Hyperelliptic curves with reduced automorphism group A5
We study genus hyperelliptic curves with reduced automorphism group and give equations for such curves in both cases where is a decomposable polynomial in or . For any fixed genus the locus of such curves is a rational variety. We show that for every point in this locus the field of moduli is a field of definition. Moreover, there exists a rational model or of the curve over its field of moduli where can be chosen to be decomposable in or . While similar equations have been given in BCGG () over , this is the first time that these equations are given over the field of moduli of the curve.
Let denote a genus hyperelliptic curve defined over an algebraically closed field of characteristic zero, its hyperelliptic involution, and its automorphism group. The group is called the reduced automorphism group of . We denote by the moduli space of genus hyperelliptic curves and by the locus in of hyperelliptic curves with automorphism group .
In previous works we have focused on the loci of hyperelliptic curves with embedded in the automorphism group , or when is isomorphic to , or ; see GS (), Sh1 (). This paper continues on the same line of thought as Sh1 () focusing instead on the case when is isomorphic to .
The second section covers basic facts on automorphism groups of hyperelliptic curves. The group is a finite subgroup of . By a theorem of Klein (see Kl ()), is isomorphic to one of the following: . We are interested in the latter case. We give a representation of the group in . The group acts on the genus zero field via the natural way. The fixed field is a genus 0 field, say . Thus, is a degree 60 rational function in which we denote by . Using this representation we compute the fixed field of . This rational function (up to a coordinate change) can be decomposed in , , or . Using computer algebra techniques (i.e, see Gu ()) we compute such decompositions and use the decomposition in , to compute an equation of the hyperelliptic curves. The equation for makes it possible to compute dihedral invariants of such curves (cf. section 4).
In section three we determine the ramification signature of the cover Using this ramification structure we are able to show that if then . Then the full automorphism group is isomorphic to or . Moduli spaces of covers are Hurwitz spaces, which we denote by . There is a map , where is the moduli space of genus algebraic curves. For a fixed there is only one signature that occurs for the cover . Hence, we denote by the image in the hyperelliptic locus . Given a curve we would like to determine if it belongs to the locus and describe points . Hence, we need invariants which determine the isomorphism classes of these curves. In the last part of section three we determine the parametric equations of such curves in all cases . Using the decompositions of we are able to compute these equations where is a decomposable polynomial in , , or .
In section four we give a brief introduction of the classical invariants of binary forms. Such invariants classify the orbits of the -action on the space of binary forms. We use transvections to discover invariants which give necessary conditions for a curve to have reduced automorphism group isomorphic to or full automorphism group isomorphic to or . Such conditions appear in the literature for the first time. Further, we compute the dihedral invariants of such curves and determine the algebraic relations among them.
In the last section we discuss the field of moduli versus the field of definition for hyperelliptic curves with reduced automorphism group . This is a problem of algebraic geometry that goes back to Weil and Grothendieck. It follows from Sh5 () or GS () that for hyperelliptic curves with reduced automorphism group the field of moduli is a field of definition. However, no rational models of the curve over the field of moduli have been known. We construct such models for all curves with . In the last part of the paper we discuss in more detail the 1-dimensional families for all cases . In these cases we prove computationally that for such loci we have , where is the fourth branch point of the cover .
There is plenty of literature on the automorphism groups of hyperelliptic curves. Among many papers we mention
BS (), BCGG (), GS (), GSS (), Sh1 (), Sh5 (). Most of these papers have studied
determining the automorphism groups of the hyperelliptic curve. The main focus of this paper is the locus of
hyperelliptic curves with reduced automorphism group isomorphic to as a subvariety of the hyperelliptic
moduli and the field of definition versus the field of moduli for curves in this locus.
Notation: Throughout this paper denotes an algebraically closed field of characteristic zero, an integer , and a hyperelliptic curve of genus . (resp., ) is the moduli space of curves (resp., hyperelliptic curves) defined over . The symbol denotes a permutation which is conjugate in to an product of -cycles.
Let be a genus hyperelliptic curve defined over an algebraically closed field of characteristic zero. We take the equation of to be , where . Denote the function field of by . We identify the places of with the points of in the natural way. Then, is a quadratic extension field of ramified exactly at places of . The corresponding places of are called the Weierstrass points of . Let . Thus, , where and .
