ON DESCENT OF MARKED DESSINS

JEROEN SIJSLING AND JOHN VOIGHT

Abstract. We revisit a statement of Birch that the field of moduli for a marked three-point ramified cover is a field of definition. Classical criteria due to Debes and Emsalem canbe used to prove this statement when the curves under consideration are smooth. We give ageometric interpretation of these results and describe methods to make them constructive.Moreover, we give explicit counterexamples for singular curves.

Contents

1. Background, notation, and statements of main results 32. Weil cocycles 73. Counterexamples 104. Branches 165. Descent of marked curves 216. Descent of marked Belyı maps 28References 31

In his Esquisse d’un Programme [11], Grothendieck embarked on a study of Gal(Q |Q),the absolute Galois group of Q, via its action on a set with a geometric description: the setof quasifinite morphisms f : X → P1 of smooth projective curves over Q unramified awayfrom {0, 1,∞}, known as Belyı maps. Belyı had proven [1, 2] that a smooth projective curveover C (or alternatively, a compact Riemann surface) can be defined over Q if and only ifit admits a Belyı map, and part of Grothendieck’s fascination with Belyı’s theorem was aconsequence of the simple combinatorial and topological characterization that follows fromit, encoded in the language of dessins (also called drawings). Progress has been made onthis program—in part led by the fecundity of certain examples—but in large part the Galoisaction on the set of Belyı maps remains mysterious.

The heart of this program is to understand over what number fields a Belyı map is definedand when two Galois-conjugate Belyı maps are isomorphic. One of the more subtle aspectsof this investigation is the issue of descent: whether or not a Belyı map is defined over itsfield of moduli, which is then necessarily a minimal field of definition. Some concrete descentproblems were studied by Couveignes [6] and Couveignes–Granboulan [7]; a more theoreticalapproach to the subject (and much more general than the case of Belyı maps) was developedby Debes–Douai [8] and Debes–Emsalem [9].

By the classical theory of Weil descent, a Belyı map with trivial automorphism groupdescends to its field of moduli. In many cases, a suitable rigidification of the Belyı map will

Date: April 10, 2015.1

eliminate nontrivial automorphisms, thus ensuring that the field of moduli (of the rigidifica-tion) becomes a field of definition. For example, a classical result due to Coombes–Harbater[5] shows that G-Galois Belyı maps (those equipped with an identification of the automor-phism group of the map with G) permit descent to the field of moduli; an explicit descentalgorithm for such rigidified Galois covers has been exhibited by Sadi [19].

In his article on dessins, Birch [3] claims that the issue of descent is overcome for asimple rigidification: he states that by equipping a Belyı map f : X → P1 with a pointP ∈ X(Q) satisfying f(P ) = ∞, the marked curve (X;P ) descends to its field of moduli[3, Theorem 2]. Birch provides several references to more general descent results, but uponfurther reflection we could not see how these implied his particular statement. Obtaininga proof or counterexample was not merely of theoretical importance: indeed, in order todetermine explicit equations for Belyı maps, one needs a handle on their field of definition [21,§7]. Moreover, if the descent obstruction indeed vanishes, then it is desirable to have generalmethods to obtain an equation over the field of moduli. The issue under consideration,and its implementation, is therefore also very relevant for practical purposes—whence ourinterest in the problem.

If we leave the immediate context of Belyı maps and consider source curves X that arenot assumed to be smooth, then descent to the field of moduli is not always possible. Forexample, we show in Section 3 that the projective plane curve with equation

X1 : i(x2 + y2)xy + x2y2 + (1 + i)x2z2 + (1− i)y2z2 + 2ixyz2 = 0

admits a map to P1C branched over {0, 1,∞} that is isomorphic to its complex conjugate but

cannot be defined over R (indeed, neither can the normalization of X1). This obstruction isintrinsic to the curve X1, in the sense that while X1 has a unique singular point P = (0 : 0 :1) ∈ X1(C), and the field of moduli of the pair (X1;P ) is R, nevertheless both the curve X1

and the pair (X1, P ) fail to descend to their common field of moduli R.The situation can be even more paradoxical. Consider the genus 4 hyperelliptic curve over

R defined by the equation

Y2 : y2 = 10x10 − 42x9 + 67x8 − 36x7 + 23x6 + 23x4 + 36x3 + 67x2 + 42x+ 10

The automorphism group of Y2 is cyclic of order 4 generated by

% : (x, y) 7→ (−1/x, y/x5).

We have Q = (1 + i, 0) ∈ Y2(C), and there is a map Y2 → X2 over R in which the pairs ofpoints {Q, %(Q)} and {Q, %(Q)} are contracted to single points P and P and such that themarked curve (X2;P ) does not descend to its field of moduli R with respect to the extensionC |R. But in this case the curve X2 itself does descend to R! The bizarre consequence: whenconsidering singular curves, adding additional rigidifying structure will not always trivializedescent obstructions, indeed it may even give rise to them. (See also Remark 3.23.)

It is crucial in our argument that a marked point P on the curve X is itself singular; if itis regular, then indeed a descent of (X;P ) does exist, and Birch’s statement holds true: amarked Belyı map (with smooth source X) indeed descends to its field of moduli. In short,

a marked dessin descends.

One can show that this result follows from more general results of Debes–Emsalem [9, §5]. Inthis paper, we provide an alternative, conceptual, and self-contained presentation of Birch’stheorem that is more explicit from the point of view of Weil cocycles. We interpret the

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results of Debes–Emsalem in terms of what we formally define to be branches of a cover; thisapproach provides a more unified treatment, avoids the use of Puiseux series in the proof,and makes it straightforward to read off a Weil cocycle from the given rigidification.

In brief, the classical proof of Birch’s theorem proceeds as follows: by rigidifying a smoothcurve X over Q with a point P , we obtain a subgroup G ≤ Aut(X) that fixes P , as wellas a field of moduli F for the rigidified curve (X;P ). Since the case where the genus of Xis at most 1 is easily dealt with (see Section 5), we may suppose that G is finite. Then thequotient V = G\X is a curve that admits a distinguished model V0 over F equipped with anF -rational point given by the image of P ; and it turns out that the existence of this markedpoint implies that X descends to F . A similar argument works for Belyı maps.

In the approach we take here using branches, it is clear that it is not necessary to demandthat the marked F -rational point of V0 is unramified in the cover X → V0. In some places inthe literature (e.g., the final section in the classical work of Debes–Douai [8]), this additionalhypothesis is stated, giving the false impression that it is necessary for descent. Yet Birchmay indeed take P ∈ X(Q) to be any point of X, even a ramified point with nontrivialstabilizer under Aut(X).

This also gives us a way to consider the structure of our singular counterexamples: inboth cases, the singular point P on X splits into two points on the normalization of X,and similarly for the point Q on the quotient V . As it turns out, this prevents us fromextracting a cocycle as in the classical approach. In a sense, then, a singular point providesless rigidification than a regular point. Marking a point on the normalization (or equivalently,marking a place of the function field) again eliminates the obstruction to descent.

Our paper is organized as follows. In Section 1, we introduce the relevant background andnotions under consideration, and we give precise statements of the main results of this article.We avoid saying anything more precise in this introduction as precision is essential in thissubject; the main statements are Theorems 1.12 and 1.14. After a review of the classical Weildescent criterion in Section 2, we give our counterexamples in the singular case in Section3. Next, in Section 4, we present the theory of branches, beginning our reinterpretation ofthe results in Debes–Emsalem [9]; we show how the presence of an F -rational point on thequotient V0 considered above gives rise to a Weil cocycle and hence a descent of X, andmoreover we show how such a cocycle can be constructed without explicitly determining V0.We show in Section 5 how to explicitly descend marked hyperelliptic curves to their field ofmoduli as well as how to descend a marked Klein quartic (X;P ) using branches. Finally,in Section 6 we consider descent of marked Belyı maps in genus zero, reexamined under thelens of branches.

1. Background, notation, and statements of main results

In this section, we set up the basic background and notation we will use in the rest of thepaper.

Let F be a field with separable closure F sep and corresponding absolute Galois groupΓF = Gal(F sep |F ). When charF = 0, we will also write F sep = F . A curve X over F is avariety over F of dimension 1. A curve is nice if it is smooth, projective, and geometricallyintegral. A map f : Y → X of curves over F is a quasi-finite morphism defined over F .

An isomorphism of maps f : Y → X and f ′ : Y ′ → X ′ of curves over F is a pair (ψ, ϕ) of

isomorphisms ψ : Y∼−→ Y ′ and ϕ : X

∼−→ X ′ over F such that ϕf = f ′ψ, i.e., such that the3

diagram

Yψ //

f��

Y ′

f ′

��X

ϕ // X ′

(1.1)

commutes; if f is isomorphic to f ′ we write f∼−→ f ′.

The absolute Galois group ΓF acts naturally on the set of curves and the set of mapsbetween curves defined over F sep—in both cases, applying the Galois action of the automor-phism σ comes down to applying σ to the defining equations of an algebraic model of X(resp. f) over F sep. The conjugate of a curve X over F sep by an automorphism σ ∈ ΓF isdenoted by σ(X), and similarly that of a map f by σ(f).

Given a curve X over F and an extension K |F of fields, we will denote by XK the baseextension of X to K. We denote the group of of automorphisms of X over K by Aut(X)(K).This latter convention reflects the fact that Aut(X) is a scheme over F , the K-rational pointsof which are the K-rational automorphisms of X. As in the introduction, we occasionallywrite Aut(X) for the group of points Aut(X)(F sep) when no confusion is possible.

We call a map f : Y → X generically Galois if the group Aut(X)(F sep) acts transitivelyon the fibers of f , or equivalently that the induced extension of function K(Y ) |K(X) isGalois. In this case, f can be identified with a quotient qG : Y → Y/G of Y by an F -rationalsubgroup G ≤ Aut(X)(F sep). Given a generically Galois map f : Y → X branching over apoint x ∈ X(F sep), the ramification indices of the points y ∈ f−1(x)(F sep) are all equal, andwe call this common value the branch index of the point x on X.

The main objects of study in this article are curves and maps equipped with extra dataof points, and accordingly we make the following definitions.

Definition 1.2. For n ∈ Z≥0, an n-marked curve (or n-pointed curve) (X;P1, . . . , Pn) over Fis a curve X equipped with distinct points P1, . . . , Pn ∈ X(F ). An isomorphism of n-marked

curves (X;P1, . . . , Pn)∼−→ (X ′;P ′1, . . . , P

′n) over F is an isomorphism ϕ : X

∼−→ X ′ over Fsuch that ϕ(Pi) = P ′i for i = 1, . . . , n.

We will often use the simpler terminology marked curve for a 1-marked curve.

Definition 1.3. For n,m ∈ Z≥0, an n,m-marked map (Y ; f ;Q1, . . . , Qn;P1, . . . , Pm) over Fis a map f : Y → X of curves over F equipped with distinct points Q1, . . . , Qn ∈ Y (F ) andP1, . . . , Pm ∈ X(F ). An isomorphism between n,m-marked maps over F is an isomorphism

of maps (ψ, ϕ) : f∼−→ f ′ over F such that ψ(Qi) = Q′i for j = 1, . . . , n and ϕ(Pi) = P ′i for

i = 1, . . . ,m.

