ii), S is cyclically p-gonally defined over its field of moduli. So, we are
done in this situation.
4.2 The case when ⟨φ⟩ is unique
We now assume that ⟨φ⟩ is unique. Set Γ = Gal(K/K). Let us consider a rational map
π : C → P1K, defined over K, which is a regular branched covering with ⟨φ⟩ as its deck
group and whose branch values are a1, ..., am ∈ C (in fact, these values belong to K). Let
the integers n1, ..., nm ∈ {1, ..., p − 1}, n1 + · · · + nm ≡ 0 mod p, be such that C is
isomorphic to a p-gonal curve E with Equation (1.1).
4.2.1 Proof of Part (1)
Let us recall that φ is already defined over K. In the next, we note that φ is defined over an
extension of K of degree at most p− 1.
Claim 4.2. The rational map φ is defined over an extension K1 of K of degree at most
p− 1.
Proof. If σ ∈ Γ, then φσ is an automorphism of order p of Cσ = C. As we are assuming
the uniqueness of ⟨φ⟩, we must have that φσ ∈ Ω := {φ,φ2, . . . , φp−1}. In particular,
the subgroup A of Γ consisting of those σ such that φσ = φ must have index at most the
cardinality of the set Ω, which is p − 1. This asserts that φ is defined over the fixed field
K1 of A, which is an extension of degree at most p− 1 of K.
Set Γ1 = Gal(K/K1). If τ ∈ Γ1, then (as the identity I : C → C = Cτ conjugates
⟨φ⟩ = ⟨φ⟩τ = ⟨φτ ⟩ to itself), there is a (unique) automorphism gτ of P1K such that π
τ =
πτ ◦ I = gτ ◦ π (see the following diagram).
C
I−−−−→ C = Cτ
π
y πτy
P1K
gτ−−−−→ P1K
As the group of automorphisms of P1K is given by Möbius transformations (i.e., ele-
ments of PGL2(K)), we must have gτ ∈ PGL2(K).
We may apply each σ ∈ Γ1 to the above diagram to obtain the following one
Cσ = C
I−−−−→ C = Cστ
πσ
y πστy
P1K
gστ−−−−→ P1K
The above permits us to obtain the following diagram
R. A. Hidalgo: On p-gonal fields of definition 7
C
I−−−−→ C = Cσ I=I
σ
−−−−→ C = Cστ
π
y πσy πστy
P1K
gσ−−−−→ P1K
gστ−−−−→ P1K
As the transformation gρ is uniquely determined by ρ ∈ Γ1, the collection {gρ}ρ∈Γ1
satisfies the co-cycle relation
gστ = g
σ
τ ◦ gσ, σ, τ ∈ Γ1.
Weil’s descent theorem [25] ensures the existence of a genus zero irreducible and non-
singular algebraic curve B, defined over K1, and an isomorphism R : P1K → B, defined
over K, so that
gσ ◦Rσ = R, σ ∈ Γ1.
Also, for σ ∈ Γ1, we have {σ(a1), ..., σ(am)} = {gσ(a1), ..., gσ(am)}, so it follows
that {R(a1), ..., R(am)} is Γ1-invariant.
Let us denote by A(nj) the set of those ak’s for which nk = nj .
Claim 4.3. Each set R(A(nj)) is Γ1-invariant.
Proof. If σ ∈ Γ1, then (as πσ = gσ ◦ π) the set gσ(A(nj)) corresponds to the set of those
σ(ak) having the same nl (for some l), that is, gσ(A(nj)) = σ(A(nl)). As φσ = φ, we
must have nl = nj , that is, gσ(A(nj)) = σ(A(nj)). This last equality implies the desired
claim.
Claim 4.4. There is an effective K1-rational divisor U ≥ 0 of degree at most two in B.
Proof. We follow similar techniques as used by Huggins in her thesis [16] (and other au-
thors). Let us consider any K1-rational meromorphic 1-form ω in B. Since B has genus
zero, the canonical divisor K = (ω) is a K1-rational of degree −2. In this way, there is a
positive integer d such that the divisor D = R(a1) + · · ·+R(am) + dK is K1-rational of
degree 1 or 2. If D ≥ 0, then we set U := D.
Let us assume D is not effective. Let us consider the Riemann-Roch space L(D),
consisting of those non-constant rational maps ϕ : B → P1K whose divisors satisfy (ϕ) +
D ≥ 0 together with the constant ones. As the divisor D is K1-rational, for every σ ∈ Γ1
and every ϕ ∈ L(D), it follows that ϕσ ∈ L(D). This, in particular, permits us to observe
that we can find a basis of L(D) consisting of rational maps defined over K1. One of
the elements of such a basis must be a non-zero constant map. As, by Riemann-Roch’s
theorem, L(D) has dimension 2 (if D has degree one) or 3 (if D has degree two), we may
find a non-constant f ∈ L(D) belonging to such a basis (defined over K1). In this case, we
may take U = (f) +D ≥ 0.
By Claim 4.4, there is an effective K1-rational divisor U of degree 1 or 2 and U ≥ 0.
We have three possibilities:
(1) U = s, where s ∈ B is K1-rational; or
(2) U = 2t, where t ∈ B is K1-rational; or
8 Ars Math. Contemp. 24 (2024) #P3.02
(3) U = r + q, where r, q ∈ B, r ̸= q, and {r, q} is Γ1-invariant.
In cases (1) and (2) we have the existence of a K1-rational point in B. In this case, we
set K2 = K1. In case (3) we have a point (say r) in B which is rational over a quadratic
extension K2 of K1.
Let b ∈ B be a K2-rational point (whose existence is provided above). By Riemann-
Roch’s theorem, the Riemann-Roch space L(b) (where b is thought of as a divisor of degree
one) has dimension 2. Similarly as above, we may choose a basis {1, L} of L(b), with each
element defined over K2. In this case, L : B → Ĉ turns out to be an isomorphism defined
over K2.
We have that Q = L ◦R ◦ π : C → Ĉ is a Galois (branched) covering with deck group
⟨φ⟩ and whose branch values are {L(R(a1)), ..., L(R(am))}. It follows that S is p-gonally
defined by
yp = F (x) =
m∏
j=1
(x− L(R(aj)))nj .
As the sets {L(R(a1)), ..., L(R(am))} and L(R(A(nj))) are Gal(K/K2)-invariant
(by Claim 4.3 and the fact that K1 is a subfield of K2), it follows that F (x) =∏m
j=1 (x− L(R(aj)))
nj ∈ K2[x]. As K2 is an extension of degree at most two of K1
and the last one is an extension of degree at most p− 1 of K, we are done.
4.2.2 Proof of Parts (2) and (3)
If φ is already defined over K then we assume K1 = K (i.e., we set Γ1 = Γ) in the above
arguments. Similarly, if in Equation (1.1) we have that n1 = · · · = nm = n, then there
will be only one set A(n). In this case, in the previous arguments, we do not need to use
Claim 4.3 (where it was needed for the choice of K1) and we may work as in the proof of
Part (1) with K instead of K1.
ORCID iDs
Ruben A. Hidalgo https://orcid.org/0000-0003-4070-2819
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