In the definition of an n,m-marked map, no relationship between the points Qj and Piunder f is assumed; moreover, the points Qj may or may not be branch points and thepoints Pi may or may not be ramification points.

To avoid cumbersome notation, we will sometimes write

(Y ; f ;Q1, . . . , Qn;P1, . . . , Pm) = (Y ;R)

and refer to R as the rigidification data.4

Remark 1.4. Marked curves are special cases of marked maps, as we can take the map f tobe the identity. One can also use the notation

(f ;Q1, . . . , Qn;P1, . . . , Pm) = (f ;R)

for a marked map as above, even though the because Y is specified by giving f . We varyour notation depending on how the rigidification naturally arises.

Let K |F be a Galois extension of fields, let X be a curve over K, and let Γ = Gal(K |F ).The field of moduli MK

F (X) of X with respect to K | F is the subfield of K fixed by thesubgroup

(1.5) {σ ∈ Γ = Gal(K |F ) : σ(X) ∼= X} .

In a similar way, one defines the field of moduli MKF (f) of a map, n-marked curve and n,m-

marked map. In this paper we will usually only consider the case where K = F sep, and wesimply call M(X) = MK

F (X) the field of moduli of X, and similarly for maps and markedvariants.

The central question we consider in this article is the following.

Question 1.6. What conditions ensure that a curve, map, or one of its marked variants isdefined over its field of moduli?

For brevity, when an object is defined over its field of moduli with respect to a separableextension K |F , we will say that the object descends.

One subtle issue concerns the existence of wild automorphisms, which interfere with theGalois-theoretic nature of descent. To address this issue, we make the following definition.

Definition 1.7. Let

(Y ; f : Y → X;Q1, . . . , Qm;P1, . . . , Pn) = (Y ;R)

be a marked map of nice curves. We say that Q ∈ Y (K) is tame (with respect to R) if π(Q)is tamely ramified in the quotient π : Y → Aut(Y ;R)\Y .

We say R is tame if a marked point Qi of R is tame with respect to R.

Because of Birch’s paper and the implications for Grothendieck’s theory of dessins, we willbe interested in the following special cases of this question.

Definition 1.8. Suppose charF = 0. A Belyı map over F is a marked map (f : X →P1;−; 0, 1,∞) over F such that X is nice and f is unramified outside {0, 1,∞}. A markedBelyı map (f : X → P1;P ; 0, 1,∞) over F is a Belyı map that is marked. A Belyı map witha marked cusp over F is a marked map (f : X → P1;P ; 0, 1,∞) over F such that f(P ) =∞;the width of the marked point P is the ramification index of P over ∞.

Remark 1.9. Replacing f by 1/f or 1/(1− f), one may realize any marked Belyı map whosemarking lies above 0 or 1 as a Belyı map with a marked cusp.

An isomorphism (ψ, ϕ) of Belyı maps fixes the target P1: since the automorphism ϕ of P1

has to fix 0, 1,∞ by the definition of an isomorphism of marked maps, we must have thatϕ is the identity. When no confusion can result, we will often abbreviate a (marked) Belyımap by simply f .

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Remark 1.10. Relaxing the notion of isomorphism of Belyı maps by removing the markedpoints 0, 1,∞ on P1 typically leads one to consider covers of conics instead of P1, as in workof van Hoeij–Vidunas [22, §§3.3–3.4]; the framework we introduce will extend to this case aswell, but we do not go into the matter here.

Let f : X → P1 be a Belyı map of degree d over Q ⊂ C. Numbering the sheets of fby 1, . . . , d, we define the monodromy group of f to be the group G ≤ Sd generated by themonodromy elements s0, s1, s∞ of loops around 0, 1,∞; the monodromy group is well-definedup to conjugation in Sd, as are the conjugacy classes C0, C1, C∞ in Sd of the monodromyelements, specified by partitions of d. We organize this data as follows.

Definition 1.11. The passport of f is the tuple (g;G;C0, C1, C∞), where g is the geometricgenus of X and G ≤ Sd and C0, C1, C∞ are the monodromy group and Sd-conjugacy classesassociated to f . The size of a passport is the cardinality of the set of Q-isomorphism classes ofBelyı maps with given genus g and monodromy group G generated by monodromy elementsin the Sd-conjugacy classes C0, C1, C∞, respectively.

For all σ ∈ Aut(C), the Belyı map σ(f) has the same ramification indices and monodromygroup (up to conjugation) as f . Therefore the index of the stabilizer of f in Aut(C) isbounded by the size of its passport. The size of a passport is finite and can be computedby combinatorial or group-theoretic means [20, Theorem 7.2.1]; thus the field of moduli off over Q is a number field of degree at most the size of the passport.

We analogously define a marked passport associated to a Belyı map with a marked cusp tobe the tuple (g;G;C0, C1, C∞; ν) where ν is the width of the marked cusp, corresponding toa cycle in the partition associated to C∞. Once more the field of moduli of a Belyı map witha marked cusp over Q is a number field of degree at most the size of its marked passport.

We are now in a position to give a precise formulation of the results motivating this paper,namely the answer to Question 1.6 for the case of a curve (resp. Belyı map) with a singlemarked point (resp. marked cusp).

Theorem 1.12. Let

(Y ; f : Y → X;Q1, . . . , Qm;P1, . . . , Pn) = (Y ;R)

be a marked map of nice curves over a separably closed field F = F sep with m ≥ 1 such thatR is tame. Then (Y ;R) descends (to its field of moduli).

Corollary 1.13. Let (X;P ) be a marked curve over a separably closed field F = F sep suchthat P is tame. Then (X;P ) descends.

Another corollary is the following theorem.

Theorem 1.14. A marked Belyı map (f : X → P1;P ; 0, 1,∞) over Q descends.

Theorem 1.14 was claimed by Birch [3, Theorem 2] in the special case of a Belyı map witha marked cusp; but his proof is incomplete. In work of Debes–Emsalem [9, §5], a proof ofTheorem 1.14 is sketched using a suitable embedding in a field of Puiseux series. We willestablish Theorems 1.12 and 1.14 as special instances of a more general geometric result thatshows how to extract a Weil cocycle from the data in the theorems above.

The following corollary of Theorem 1.14 then follows immediately.6

Corollary 1.15. A Belyı map f : X → P1 is defined over a number field of degree at mostthe minimum of the sizes of the marked passports (f ;P ; 0, 1,∞), where P ∈ f−1({0, 1,∞}).

Corollary 1.15 often enables one to conclude that the Belyı map itself descends, as thefollowing result shows.

Corollary 1.16. Let f be a Belyı map over Q with a marked cusp P whose ramificationindex is unique in its fiber f−1(∞). Then the Belyı map (f ; 0, 1,∞) descends.

Proof. Let F be the field of moduli for the marked map (X; f ;−; 0, 1,∞) with respect to theextension Q |Q. Let σ ∈ ΓF and consider the conjugate Belyı map (σ(X);σ(f);−; 0, 1,∞).Since F is the field of moduli, there exists an isomorphism

(ψσ, ϕσ) : (σ(X);σ(f);−; 0, 1,∞)∼−→ (X; f ;−; 0, 1,∞)

with ϕσ the identity map (as noted after Definition 1.8).We see that

(1.17) f(ψσ(σ(P ))) = σ(f)(σ(P )) = σ(∞) =∞ = f(P )

so ψσ(σ(P )) ∈ f−1(∞). The ramification index of f = σ(f)ψ−1σ at ψσ(σ(P )) is equal to

the ramification of σ(f) at σ(P ), which in turn equals that of f at P . By the uniquenesshypothesis, we must have (ψσ, ϕσ) sends σ(P ) to P .

Since σ was arbitrary, this means that the field of moduli of the marked Belyı map(X; f ;P ; 0, 1,∞) coincides with that of (X; f ;−; 0, 1,∞). By Theorem 1.14, (X; f ;P ; 0, 1,∞)descends to this common field of moduli and thus so does (X; f ;−; 0, 1,∞). �

Remark 1.18. The hypothesis of Corollary 1.16 is very often satisfied. When it is not, one canstill try to obtain a model of a Belyı map over a small degree extension of its field of moduliby ensuring that marking a point does not make the size of the passport grow too much; forexample, if P is a point of maximal ramification index then the automorphism group of themarked tuple (X; f ;P ; 0, 1,∞) may be of small index in that of (X; f ;−; 0, 1,∞).

We now consider to what extent the above considerations remain true when the objectsinvolved are singular.

Theorem 1.19. There exists a marked map (f : X → P1;P ; 0, 1,∞) with X projective andf unramified outside {0, 1,∞} whose field of moduli for the extension C |R is equal to R butthat does not descend to R.

Theorem 1.14 shows that any map f : X → P1 as in Theorem 1.19 has the property thatX is singular. More precisely, we will see that P is a singular point on X for any suchcounterexample.

We will give a proof of Theorem 1.19 in Section 3 by constructing two explicit counterex-amples, one on a curve X that descends to R and another on a curve that does not. In thelanguage of branches, we explain in both cases why extracting a Weil cocycle is not possible.

2. Weil cocycles

Our main tool for the construction of examples and counterexamples is the Weil cocyclecriterion; we give a summary exposition of this criterion in this section, first in the case ofnice curves and then in somewhat greater generality. For more details, we refer to Huggins’sthesis [13], which is an excellent exposition on descent of curves.

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Let K |F be a Galois extension and let X be a curve over K. Enlarging the base field Fif necessary, we may and do suppose that the field of moduli MK

F (X) of X is equal to F .

Thus, for σ ∈ Γ = Gal(K |F ) there exist isomorphisms ϕσ : σ(X)∼−→ X over K. For such

curves Weil proved the following result.

Theorem 2.1 (Weil cocycle criterion). X descends (to F ) if and only if there exist isomor-phisms

(2.2) {ϕσ : σ(X)→ X}σ∈Γ

over K such that

(2.3) {σ ∈ Γ : ϕσ = 1} ≤ Γ has finite index in Γ

and moreover the cocycle condition

(2.4) ϕστ = ϕσσ(ϕτ ) for all σ, τ ∈ Γ

holds. More precisely, if (2.4) is satisfied for isomorphisms (2.2), then there exists a descent

X0 of X to F and an isomorphism ϕ0 : X∼−→ X0 over K such that ϕσ is given as the

coboundary

(2.5) ϕσ = ϕ−10 σ(ϕ0).

Remark 2.6. We warn the reader not to confuse the descent ϕ0 with the isomorphisms ϕσfor σ ∈ Γ.

An important corollary of Theorem 2.1 is the following.

Corollary 2.7. Let X be a curve over K with field of moduli F . If Aut(X)(K) is trivial,then X descends.

Proof. First, we claim that without loss of generality we may take [K : F ] < ∞ and allisomorphisms ϕσ in (2.2) defined over K. Since X is of finite type, it is defined over K ′

where F ⊆ K ′ ⊆ K and [K ′ : F ] <∞. The set of all conjugates σ(X ′) with σ ∈ Gal(K |F )is the same as the set of conjugates σ′(X) where σ′ ∈ Gal(K ′ | F ); these conjugates arewell-defined by our choice of K ′. Replacing X by X ′ and enlarging K ′ if necessary, we canalso suppose that the finite number of isomorphisms σ(f) : σ(X)

∼−→ X are defined over K ′.Of course, if Aut(X)(K) is trivial then so is Aut(X)(K ′).

The corollary then follows since both sides of (2.4) define an isomorphism στ(X)∼−→ X

over K; their compositional difference belongs to Aut(X)(K), which is trivial, so they mustbe equal. �

The Weil cocycle criterion is especially concrete when the base field F equals R. In thiscase we only have to find an isomorphism ϕ : X

∼−→ X between X and its complex conjugateX and test the single cocycle relation

(2.8) ϕϕ = 1

that corresponds to the complex conjugation being an involution. Given any ϕ as above, allother are of the form αϕ, where α ∈ Aut(X)(C).

Remark 2.9. More generally, if the extension K | F is finite, there is a method to find allpossible descents of X with respect to the extension K |F ; see Method 5.1.

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In general, using Theorem 2.1 requires some finesse; when the extension K |F is infinite,it is not immediately clear through which finite subextensions a descent cocycle could factor.However, in Corollary 4.17 below we shall see that for marked curves or marked Belyı maps itis possible to reduce considerations to an explicitly computable finite subextension of K |F .

Now let (X;R) be a marked curve, with R the rigidification data. In this case (as men-tioned by Debes–Emsalem [9, §5]), Theorem 2.1 still applies after replacing Aut(X) byAut(X;R). Moreover, in case a Weil cocycle exists the rigidified data then descend alongwith X to give a model (X0;R0) of (X;R) over F .

Example 2.10. Let (X;P ) be a marked curve over K. Suppose that X has field of moduli

F , with ϕσ : σ(X)∼−→ X in (2.4) having the additional property that ϕσ(σ(P )) = P . Let

X0 be a descent of X to F , with ϕ0 : X∼−→ X0 over K. Let P0 = ϕ0(P ) ∈ X0(K). Then by

(2.5), for all σ ∈ Γ we have

(2.11) σ(P0) = σ(ϕ0)(σ(P )) = ϕ0ϕσ(σ(P )) = ϕ0(P ) = P0.

By Galois invariance, we see that P0 ∈ X0(F ). Therefore the marked curve (X;P ) descendsto F .

The following analogue of Corollary 2.7 is then clear from Example 2.10.

Corollary 2.12. Let (Y ;R) be a marked map over K with field of moduli F . If Aut(Y ;R)(K)is trivial, then (Y ;R) descends.

This already shows a special case of Theorem 1.12.

Corollary 2.13. Let

(Y ; f : Y → X;P1, . . . , Pn;Q1, . . . , Qm) = (Y ;R)

be a marked map over K with n ≥ 1. Suppose that Pi is not a ramification point of f . Then(Y ;R) descends.

Proof. We have Aut(Y ;R)(K) ≤ Aut(Y ; f ;P )(K) ≤ Aut(Y, f)(K) and the latter acts freelyon its fibers over non-branch points, so Aut(Y ; f ;P )(K) is trivial. �

The moral is that when rigidification trivializes the automorphism group, then the ob-struction to descent vanishes. At the other extreme, when the original curve descends andthe rigidified curve has the same automorphism group, then the rigidified curve descends aswell.

Proposition 2.14. Let Y be a curve over K, and let (Y ;R) be a marked map over K.Suppose that Aut(Y ;R)(K) = Aut(Y )(K) and that Y descends to its field of moduli. Then(Y ;R) descends to its field of moduli, which coincides with that of Y .

Proof. Let Y0 be a descent of Y to its field of moduli F with isomorphism ψ : Y∼−→ Y0 over

K. Let F ′ be the field of moduli of (Y ;R) and let σ ∈ Gal(K | F ′). By hypothesis, there

exists an isomorphism ϕσ : σ(Y )∼−→ Y over K that sends the conjugate data σ(R) to R.

Composing, we get an isomorphism

(2.15) ϕ′σ = Y0ψ−1

−→ Yϕσ−→ Y

ψ−→ Y0.

Since Y0 is defined over F so that σ(Y0) = Y0, the isomorphism ϕ′σ actually belongs toAut(Y0)(K). By hypothesis, these automorphisms all fix the dataR. Hence in fact σ(R) = Rfor all σ, and therefore R0 = R descends on Y0. �

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Remark 2.16. In case of proper inclusions {1} ( Aut(X;R)(K) ( Aut(X)(K), it is possiblefor X to descend to its field of moduli whereas this is not possible for (X;R): we will see anexample of this in the next section which is minimal in the sense that the chain of inclusions

{1} ( Aut(X;R)(C) ∼= Z/2Z ( Aut(X)(C) ∼= Z/4Z

is as small as possible.

3. Counterexamples

In this section, we consider two (counter)examples that show how descent in the presenceof singularities can exhibit quite different behavior than when the underlying curves are nice;we exhibit pointed curves (and associated pointed Belyı maps) that are defined over C andhave field of moduli R with respect to the extension C |R, yet do not descend to R. In thefirst example, neither the singular curve nor its normalization descends to R; thus one canequip extra data that can be compatibly identified by the complex conjugate that maintainsthe obstruction to descent. In the second example, the curve itself is defined over R butthe marked curve is not, so rigidifying by marking a point creates a descent problem wherepreviously none existed.

First example: the curve does not descend. We will employ the criterion (2.8) of Weildescent with respect to the extension C |R.

Lemma 3.1. Let W be a variety over R and let % ∈ Aut(W )(R) have order 4. Suppose thatX ⊆ W is a curve over C such that %(X) = X and Aut(X)(C) = 〈%2|X〉 ∼= Z/2Z. SupposeP ∈ W (R) ∩ X(C) has %(P ) = P . Then both X and (X;P ) have field of moduli R withrespect to the extension C | R but neither descends to R.

Proof. By hypothesis, the map ϕ = %|X gives an isomorphism ϕ : X∼−→ X, and ϕ(P ) =

%(P ) = P = P since P ∈ W (R) and % is defined over R. We have ϕϕ = (%2)|X 6= 1 on bothW and X. Therefore ϕ does not give rise to a descent of X to R. Neither does ϕ′ = %3|X ,since again it similarly satisfies ϕ′ϕ′ = (%2)|X 6= 1.

To show that (X;P ) does not descend to R, it is enough to show that X does not descend

to R. Suppose that X0 is a descent; then we would have an isomorphism ϕ0 : X∼−→ (X0)C.

Let ω = ϕ−10 ϕ0, defined via the composition

(3.2) Xϕ0−→ (X0)C = (X0)C

ϕ−10−−→ X.

Then the cocycle condition ωω = 1 is satisfied, since ω = ϕ0−1ϕ0. But we also have

%|−1

Xω ∈ Aut(X)(C), and by hypothesis, we have Aut(X)(C) = 〈%2|X〉 ∼= Z/2Z; so either

ω = %|X = ϕ or ω = %3|X = ϕ′, and in either case we obtain a contradiction from the firstparagraph. �

Remark 3.3. It follows from Theorem 1.12 that a curve X satisfying the hypotheses ofLemma 3.1 cannot be nice. The theory of branches developed below, and more precisely theargumentation in Propositions 4.11 and 4.13, will show that a Weil cocycle can be extractedas long as the point P is regular.

10

To give an explicit example of Lemma 3.1, we take W = P2 and the automorphism

% : P2 → P2

(x : y : z)→ (y : −x : z).(3.4)

Lemma 3.5. We have

(3.6) {P ∈ P2(C) : %(P ) = P} = {(0 : 0 : 1), (±i : 1 : 0)}.

Proof. In the affine open where z = 1 we get the equations y = x = −x, whose uniquesolution is given by x = y = 0. On the other hand, when z = 0 we find (x : y) = (y : −x) ∈P1(C). This implies x = iy, and the result follows. �

By Lemma 3.5, the only point P ∈ P2(R) with %(P ) = P is P = (0 : 0 : 1). Our curve X ⊆P2 is a projective plane curve, defined by a homogeneous polynomial h(x, y, z) ∈ R[x, y, z]with h(0 : 0 : 1) = 0. The condition that %(X) = X is equivalent to the condition thath(y : −x : z) is a scalar multiple of h(x : y : z); for simplicity, we assume that

h(y : −x : z) = h(x : y : z).

The condition that %2|X ∈ Aut(X)(C) implies that

h(x : y : −z) = h(x : y : z).

We take deg h = 4 so that Aut(X)(C) is finite. The above conditions then imply that

(3.7) h(x, y, z) = ax4 + ay4 + (bx2 − by2)xy + (cx2 + cy2)z2 + rx2y2 + sixyz2

with a, b, c ∈ C and r, s ∈ R. We observe that P = (0 : 0 : 1) is a singular point of X and Pis a nodal double point if and only if s2 6= −4|c|2.

To apply Lemma 3.1, we need to select coefficients of h such that X has no automorphismsbeyond %2|X ; we will show this is true for a particular choice, which will then in fact implythat the resulting computations are true for a general choice, i.e., for the curve defined byequation (3.7) over C[a, b, c, r, s,∆−1] where ∆ is the discriminant of h. We consider thecurve with

(a, b, c, r, s) = (0, i, 1 + i, 1, 2)

so that X is described by the homogeneous equation

(3.8) X : i(x2 + y2)xy + ((1 + i)x2 + (1− i)y2)z2 + x2y2 + 2ixyz2 = 0.

Standard techniques in computational algebraic geometry (we performed our computationin Magma [4]) show that P is the only singular point of X, so that X has geometric genus2. We verify that Aut(X)(C) ∼= Z/2Z by computing that the normalization of X over R isgiven by

(3.9) Y : y2 = (i+ 1)x5 + (−i− 1)x4 + 4x3 + (−i+ 1)x2 + (−i+ 1)x

and by a computation using invariant theory (again a computation in Magma [15]) wesee that indeed Y has automorphism group Z/2Z generated by the hyperelliptic involution.Since every automorphism group of a singular curve lifts to its normalization, we see thatindeed Aut(X)(C) ∼= Z/2Z. Lemma 3.1 therefore applies to show that 3.8 is an explicitcounterexample.

Alternatively, one can compute that the Igusa–Clebsch invariants of Y are defined over R,so that the field of moduli is R, but that there is an obstruction to the curve Y being defined

11

over R, as described by Mestre [17]. On the other hand, one shows that the existence of adescent of Y is equivalent with that for X since P is the unique singular point of X.

Remark 3.10. In this construction, the canonical model over R of Aut(X)\X from the up-coming Definition 4.20 has an R-rational point (the singular image of the singular point P ),so the rational point criterion Theorem 4.27 that we will establish in the next section failsfor singular curves.

Remark 3.11. With the equation (3.9) in hand, one sees how to recover this class of examplesin a different way, as follows. Let W0 be the conic defined by x2 + y2 + z2 = 0 in P2 overR. Then W0(R) = ∅. Let S ⊆ W0(C) be a subset of 6 distinct points over C forming 3complex conjugate pairs. Then there exists a nice hyperelliptic curve X ′ over C branchedover the set S that does not descend to R: this follows from work of Mestre [17], but it isalso a rephrasing of the results by Shimura and Earle reviewed by Huggins [13, Chapter 5].

Now let Q1, Q2 ∈ S be such that Q2 = Q1 and let P1, P2 ∈ X ′(C) be their ramifiedpreimages. By the cocycle condition, the points P1, P2 ∈ X ′(C) are mapped to P1, P2 ∈X ′(C) under the isomorphism between X ′ and X ′. Now pinch together the points P1, P2 toobtain a curve X over C with a double point P . Then P ∈ X(C) is mapped to its conjugateP on X (“itself” on W0) by construction, and so (X;P ) also has field of moduli R.

To obtain a map f : X → P1 unramified away from {0, 1,∞}, thereby completing the firstproof of Theorem 1.19, we first choose a function π on W defined over R that is invariantunder % and that is nonconstant on X. This yields a morphism π|X : X → P1, and because %maps X to X and π is invariant under %, the branch locus of π|X is invariant under complexconjugation.

In the specific example (3.8) above, we take π = x2 +y2 and compute that the ramificationlocus is given by the vanishing of the homogeneous polynomial

(3.12) 3072u7v + 4352u6v2 + 5840u5v3 + 3424u4v4 + 920u3v5 + 104u2v6 + 5uv7,

which shows that it is in fact even stable under Gal(Q |Q).We can now apply the following slight strengthening of the result of Belyı.

Theorem 3.13 (Belyı [2]). Let g : X → P1 be a map of curves over Q whose branch locusin P1(Q) is defined over a number field F ⊂ Q. Then there exists a morphism α : P1 → P1

defined over F such that f = α ◦ g is a Belyı map.

Theorem 3.13 applies equally well to the extension C |R, essentially since every Belyı mapis already defined over Q. We apply the above for g = π, whose branch locus is R-rational,and let f = απ : X → P1 be the map thus obtained, unramified away from {0, 1,∞}. Nowf is %-invariant since π is, and because we have π = π% = π on W we also have

(3.14) f% = απ% = απ% = απ%2 = απ = f.

So indeed the field of moduli of the pointed map (X; f : X → P1;P ; 0, 1,∞) with respectto the extension C |R equals R. However, because X does not descend to R, neither does(X; f ;P ; 0, 1,∞).

Rigidifying by a Belyı map. We pause before continuing to our second counterexampleto consider an issue exemplified by the first: rigidifying a curve by a Belyı map does not

12

suffice to eliminate the descent obstruction. This is known in the genus 0 case [6]. Here weshow how to concoct such examples in more greater generality.

Consider a genus 2 hyperelliptic curve X : y2 = p(x) defined over K = C that doesnot descend to its field of moduli F = R relative to Gal(K | F ). Suppose further thatAut(X) ∼= Z/2Z is the hyperelliptic involution, and the isomorphism of X with its conjugateX is given by ϕ(x, y) = (−1/x, y/x3). Such curves exist due to the results mentioned inRemark 3.11.

Consider the projection q : X → W = 〈ι〉\X = Aut(X)\X. The curve W is isomorphicto P1 with coordinate x. In what follows, we will consider W as being defined over R, bytaking W0 = P1

R as a model, with the same coordinate x; this is not the canonical model ofW from Definition 4.20, which is in fact the non-trivial conic over R.

Let g0 : W0 → P1 be the map with g0(x) = x − 1/x. By composing, we get a mapf = g0q : X → P1 defined over C whose branch locus is the union of the branch locus {±i}of g0 and the image (g0)∗(D) under g0 of the branch locus D ⊂ W0 of q.

We claim that (g0)∗(D) is stable under Gal(C | R). This follows because D = α∗(D)for the automorphism α : x → −1/x of W0, which in turn holds because of the field ofmoduli condition on X. In fact α can be seen as the isomorphism W → W induced byϕ : X → X, so that qϕ = αq. Now α generates the automorphism group of the genericallyGalois morphism g0, so that indeed

(3.15) (g0)∗(D) = (g0)∗(D) = (g0)∗(D) = (g0)∗(α∗(D)) = (g0α)∗(D) = (g0)∗(D).

As a result the branch locus {±i} ∪ (g0)∗(D) of g0q is also R-rational.Using Theorem 3.13 we see that there exists a h0 : W → P1 defined over R such that

f = h0g0q : X → P1 is a Belyı map over C. By construction the pair (X, f) has field ofmoduli R with respect to the extension C |R; the isomorphism ϕ sends X to X and since g0

and h0 are both defined over R we also have

(3.16) fϕ = h0g0qϕ = h0g0αq = h0g0q = h0g0q = f.

On the other hand, a descent of (X, f) to R cannot exist, because then certainly X woulddescend to R.

In conclusion, X admits a Belyı map whose field of moduli is R and yet X still cannot bedefined over R.

Second example: the curve descends. In this second example, the curve itself descendsbut the marked curve does not. The setup is described by the following lemma.

Lemma 3.17. Let X be a curve over R such that

(3.18) Aut(X)(C) = Aut(X)(R) = 〈%〉 ∼= Z/4Z.

Suppose that P ∈ X(C) \X(R) satisfies P = %(P ). Then (XC;P ) has field of moduli R withrespect to the extension C |R but does not descend to R.

Proof. The field of moduli claim follows by taking ϕ = % to be the identity map, since X = Xand %(P ) = P as % is defined over R. Moreover, we have

(3.19) P = P = %(P ) = %(P ) = %2(P ).13

It remains to show that (X;P ) does not descend to R. For purposes of contradiction,

suppose that (X0;P0) is a descent. Then we have an isomorphism ϕ0 : XC∼−→ (X0)C such

that ϕ0(P ) = P0 satisfies P0 = P0. The composition ϕ−10 ϕ0 = ω gives a map

(3.20) XC = XCϕ0−→ (X0)C = (X0)C

ϕ−10−−→ XC

so ω ∈ Aut(X)(C), and since ω = ϕ0−1ϕ0 we have ωω = 1. But since Aut(X)(C) =

Aut(X)(R), we have that ω = ω. Therefore the cocycle condition implies that ω is aninvolution. If ω = 1 then ϕ0 = ϕ0 is defined over R so that under this isomorphism

(3.21) ϕ0(P ) = ϕ0(P ) = ϕ0(P ) = P0 = P0 = ϕ0(P ),

a contradiction with our hypothesis P ∈ X(C) \ X(R). So we must have ϕ0 = ϕ0%2. But

then by (3.19) and rationality of % we would again have

(3.22) ϕ0(P ) = ϕ0(%2(P )) = ϕ0(%2(P )) = ϕ(%2(P )) = ϕ0(P ) = ϕ0(P ) = P0 = ϕ0(P ).

Therefore no descent of (X;P ) exists. �

Remark 3.23. Admitting for a moment that curves as in Lemma 3.17 do exist, we see thatrigidifying by marking a point may very well lead to a descent problem where previouslynone existed: while both (XC, P ) and a fortiori XC have field of moduli R with respect tothe extension C |R, the obstruction vanishes for the non-rigidified curve XC, whereas it isnon-trivial for the pair (XC, P ). So while non-rigidified curves XC may not descend, as wasnoted by Birch [3], neither may a given rigidification of XC, as we saw in the last subsection,and in fact it may even complicate descent matters further when the automorphism groupremains nontrivial.

To exhibit an explicit curve meeting the requirements of Lemma 3.17, we first constructa nice curve Y defined over R with

Aut(Y )(R) = Aut(Y )(C) ∼= Z/4Z.To do this, we choose distinct x1, x2 ∈ C and let

(3.24) p(x) = (x2 + 1)2∏i=1

(x− xi)(x− xi)(x+ 1/xi)(x+ 1/xi) ∈ R[x].

We see that p(−1/x) = p(x)/x10. One then expects that generically the hyperelliptic curve

Y : y2 = p(x)

will have automorphism group generated by %(x, y) = (−1/x, y/x5). It is indeed possible toverify [15] that this indeed happens for the choice x1 = 1+ i, x2 = 2+ i, which (after scaling)yields the genus 4 hyperelliptic curve

(3.25) Y : y2 = 10x10 − 42x9 + 67x8 − 36x7 + 23x6 + 23x4 + 36x3 + 67x2 + 42x+ 10.

Consider the point Q = (1+i, 0) ∈ Y (C). To get Lemma 3.17 to apply, we need Q = %(Q).Of course this cannot be the case on Y in light of Theorem 1.12, but we can make it true byconstruction if we pinch together the points Q and %(Q). Moreover, if we take care to pinchQ with %(Q) similarly, then the resulting contraction morphism c : Y → X will be definedover R. If we let P = c(Q), then by construction, the pair (X;P ) will satisfy the conditionsof Lemma 3.17. Indeed, any automorphism of a singular curve lifts to its normalization,

14

and conversely the automorphisms of Y all transfer to X because of our construction of thecontraction, so that indeed

Aut(X)(R) = Aut(X)(C) ∼= Aut(Y )(C) ∼= Z/4Z;

the stabilizer of P again corresponds to the subgroup of Z/4Z of order 2.A contraction morphism c can be constructed in many different ways, none of which is

particularly pleasing from the point of view of finding an equation. The first approach is tofind a sufficiently ample linear system and extract the sublinear system of sections whosevalues at Q and %(Q) (resp. Q and %(Q)) coincide. This leads to ambient spaces that aretoo large to give rise to attractive equations.

An alternative approach is as follows. Define

q(x) = (x− x1)(x− x1)(x+ 1/x1)(x+ 1/x1) = x4 − x3 +1

2x2 + x+ 1,

r(x) =2

3x3 − 2x2 + x.

(3.26)

and let c : A2 → A4 be given by

(3.27) c(x, y) = (q(x), xq(x), r(x), y) = (t, u, v, w).

Given a point in the image of c, we can always read off the original y-coordinate from w. Wecan also recover the x-coordinate u/t as long as t 6= 0. The latter only occurs if x is a root ofq; these roots all have the property that t = u = w = 0. On the other hand, one verifies thatr assumes exactly two distinct values on this set of roots; moreover, the preimage of one ofthese values is given by {Q, %(Q)}, and the other by {Q, %(Q)}. We see that the morphismc constracts the pairs {Q, %(Q)} and {Q, %(Q)} exactly in the way that we wanted.

Of course this method gives only an affine open of X, but this open contains all the singularpoints that we are interested in, so that completing the corresponding model smoothly atinfinity will give the model of X that is desired. On the patch t 6= 0 we can describe theimage of c by the following equations:

(3.28)

t5 − t4 − t3u− 1/2t2u2 + tu3 − u4 = 0,

t3v − t2u+ 2tu2 − 2/3u3 = 0,

10t10w2 − 10t10 − 42t9u− 67t8u2 − 36t7u3 − 23t6u4

−23t4u6 + 36t3u7 − 67t2u8 + 42tu9 − 10u10 = 0.

Adding a few more coordinates, we can also describe the automorphism %: using thecontraction

c(x, y) = (q(x), xq(x), r(x), y, q(−1/x),−1/xq(−1/x), r(−1/x), y/x5)

= (t, u, v, w, t′, u′, v′, w′),(3.29)

it can be described by

(3.30) %(t, u, v, w, t′, u′, v′, w′) = (t′, u′, v′, w′, t, u, v,−w).

In both cases, the full ideal of the image of c can be recovered by using Grobner bases.To conclude our argument; by Theorem 3.13, we can find a map f : X → P1 unramified

outside {0, 1,∞} such that (f ;P ) has the same automorphism group: one takes any function15

ϕ defined over R that is invariant under Aut(X,P ) and postcompose with a function hfollowing Belyı to get a map ϕ = hϕ0 with the same field of moduli and the same obstruction.

Remark 3.31. Whereas the curve in the first counterexample had exactly one singular point—in a sense, it was “marked” anyway. By contrast, the curve in this subsection has two singularpoints, which makes it important to keep track of the chosen marking.

Remark 3.32. There should be many more ways to construct singular curves over R for whomthe field of moduli is no longer a field of definition after rigidification by a singular point.

For example, if we take the automorphism % : P3 → P3 given by the cyclic permutationof the coordinates %(x : y : z : w) = (y : z : w : x) and considers the generic completeintersection X of two polynomials invariant under % containing the point P = (1 : i : 1 : i)with %(P ) = P , then we expect that the curve X will satisfy the hypotheses above. Forexample, if we define

(3.33) %(m) = m+ %∗(m) + (%∗2)(m) + (%∗3)(m)

for m ∈ k[x, y, z, w] a monomial, then we believe that the curve X of genus 7 defined by

%(x2) = x2 + y2 + z2 + w2 = 0

5%(x4) + %(x3y) + %(x3z)− 4%(x2yw)− 2%(x2zw) + 3%(x2y2) + 7%(xyzw) = 0(3.34)

has this property, as we can verify that Aut(X)(Fp) ∼= Z/4Z for a number of large primes p;however, it is much more involved to provably compute the automorphism group for such acurve, which is why we have preferred to stick with this admittedly more special case.

Remark 3.35. The methods from this section can be used to show that given any number nof points to be fixed, there exist tuples (X,P1, . . . , Pn) that give rise to a descent obstruction.Conversely, for any given curve there exists a number n such that the descent obstructionfor the marked curve vanishes as soon as they are rigidified by n distinct points (since thenumber of orbifold points on the quotient Aut(X)\X is finite).

4. Branches

The geometric equivalent of a fundamental tool of Debes–Emsalem [9] is the considerationof the branches of a morphism of non-singular curves over a point P . One can interpretbranches embeddings into certain power series rings [9]. We revisit this definition in a moregeometric context, allowing for a uniform way of constructing Weil cocycles: for nice curves,the number of branches above geometric points has constant value, even though the numberof points in the fiber may be only lower semi-continuous.

Let X be a curve over F . We first consider the case of branches over a smooth point, andwe let x ∈ X(F sep) be a smooth geometric point. Let OXet,x be the local ring of X at xfor the etale topology, and let PX,x be the integral closure of OXet,x in Frac(OXet,x)

sep, theseparable closure of its quotient field, so that there is a canonical morphism Spec PX,x → X.

Definition 4.1. Let f : Y → X be a separable map of curves over F and let x ∈ X(F sep)be a smooth geometric point. A branch of f over x is a morphism b : Spec PX,x → Y such

16

that the following diagram commutes:

(4.2)

Spec PX,xb //

%%

Y

f��X

The set of branches of f over x is denoted B(f, x).

Concretely, the ring OXet,x is isomorphic to the integral closure of the polynomial ringF sep[t] in the ring of power series F sep[[t]], for t a uniformizer at x. Therefore PX,x isisomorphic to the valuation ring of the field of Puiseux series F sep((t1/∞)), which consistsof those elements of F sep((t1/∞)) whose monomials all have non-negative exponent. (Thisisomorphism explains our notation for the ring PX,x.)

Remark 4.3. A branch of f can be seen as an equivalence class of sections V → Y , whereV → X is a “tame neighborhood” of x in the sense that it factors as V → U → X, whereV → U is tamely ramified and U → X is the inclusion of some open neighborhood of x.

By taking the fiber over the closed point of Spec PX,x (thinking of this as the valuation ringin a field of Puiseux series, we are setting t = 0), we recover a lift of the point x ∈ X(F sep).We can see b as an infinitesimal thickening of this lift; the fact that we cannot use the localring for the etale topology reflects that we need slightly thinner thickenings than that in thistopology to obtain enough sections.

The above definition of branch generalizes to singular points as follows. Now let x ∈X(F sep) be a geometric point of X (possibly singular). Let X → X be the normalization of

X. Then for any x ∈ X over x there is a canonical composition of maps

OXet,x→ X → X

whence also the composition

PXet,x→ X → X.

Definition 4.4. Let f : Y → X be a separable map of curves over F and let x ∈ X(F sep) be

a geometric point. A branch of f over x is a morphism b : Spec PX,x → Y with x ∈ X(F sep)lying over x, making the following diagram commute:

(4.5)

Spec PX,x

b //

��

Y

f

��X // X

Remark 4.6. Alternatively, let Y be the normalization of Y . Then f lifts to a map f : Y → X.

One can then equivalently define a branch of f at x to be a branch of f at some point

x ∈ X(F sep) over x.

We can recover the classical terminology as follows.

Definition 4.7. Let X be a curve over F . A branch of a curve X over x ∈ X(F sep) is a

branch of the normalization map X → X over x.17

Remark 4.8. Equivalently, one can define a branch over x to be a branch of the identity mapX → X over x.

Given an element σ of the absolute Galois group Gal(F sep |F ), there is a canonical mapof sets

B(f, x)→ B(σ(f), σ(X))

b 7→ σ(b)(4.9)

Here σ(f) is the morphism σ(Y )→ σ(X) induced by the action of σ on the coefficients of f .The map on branches is defined as follows. Given a branch b : Spec PX,x → Y of f over x,we can conjugate it to obtain a morphism σ(b) : Specσ(PX,x) → σ(Y ). We compose withthe canonical isomorphism

(4.10) σ(PX,x) ∼= Pσ(X),σ(x)

arising from the fact that both sides are abstractly isomorphic to the ring PX,x, and in bothcases the F sep-algebra structure on this ring has been changed by conjugation with σ.

The set B(f, x) has two very pleasant properties, which should be thought of as general-izing the properties of usual fibers over non-branch points. The first of these properties isthat the number of branches of f : Y → X over a given point is constant on the target X:

Proposition 4.11. Let f : Y → X be a map of nice curves of degree d, and let x ∈ X(F sep)be a geometric point of X that is tamely ramified in Y . Then the set B(f, x) has cardinalityd.

Proof. Everything being local for the etale topology, we can reduce to the case where f isthe map A1 → A1 defined by t 7→ tn and where the base point x is given by t = 0. In thiscase everything is clear, since the sections b in (4.2) correspond to the maps t 7→ ζnt; by thetameness hypothesis, we have gcd(n, charF ) = 1, so these branches are all distinct. �

Remark 4.12. Intuitively, the above Proposition shows that when describing a map f by theassociated representation on its unramified fibers, then the branches of f over any point canbe thought of as the set of unramified fibers; in a sense, branches can be extended over thebranch locus of f where regular points cannot.

The second fundamental property, in line with the preceding remark, is that we can readoff the action of the automorphism group of f from that on the branches above any point ofX.

Proposition 4.13. Let f : Y → X be a map of nice curves, and let x ∈ X(F sep) be ageometric point of X that is tamely ramified in Y . Then the action of Aut(f) on B(f, x) isfaithful.

Proof. The statement is true over non-branch points. The general case follows as in the proofof Proposition 4.11: any morphism acting trivially on a ramifying fiber would locally be givenby t 7→ t, and would therefore also act trivially on the fibers in an etale neighborhood of0. This contradicts the fact that such a neighborhood contains a non-branch point overF sep. �

Remark 4.14. As in Birch’s original article [3] we can consider the case where X is a modularcurve associated with a subgroup SL2(Z), which we suppose to be defined over some number

18

field F . Using a cusp of X then allows one to use q-expansions with respect to a uniformizerq of the appropriate width with respect to the cusp.

Echelonizing a basis of modular forms gives a defining equation for X over F ; this doesindeed turn out to lead to F -rational q-expansions, with the slight subtlety that one mayneed to twist X to have a rational branch over F .

Suppose now that X is a curve over F sep whose field of moduli equals F , and let Q ∈X(F sep). Let G = Aut(X)(F sep) be the geometric automorphism group of X. Consider thequotient V = G\X of X by G, let π : X → V be the quotient map, and let P = π(Q) ∈V (F sep). We then have the following more explicit version of Theorem 1.12.

Theorem 4.15. If P is tamely ramified in π : X → V , then the set of branches B(π, P ) isa torsor under G.

Let b ∈ B(π, P ), and given σ ∈ Γ = Gal(F sep | F ), let ϕσ be the unique morphism

σ(X)∼−→ X such that σ(b) is sent to b. Then {ϕσ}σ∈Γ defines a Weil cocycle.

Proof. That the set B(π, P ) is a G-torsor follows by combining Propositions 4.11 and 4.13;the map X → B is generically Galois. The existence of ϕσ follows because choosing just anyϕ′σ : σ(X) → X maps σ(b) to some branch over P , which we can then modify to be b byinvoking the transitivity part of being a torsor. The ϕσ give a Weil cocycle because of theuniqueness part of being a torsor; both ϕστ and ϕσσ(ϕτ ) send b to στ(b). �

Remark 4.16. Theorem 4.15 in fact shows that there exists a descent (X0, R0) with a branchat R0 that is defined over F . Indeed, the coboundary corresponding to the Weil cocycle ϕσmaps b to a F -rational branch by a similar argument as that in (2.11).

The proof of Theorem 4.15 also shows how to obtain a finite extension of F over which aWeil cocycle can be constructed.

Corollary 4.17. Let (X;P ;R) be a tame marked map. Let K be the Galois closure of thefield generated over F by the coefficients of defining equations of X, the elements of the groupscheme Aut(X), and the branches of the canonical morphism π : X → G\X at π(P ). Thena Weil cocycle for X can be constructed relative to the extension K |F .

Remark 4.18. As we will see explicitly later, computing the splitting field of the branches ata given point requires no more than determining the leading term in a certain power series,which in turn has known order.

Remark 4.19. The counterexamples in Section 3 all share the property that the number ofbranches over the points of the base curve is not constant, because of the merging of thesebranches over the singular points, which are (crucially) the only rational points of the basecurve. This makes it impossible to read off a uniquely determined cocycle as in Theorem4.15.

To conclude this section, we state our results using a classical notion from Debes–Emsalem[9].

Let (X;R) be a rigidified curve over F sep with field of moduli F and automorphism groupG = Aut(X;R), and let V = G\X. Then by construction any choice of isomorphisms (2.2)gives rise to a Weil cocycle on V .

Definition 4.20. The canonical model V0 is the model of V defined over F determined byany choice of Weil cocycle.

19

The canonical model depends only on the isomorphism class of (X;R) over F sep. In the

canonical model, we denote the quotient morphism π : X → G\X and let ψ0 : V∼−→ V0 be

a coboundary corresponding to the uniquely determined collection of isomorphisms {ψσ}σ∈Γ

induced by (2.2). Set π0 = ψ0π.

Proposition 4.21. Let P ∈ X(F sep). Then the field of moduli of (X;R;P ) is the field ofdefinition of the point π0(P ) on V0.

Proof. Let F ′ = F (π0(P )) be the field of definition of π0(P ) on V0.First, we claim that F ′ is contained in the field of moduli M of (X;R;P ). By Theorem

1.12, we may assume without loss of generality that (X;R;P ) = (X0;R0;P0) is definedover M . Thus P ∈ X(M). This implies π0(P ) ∈ V0(M): we have V0 = G\X whereG = Aut(X,R) so that π0 coincides with the natural projection X → V = G\X definedover M . Thus M ⊇ F ′ = F (π0(P )).

To show that M ⊆ F ′, without loss of generality we may assume M = F . We then haveto show that for all σ ∈ Γ′ = Gal(F sep |F ′), there exist isomorphisms

(4.22) ϕσ : (σ(X);σ(R);σ(P ))∼−→ (X;R;P )

so that M is contained in F ′. We are given that (X;R) has field of moduli F , so we have

isomorphisms ϕσ : (σ(X);σ(R))∼−→ (X;R) for all σ ∈ Gal(F sep |F ).

Let σ ∈ Γ′. We will show that ϕσ can be chosen in such a way that additionally ϕσ(σ(P )) =P . Let ψσ = ψ−1

0 σ(ψ0). Then by construction of the canonical Weil cocycle for V we knowthat πϕσ = ψσσ(π). We obtain the following commutative diagram:

(4.23) σ(X)

σ(π0)00

ϕσ //

σ(π)��

X

π

��

π0

pp

σ(V )ψσ //

σ(ψ0) ""

V

ψ0��V0

The rationality of π0(P ) over F ′ implies that

(4.24) π0(P ) = σ(π0(P )) = σ(π0)(σ(P )).

We claim that P, ϕσ(σ(P )) ∈ X(F sep) are in the same fiber of the map π0. Indeed, from(4.24) and tracing through the diagram (4.23), we obtain

(4.25) π0(P ) = σ(π0)(σ(P )) = π0(ϕσ(σ(P ))).

Because Aut(X;R) acts transitively on the fibers of π and thus on those of π0, we see thatwe can compose the chosen ϕσ with an element of this group to obtain an isomorphism asin (4.22). �

Theorem 4.26. Let P ∈ X(K). Then P is F -rational on some descent of (X;R) to F ifand only if π0(P ) is F -rational.

As a corollary, we obtain the following theorem, alluded to by Debes–Emsalem [9, §5].

Theorem 4.27. Suppose that V0(F ) 6= ∅. Then (X;R) descends.20

Proof. Let P ∈ X(F sep) be such that π0(P ) ∈ V0(F ). Then the field of moduli of (X;R;P )equals F by Proposition 4.21. We get a descent (X0;R0;P0) of (X;R;P ) to F by Theorem1.12, which also gives a descent of (X;R) to F . �

Remark 4.28. Theorem 4.27 applies in particular when X is not equipped with a rigidifica-tion. Consider for example the case where X is hyperelliptic, with hyperelliptic involution ι;one can then show [16] that a descent to F induced by a point P as in Theorem 4.26 alwaysexists, except possibly if g(X) and #〈ι〉\Aut(X) are both odd. That is, except in thesespecial cases any descent can be obtained from some rigidification of X by marking.

5. Descent of marked curves

This section focuses on the explicit descent on marked curves, notably hyperelliptic curvesand plane quartics. Before treating these examples, we indicate a general method for de-scending rigidified curves.

Determining a Weil descent. Suppose that the extension K | F is finite and that theautomorphism group G = Aut(X)(K) is finite. (See below for the case when G is infinite.)Then the following method finds all possible descents of X with respect to the extensionK |F .

Method 5.1. This method takes as input a rigidified curve (X;R) over K and produces asoutput all descents (X0;R0) of (X;R) to F .

1. Compute a presentation Γ = 〈Σ | Π〉 of the Galois group Γ = Gal(K |F ) in finitelymany generators Σ and relations Π.

2. Compute G = Aut(X;R)(K).

3. For all σ ∈ Σ, compute an isomorphism ψσ : (σ(X), σ(R))∼−→ (X;R) over K. If

for one of these generators no such isomorphism exists, output the empty set andterminate.

4. For all tuples (ϕσ)σ∈Σ in the product∏

σ∈Σ Gψσ, use the relation (2.4) to determinethe value of the corresponding cocycle on the relations in Π; retain a list L of thosetuples for which the corresponding cocycle is trivial on all elements of Π.

5. For the tuples (ϕσ)σ∈Σ in L, construct a coboundary morphism ϕ0 : (X;R)∼−→

(X0;R0) to obtain a descent of (X;R). Output this set of descents.

Remark 5.2. Method 5.1 applies to rigidified curves over the separable closure F sep to detectthe existence of a descent: by Corollary 4.17, we can always reduce the existence of a descentfrom F sep to a finite extension K |F . (There may be infinitely many descents of (X;R) overF sep, but only finitely many coming from a given finite extension K |F .)

Remark 5.3. In step 1 of the above method, one can work instead with the full group Γitself rather than generators and relations; then one loops over the elements of the groupand checks compatibility with products.

The correctness of this method follows by the general approach in the previous section.For now, we present only a ‘method’ rather than a true algorithm, as each of these stepsmay involve a rather difficult computer algebra problem in general; however, in cases such ashyperelliptic or nice plane curves, efficient algorithms for finding isomorphisms exist, so thatall but the final step of the method are algorithmic. The construction of a coboundary is

21

somewhat more involved but can be obtained in the aforementioned cases as well, essentiallyby finding points on a certain variety over the ground field. We do not go into the generalproblem here, as they form a substantial challenge of their own, but we do indicate how toproceed in special cases below.

Remark 5.4. If X is already defined over F , the method above will find the twists of X overK, that is, the set of F -isomorphism classes of curves over F that become isomorphic to Xover K. In this case the isomorphisms ϕσ are themselves already in G, so this computationis slightly easier.

Descent of marked curves of genus at most one. We begin by disposing of two easycases in low genus.

Lemma 5.5. Let (X;P ) be marked curve over F sep with X of genus 0. Then (X;P ) hasfield of moduli F , and a descent to F is given by (X0;P0) = (P1;∞).

Proof. Immediate. �

Proposition 5.6. Let (X;P ) be marked curve over F sep with X of genus 1. Then (X;P )descends to its field of moduli, equal to F (j(Jac(X))).

Proof. By translation on the genus 1 curve X, we can take (X0;P0) = (J ;∞), where J =Jac(X) is the Jacobian of X with origin ∞, which is defined over F (j) where j is thej-invariant of J . �

Hyperelliptic curves. We now pass to a class of examples that can be treated in somegenerality, namely that of hyperelliptic curves. Throughout, we suppose that the base fieldF does not have characteristic 2.

Let X be a nice hyperelliptic curve over F sep and let ι : X → X be the hyperellipticinvolution. Let W = 〈ι〉\X and V = Aut(X)\X. We get a sequence of natural projectionmaps

(5.7) Xq−→ W

p−→ V ;

the map p is the quotient of W by the reduced automorphism group G = Aut(X)(F sep)/〈ι〉of W (or of X, by abuse).

We can identify G with a subgroup of PGL2(F sep) = Aut(W )(F sep). For simplicity, wealso suppose that G does not contain any non-trivial unipotent elements. This hypothesisimplies that G is isomorphic to one of the finite groups Cn, Dn, A4, S4 and A5.

The reduced automorphism group G is also described as G = Aut(W ;D)(F sep), where Dis the branch divisor of q, or equivalently, image on W of the divisor of Weierstrass points onX; thus X can be recovered from (W ;D) over F sep by taking a degree 2 cover of W ramifiedover D.

Remark 5.8. Given (W ;D) = (W0;D0) over F , it is not always possible to construct a coverX0 over F of (W0;D0) whose ramification locus equals D0. Indeed, suppose X has field ofmoduli F and Aut(X)(F sep) = 〈ι〉. Then W = V , and there exists a canonical descent W0

of W . Moreover, by the same argument that we used to obtain (2.11), D transforms to aF -rational divisor D0. If the aforementioned reconstruction were possible over F , then Xwould descend. But as we have seen in Remark 3.11, there exist hyperelliptic curves of genus2 with generic automorphism group that do not descend.

22

Let R ∈ X(F sep) and consider the marked curve (X;R) over F sep. Let Q = q(R) bethe image of R on W , and similarly let P = p(q(R)) be its image on V . Then we canalso reconstruct (X;R) from the datum (W ;Q;D) obtained by rigidifying (W ;Q) with thebranch divisor D of q.

Proposition 5.9. With the above notation, let X ′ be a curve over F sep obtained as a degree2 cover of W ramifying over D, and let R′ be any preimage of Q under the correspondingcovering map. Then (X ′, R′) is isomorphic to (X,R) over F sep.

Proof. This follows from the fact that a hyperelliptic curve over F sep is uniquely determinedby the branch locus of its hyperelliptic involution ι, combined with the fact that ι actstransitively on the fibers of the quotient X → W . �

We now suppose that (X;R) has field of moduli F , which we can do without loss ofgenerality. By Theorem 1.12, we know that (X;R) descends to a marked curve (X0;R0)over F , which gives rise to a sequence of marked curves and morphisms

(5.10) (X0;R0)p0−→ (W0;Q0)

q0−→ (V0;P0).

over F . In what follows, we will show how to calculate the descent (X0;R0) explicitly. Ourmethods do not yet need branches; for pointed hyperelliptic curves, it suffices to use moreelementary techniques involving morphisms between curves of genus 0.

We will first construct the descent (V0;P0) of (V, P ). This is done as in Debes–Emsalem[9]; we start with any collection of isomorphisms {ϕσ : σ(X) → X}σ∈ΓF and consider the

induced maps χσ : σ(V )∼−→ V . By construction, the collection {χσ}σ∈ΓF satisfies the Weil

cocycle relation. Therefore there exists a map χ0 : V∼−→ V0 to a descent V0 of F such that

χσ = χ−10 σ(χ0). By the same argument that we used to obtain (2.11), we see that P0 = χ0(P )

becomes an F -rational point on V0.The isomorphisms ϕσ : σ(X)

∼−→ X can be explicitly calculated [15]. After determining the

induced maps χσ : σ(V )∼−→ V between genus 0 curves, we can explicitly calculate the map

χ0, and with it the point P0 [12, 19]. We therefore assume the descent χ0 : (V ;P )∼−→ (V0;P0)

to be given, and will now discuss how to reconstruct a descent (X0;R0) of (X;R) from it.In one case, determining (X0;R0) turns out to be particularly easy. Let G0 = Aut(W0;Q0).

Suppose that G0 is trivial, so that (W0;Q0) = (V0;P0). Then the image D0 of the branchdivisor D on W0 is defined over F . Since V0 admits the point P0 over F , it is isomorphic toP1 over F .

Suppose the reduced automorphism group of the pair (X;R) is trivial. Let W0∼−→ P1 be

any isomorphism over F that sends R0 to ∞, and let p0 be a monic polynomial cutting outthe image of Supp(D0)\∞. Let

(5.11) X0 : y2 = p0(x)

If R is not a Weierstrass point of X, then ∞ /∈ Supp(D0), so that p0 is of even degree. Inthis case we let R0 be the point at infinity corresponding to the image of (0, 1) under thechange of coordinates (x, y)← (1/x, y/xg+1). On the other hand, if R is a Weierstrass pointof X, then ∞ ∈ Supp(D0) and p0 is of odd degree, in which case we let R0 be the pointcorresponding to (0, 0) under the aforementioned change of coordinates.

The following proposition is then immediate.23

Proposition 5.12. Let (X;R) be a marked hyperelliptic curve over F sep whose reducedautomorphism group is trivial. Then the pair (X0;R0) defined in (5.11) is a descent of(X;R).

Remark 5.13. In Proposition 5.12, the indicated point P0 on X0 admits a rational branch. Inthe case when R is a not a Weierstrass point of X, this is because P0 is unramified, whereasin the latter case we get a rational branch for the map (x, y) 7→ x. This follows from alocal power series expansion at infinity; since y2 = x + O(x2), we get the rational branchcorresponding to the root y =

√x+O(x).

The quadratic twist defined by c0y2 = p0(x) also gives a descent of the marked curve

(X;R); but if c0 ∈ F× \ F×2, the map (x, y) → x does not admit a rational branch at thepoint at infinity, rather the morphism (x, y) 7→ c0x gives a rational branch. In general, thepassage from points to branches gives rise to further cohomological considerations, and anybranch on a descended curve (X0;R0) for the quotient map Aut(X0;R0)\(X0;R0) becomesrational on some uniquely determined twist of (X0;R0).

Now suppose that the pair (X;R) has nontrivial reduced automorphism group G =Aut(X)/〈ι〉. This automorphism group is then cyclic, of order n = #G. We will nowshow how to obtain a descent of (X;R) in this case.

Let α be a generator of G, and let Q = q(R) be the image of R on the quotient W = 〈ι〉\X.Since we assumed that α was not unipotent, we can identify W with a projective line P1

with affine coordinate x in such a way that Q corresponds to the point ∞ ∈ P1(F sep) andsuch that α is defined by

(5.14) α(x) = ζnx

with ζn a primitive nth root of unity in F sep. Let p be the polynomial defining X as a coverof W . Then because of our normalizations, we either have

(5.15) p(x) = π(xn)

or

(5.16) p(x) = xπ(xn).

for some polynomial π. In the latter case, there is a single point at infinity that is a Weier-strass point of X, whereas in the former there are two ordinary points at infinity.

We normalize π to be monic by applying a scaling in x (which does not affect the aboveassumptions on α and Q) and denote its degree by d. By our assumption that the reducedautomorphism group is of order n, if we write

(5.17) π(x) =d∑i=0

πixd−i = xd + π1x

d−1 + π2xd−2 + · · ·+ πd−1x+ πd

then the subgroup of Z generated by {i : πi 6= 0} equals Z.Given a monic polynomial π as above, we define the hyperelliptic curves Xπ and X ′π by

(5.18) Xπ : y2 = π(x)

and

(5.19) X ′π : y2 = xπ(x).24

The former curve admits a distinguished point Rπ = (0 : 1) at infinity after the coordinatechange (x, y) ← (1/x, y/xg+1), whereas the latter admits a Weierstrass point at infinity(which corresponds to R′π = (0, 0) under the same coordinate transformation).

Our only freedom left to modify the coefficients of π is scaling in the x-coordinate: thisinduces an action of F sep on the coefficients πi, given by

(5.20) α(π1, π2, . . . , πd) = (α1π1, α2π2, . . . , α

dπd).

We can interpret the coefficients of π as a representative of a point in the correspondingweighted projective space. Now there is a notion of a normalized representative of such apoint [14]. We call a polynomial π normalized if its tuple of coefficients is a normalized.

The uniqueness of the normalized representative then shows the following theorem.

Theorem 5.21. Let (X;R) be a marked hyperelliptic curve over F sep whose reduced auto-morphism group is tamely cyclic of order n. Then (X;R) admits a unique model of the form(Xπ, Rπ) or (X ′π, R

′π), with π normalized, depending on whether R is a Weierstrass point of

X or not.

Proof. It is clear from the above that (X;R) admits a model of the indicated form. Onthe other hand, let σ ∈ GF and consider the polynomial σ(π) obtained by conjugating thecoefficients of π by σ. If the coefficients of pπ were not stable under Galois, then we wouldfind two distinct normalized representatives of the same weighed point, a contradiction.Therefore σ(π) = π, and since σ was arbitrary, our theorem is proved. �

Remark 5.22. By normalizing as above, one in fact obtains a representative family of markedhyperelliptic curves.

The Klein quartic. We now apply the method of branches to a plane quartic, namely theKlein quartic, defined by

(5.23) X : x3y + y3z + z3x = 0.

By classical results (see the complete exposition by Elkies [10]), the curve X admits Belyımap of degree 168 that realizes the quotient map X → Aut(X)\X. In what follows, we willdetermine a model of X that admits a rational ramification point for this morphism thatis of index 2. The geometry of the Klein quartic is exceedingly well-understood due to itsmoduli interpretation. As such, our result is not new; it can be found in a different formin work of Poonen–Schaefer–Stoll [18]. However, what makes our method new is the purelyalgebraic approach to the problem, which is available in rather greater generality.

Before starting, we modify our model slightly and instead consider the rational S3 modelfrom Elkies [10], which is given by

(5.24) X ′ : x4 + y4 + z4 + 6(xy3 + yz3 + zx3)− 3(x2y2 + y2z2 + z2x2) + 3xyz(x+ y+ z) = 0.

Consider the field K = Q(i, θ) where θ4 = −7. Over K, the curve X ′ admits the involutiondefined by the matrix

(5.25)

2 −3 −6−3 −6 2−6 2 −3

∈ PGL3(Q) = Aut(P2)(Q)

This involution has the fixed point

(5.26) P = ((i+ 1)θ3 + 12iθ2 − 21(i− 1)θ+ 66 : 3(i+ 1)θ3 + 36iθ2 − 63(i− 1)θ− 50 : 124).25

Using branches, we will construct an isomorphism between the curve (X;P ) and its conju-gates that satisfies the cocycle relation.

Let σ be the automorphism of K such that σ(θ) = iθ and σ(i) = i; and similarly letτ(θ) = θ and τ(i) = i. Then σ and τ satisfy the standard relations for the dihedral group oforder 8, the Galois group of the extension K.

One surprise here is that it is impossible to construct a Weil cocycle for the full extensionK |Q. In fact there exist isomorphisms ϕσ : (σ(X);σ(()P )) and ϕσ : (τ(X); τ(()P )) but nomatter which we take, the automorphism

(5.27) σ3(Tσ)σ2(Tσ)σ(Tσ)Tσ

of X that corresponds to the cocycle relation for σ is never trivial, but always of order 2.This strongly suggests that we should find an extension L of Q with Galois group dihedral

of order 16 into which K embeds. Such extensions do indeed exist; one can be found inthe ray class field of Q(i) of conductor 42, obtained as the splitting field of the polynomial63t8 − 70t4 − 9 over Q.

The above extension is exactly the one that shows up when considering the branches ofthe Belyı map ϕ′ : X ′ → Aut(X ′)\X ′, which can be calculated by using the methods inPoonen–Schaefer–Stoll [18]. Note that it would have sufficed for our purposes to quotientout by the involution fixing P ; however, we use the Belyı map to show that it, too, can beused for purposes of descent.

We use x = s as a local parameter at the point P , and modify the coordinate onAut(X ′)\X ′ such that P is sent to 0. After dehomogenizing by putting z = 1 and de-termining y = y(s) as a power series in s, we obtain a power series expression in s for ϕ′ bycomposition. This power series looks like

(5.28) s→ c2s2 +O(s3)

The leading coefficient c2 of this power series has minimal polynomial

(5.29) 27889t4 − 1869588t3 − 18805122t2 + 1869795900t− 25658183943

over the rationals. Since the branch is an inverse of the power series above under compositionleading term in the branch, it will look like

(5.30) s→ (1/√c2)s1/2 +O(s).

The field generates by its leading coefficient can therefore be obtained as the splitting fieldof the polynomial (5.29) evaluated at t2. This is indeed the extension L constructed above:we have

(5.31)1√c2

=1

16032(211239η7 − 66339η5 − 163835η3 + 98343η)

where η is a root of 63t8 − 70t4 − 9.We now evaluate all power series for x, y, z involved in the branch (1/

√c2)s1/2 + O(s)

instead of s. This allows us to study the Galois action; by dehomogenizing and comparingthe conjugated power series (σ(X)(s), σ(y)(s)) with the fractional linear transformation of(x, y) under the automorphisms of X ′, we can read off which ασ maps the former branch tothe latter.

There exist elements σ, τ satisfying the relations for the dihedral group of order 16, gener-ating the Galois group of L, and restricting to σ, τ on K. For these elements, the conjugate

26

branch (σ(x)(s), σ(y)(s)), respectively (τ(x)(s), τ(y)(s)), is sent to (x(s), y(s)) by the ambi-ent transformation

(5.32) ϕσ =

−6 −2θ2 + 2 θ2 + 112θ2 + 2 −10 3θ2 + 1−θ2 + 11 −3θ2 + 1 2

,

respectively

(5.33) ϕτ =

−6 2θ2 + 2 −θ2 + 11−2θ2 + 2 −10 −3θ2 + 1θ2 + 11 3θ2 + 1 2

.

In particular σ(P ), respectively τ(P ), is sent to P by ϕσ, respectively ϕτ . By uniqueness ofασ and ατ when considering its action on the branches, we do get a Weil cocycle this time.Moreover, the corresponding descent leads to a rational branch of the Belyı map ϕ′. Aftercalculating a coboundary and polishing the result, we get the model

(5.34) X0 : x4 + 2x3y + 3x2y2 + 2xy3 + 18xyz2 + 9y2z2 − 9z4 = 0

as in Poonen–Schaefer–Stoll [18]. The curve X0 admits the point (0 : 1 : 0), which is arational branch for the quotient map (x : y : z) 7→ (x : y : z2) to the curve

(5.35) x4 + 2x3y + 3x2y2 + 2xy3 + 18xyz + 9y2z − 9z2 = 0

in P(1, 1, 2).We do not give explicit isomorphisms with the models (5.23) and (5.24) here, as they are

slightly unwieldy to write down; a matrix β inducing an isomorphism X ′∼−→ X0 can be found

by solving the equations

βασ = σ(β)

βατ = σ(β)(5.36)

up to a scalar. These equations are none other than (2.5). Once we lift the PGL3(L)-cocycleσ 7→ ασ to GL3(L), then we can forget about the scalar factor, so that (5.36) can be solvedby expressing the entries of β as combinations of a Q-basis of L. This in turn reduces solving(5.36) to solving a system of linear equations; it then remains to find a solution of relativelysmall height to find a reasonable descent morphism.

Remark 5.37. In fact lifting the cocycle σ 7→ ασ mentioned above to GL3(L) is not allthat trivial; for general plane curves it requires modifying an arbitrary set-theoretic lift bysuitable scalars, which comes down to determining the rational points on a certain conic.For plane quartics, however, things are slightly easier, since we can consider the morphismon the space of differentials that our cocycle induces.

The case of a ramification point of index 3 is considerably easier and yields the model

(5.38) 7x3z + 3x2y2 − 3xyz2 + y3z − z4 = 0,

on which the points (1 : 0 : 0) and (0 : 1 : 0) are fixed for the automorphism (x : y : z) 7→(ω2x : ωy : z) where ω is a primitive cube root of unity. In this case, we do not insist on thebranch being rational in order to get a nicer model for the marked curve.

Finally, the original model (5.23) already admits the rational point (0 : 0 : 1), which isstable under the automorphism (x : y : z)→ (ζ4

7x : ζ27y : ζ7z) of order 7.

27

Remark 5.39. Yet another way to find the descent above is to diagonalize the automorphism5.25 to the standard diagonal matrix with diagonal (1, 1,−1), which can be done by a Q-rational transformation. The 4 fixed points of the involution are then on the line z = 0. Bytransforming the coordinates x and y over Q we can ensure that the involution remains givenby the diagonal matrix (1, 1,−1) and the set 4 points remains still defined over Q, while alsoputting one of these points at (0 : 1 : 0). This normalization reduces the automorphismgroup sufficiently to make the rest of the descent straightforward.

6. Descent of marked Belyı maps

As mentioned in the previous section, in the absence of unipotent automorphisms a gener-ically Galois map W → V between curves of genus 0 over F sep is always a quotient by oneof the groups Cn, Dn, A4, S4, A5. In fact all of these quotients are Belyı maps over F sep, andin what follows we wish to consider some special descents of these maps. This will expandon the results in Couveignes–Granboulan [7], in which quite pleasing forms of these Belyımaps were already given.

We will slightly broaden our notion of Belyı maps over F in order to allow their branchlocus to be an arbitrary F -rational divisor instead of merely a union of points over F .

Definition 6.1. A weak Belyı map over F is a map f : X → P1 of curves defined over F thatis ramified above at most three points. A marked weak Belyı map over F is a pair (X; f ;P ),where f : X → P1 is a weak Belyı map over F and P ∈ X(F ) is a point of X over F .

By Birch’s theorem, any marked weak Belyı map descends to its field of moduli. Inprinciple we could once more use the methods of branches to descend marked weak Belyımaps. However, it turns out that if we consider the case where X has genus 0, then thingsare much easier, essentially because we have an obvious descent (P1,∞) of (X;P ) that wecan exploit. This leads to a useful and easily memorizable trick that resembles our approachto hyperelliptic curves in the previous section.

In what follows, we suppose that f : X → P1 is a weak Belyı map with X of genus 0 suchthat f is generically Galois with field of moduli F with respect to the extension F sep | F .We suppose P is a ramification point of f of non-trivial order n ∈ Z≥2, which we supposemaps to 0 ∈ P1 for the sake of simplicity. Our reasoning will now be very similar to that inSection 5.

Using an automorphism of P1 if necessary, we can obtain a model P1 of X such that notonly doesR correspond to the point (1 : 0) ofX at infinity but additionally the automorphismgroup of (f ;P ) is generated by (x : z) 7→ (ζnx : z) with ζn a primitive nth root of unity.This means that if we let t be the affine coordinate z/x the function f can be identified witha power series expansion

(6.2) f(t) =∞∑i=1

citin.

Note that ci = 0 for i ≤ 0 because we imposed that f(R) = 0.Since Aut(f ;P )(F sep) ∼= Z/nZ, the set {i ≥ 1 : ci 6= 0} generates Z as a group. So

there exists a finite subset I of this set with this property as well. Let cI = (ci)i∈I be thecorresponding point in the projective space weighted by its indices, and choose a normalizedrepresentative cI,0 of cI [14]. There exists a scaling t 7→ αt of t that transforms cI into cI,0,

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which is uniquely determined up to a power of the automorphism t 7→ ζnt since the elementsin I generate Z.

Proposition 6.3. With notations as above, let

(6.4) f0(t) = f(αt) =∞∑i=1

ci,0tin.

Then f0 has coefficients in F and (f0;∞) is a descent of (f ;P ).

Proof. Any isomorphism between f0 and its conjugate σ(f0) normalizes the group of auto-morphisms generated by (x : z) 7→ (ζnx : z) and fixes (1 : 0) ∈ P1(x : z). Therefore it isgiven by a scaling t 7→ ασt. The subset cI is defined over F , and in particular it is fixed byGalois conjugation. By uniqueness of the normalized representative, we therefore see thatασ is a power of ζn, and in particular that f0 is stable under conjugation by σ. Since σ wasarbitrary, we see that indeed f0 is defined over F . �

Remark 6.5. If we do not happen to know the field of moduli of the pair (f ;P ), then theapproach above calculates it as well. Indeed, MF sep

F (f ;P ) is generated by the coordinates ofthe normalized representative cI,0.

In what follows, we exhibit descents of all the generically Galois weak Belyı maps of genus0, ordered by their automorphism group G under the hypothesis that G does not contain anyunipotent elements. We impose an ordering of the corresponding branch indices b0, b∞, b1.Then we give a number of equalities of the form

f0 − f∞ = f1.

These equations will represent a rational function f = f0/f∞ that is ramified over three in-dividually F -rational points, which we may and do put at 0,∞, 1. The polynomial functionsf0, f∞, f1 will show the ramification behavior over these three points, which will be uniformof branch index b0, b∞, b1. In some cases, we will only give a single triple, usually when thattriple already has obvious rational points over 0,∞, 1. But when normalizing requires morework, we will give three triples f0, f∞, f1.

The branch point on P1 that is the image of the marked point P will always be rationalover the field of moduli; this was also the reason that we did not consider conics as the targetspace in our Definition 6.1. However, by our relaxation of the condition in Definition 6.1 it ispossible that the other two branch points are conjugate over F if they have the same branchindices, which leads to further twists. In this case, we put the points with coinciding branchindices at 0 and∞ in the first triples. We then change f to

√d(f + 1)/(f − 1), which moves

the branch point 1 to ∞ and the pair 0,∞ to ±√d. Normalizing again, we get our twists.

Case G ∼= Cn. We have the obvious triple with branch indices n, n, 1:

(6.6) tn − 1 = (tn − 1).

Twisting this map to get it to ramify over ±√d is actually not so straightforward as the

procedure above, essentially because there is no branching over 1, so that we cannot applyour normalization argument. However, a corresponding weak Belyı map was constructed byLercier–Ritzenthaler–Sijsling [16, Proposition 3.16]. We pass over this case here since thecorresponding points have trivial ramification index.

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Case G ∼= Dn. The branch indices are now equal to n, 2, 2. Putting the indices 2 at 0,∞and n at 1, we get the triples

(6.7) f(t) =(tn − 1)2

(tn + 1)2.

This map admits two rational points 0,∞ over 1. Moreover, there is the rational point 1over 0, and by exchanging f0 and f∞ we get a rational point over ∞. Twisting gives thefunction

(6.8) f(t) =dt2n + 1

2tn

which indeed branches of index n over ∞ and moreover of index 2 over ±√d since

(6.9) (dt2n + 1)± 2√dtn = (±

√dtn + 1)2.

Case G ∼= A4. This case has branch indices 2, 3, 3. We put the indices 3 at 0,∞ and 2 at 1to get

(6.10) (t(t3 + 8))3 − 26(t3 − 1)3 = (t6 − 20t3 − 8)2.

Permuting f0 and f∞ gives rise to a rational point over 0, while a triple that has the rationalpoint ∞ over 1 is given by

(6.11) (3t4 − 6t2 − 1)3 − (3t4 + 6t2 − 1)3 = −2232(t(3t4 + 1))2.

The corresponding family of twists branching over ∞ and ±√d is given by

(6.12) f(t) =−d3t12 − 99d2t8 + 297dt4 + 27

18(d2t10 + 6dt6 + 9t2).

Case G ∼= S4. For this case, the branch indices 2, 3, 4 are distinct, so we do not get extratwists, but only the three following triples:

(t8 + 14t3 + 1)3 − 2233(t(t4 − 1))4 = (t12 − 33t8 − 33t4 + 1)2,(6.13)

−28(t7 + 7t4 − 8t)3 − (t6 − 20t3 − 8) = (t12 + 88t9 + 704t3 − 64)2,(6.14)

and

(6.15)(3t8 + 28t6 − 14t4 + 28t2 + 3)3 − 33(t6 − 5t4 − 5t2 + 1)4

= 24(9t11 + 11t9 + 66t7 − 66t5 − 11t3 − 9t)2.

Case G ∼= A5. Once more the branch indices 2, 3, 5 are distinct. We get the following triples:

(6.16)(t20 + 228t15 + 494t10 − 228t5 + 1)3 − 2633(t(t10 − 11t5 − 1))

= t30 − 522t25 − 10005t20 − 10005t10 + 522t5 + 1,

and

(6.17)

− 53(t(5t6 − 5t3 + 8)(40t6 + 5t3 + 1)(5t6 + 40t3 − 1))3

− 26(25t12 − 275t9 − 165t6 + 55t3 + 1)5

= −(5t6 + 1)(200t12 + 500x9 + 2055t6 − 100t3 + 8)·

(25t2 + 1750t9 − 2190t6 − 350t3 + 1))2,

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and finally

(6.18)

((3t4 − 10t2 + 15)(t8 − 60t6 − 370t4 − 300t2 + 25)(t8 + 70t4 + 25)

)3

− 22(t(t4 − 5)(9t4 + 10t2 + 5)(t4 + 10t2 + 45)·

(t8 − 20t6 + 470t4 − 500t2 + 625)(5t8 − 20t6 + 94t4 − 20t2 + 5))2

= 33((t4 + 2t2 + 5)(t8 + 20t6 − 210t4 + 100t2 + 25)

)4.

The above method extends to curves and dessins of genus 0 with unipotent automorphismgroup, and to higher genus (hyperelliptic or not). We expect this will be useful in buildingdatabases of dessins with small defining equations, a topic for future work.

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[14] Reynald Lercier and Christophe Ritzenthaler, Hyperelliptic curves and their invariants: geometric,arithmetic and algorithmic aspects. Journal of Algebra, 372(0):595-636, December 2012.

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[16] Reynald Lercier, Christophe Ritzenthaler, and Jeroen Sijsling, Explicit Galois obstruction and descentfor hyperelliptic curves with tamely cyclic reduced automorphism group, arXiv:1301.0695, 2013.

[17] Jean-Francois Mestre, Construction de courbes de genre 2 a partir de leurs modules, in Effective methodsin algebraic geometry, volume 94 of Prog. Math., pages 313–334, Boston, 1991. Birkauser.

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[18] Bjorn Poonen, Edward F. Schaefer and Michael Stoll, Twists of X(7) and primitive solutions to x2 +y3 = z7, Duke Math. J. 137 (2007), no. 1, 103–158.

[19] Bounab Sadi, Descente effective du corps de definition des revetements galoisiens, J. Number Theory77(1), 1999, 71–82.

[20] Jean-Pierre Serre, Topics in Galois theory, Research Notes in Mathematics 1, Jones and Bartlett, 1992.[21] Jeroen Sijsling and John Voight, On computing Belyi maps, Publ. Math. Besancon: Algebre Theorie

Nr. 2014/1, Presses Univ. Franche-Comte, Besancon, 73–131.[22] Mark van Hoeij and Raimundas Vidunas, Algorithms and differential relations for Belyi functions,

arxiv:1305.7218v1, 2013.[23] Andre Weil, The field of definition of a variety, Amer. J. Math. 78, 1956, 509–524.

Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK;Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755,USA

E-mail address: sijsling@gmail.com

Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, NH 03755,USA

E-mail address: jvoight@gmail.com

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