Let . Since is the only genus 0 subfield of degree 2 of , then fixes . Thus, , with , is central in . We call the reduced automorphism group of the group . Then, is naturally isomorphic to the subgroup of induced by . We have a natural isomorphism . The action of on the places of corresponds under the above identification to the usual action on by fractional linear transformations . Further, permutes . This yields an embedding .
Because is the unique degree 2 extension of ramified exactly at , each automorphism of permuting these places extends to an automorphism of . Thus, is the stabilizer in of the set . Hence under the isomorphism , corresponds to the stabilizer in of the -set .
By a theorem of Klein, is isomorphic to one of the following: , , , or . We are interested in the latter case. The branching indices of the corresponding cover are respectively; see Sh1 () for details of the general setup. That means that is given as where and have orders 2, 5, 3. How lift in the extension of will determine . In the next section, we will determine the cover explicitly. The lifting of the elements will determine the group and the equation of the hyperelliptic curve.
Let and , where and is a primitive root of unity. Then have orders 2 and 5 respectively and has order 3. This gives an embedding of in in the following way: . In the next section we will find the fixed field of under the action and study intermediate fields of the extension .
The group given above acts on via the natural way. The fixed field is a genus 0 field, say . Thus, is a degree 60 rational function in , say . In this section we determine and its decompositions.
Let be a finite subgroup of . Let us identify each element of with the corresponding Moebius transformation and let be the -th elementary symmetric polynomial in the elements of , . Then any non-constant generates .
It is easy to check that the are the coefficients of the minimum polynomial of over . It is well-known that any non-constant coefficient of this polynomial generates the field. ∎
The fixed field of is generated by the function
Apply the theorem to the embedding of given above. ∎
The branch points of are 0, 1728 and . These correspond respectively to the elements in the monodromy group (cf. Section 3.1). At the place the function has the following ramification:
We denote the following by
As we will see in the next section these functions will be instrumental in determining the equation of the hyperelliptic curves.
2.1 Decomposition of
The automorphism group of is the embedding of detailed before. As , there is a degree-preserving correspondence between subgroups of and intermediate fields in the extension. By Lüroth’s Theorem, each of those fields is for some rational function . Now, it is clear that, in general, . Thus, we can use computer algebra techniques to find all the decompositions of and describe the lattice of intermediate fields.
It is clear from the expression of that there is a decomposition . This comes also from the fact that the subgroup of corresponds to the field generated by
It is also possible to find decompositions involving or for functions that are equivalent to . Namely, for any , a generator of the field fixed for the conjugate group is . If is chosen in such a way as having , then will be an intermediate field by Lemma 1. This can be accomplished by conjugating any involution of into . In the same manner, if an element of order 3 in is conjugated into , where is a primitive cubic root of , the resulting function can be written in terms of
We present the former case here, as it will be used later. The element satisfies where Therefore, will have as a component. Indeed,
3 Automorphism groups and the corresponding loci
In this section we determine the automorphism group of and the ramification structure of the cover . Further, we will discuss the locus of such curves in the variety of moduli.
The automorphism group of the hyperelliptic curve is a degree 2 central extension of . The following lemma is proved in GS ().
Let , and its image in with order . Then,
i) if and only if it fixes no Weierstrass points.
ii) if and only if it fixes some Weierstrass point.
Thus, is the monodromy group of a cover with signature as in section 2; see §2 in Sh1 () for further details. We fix the coordinates in as and respectively and from now on denote the cover . Thus, is a rational function in of degree . We denote by the corresponding branch points of . Let be the set of branch points of . Clearly . Let denote the images in of Weierstrass places of and .
Let , where . For each branch point , we have the degree equation where the multiplicity of the roots correspond to the ramification index for each (i.e., the index of the normalizer in of ). We denote the ramification of , by , where the subscript denotes the degree of the polynomial.
Let . The points in the fiber of a non-branch point are the roots of
the equation: To determine the equation of the curve we simply need to determine
the Weierstrass points of the curve. For each
fixed there are the following cases:
From the above lemma we have that if the places in the fiber , , , are Weierstrass points then lift in to elements of order 4, 6, and 10 respectively. The first four cases give the group and the other four cases give the group . We have the following table. The column containing the dimension of the corresponding spaces will be explained in the next subsection.
In the Table we give the ramification structure of . The tuple corresponding to this signature is such that and . We call this tuple the signature tuple of the covering (cf. Section 3.1 for details).
Let be a genus hyperelliptic curve with reduced automorphism group isomorphic to . If is odd then , otherwise .
3.1 Hurwitz spaces
In this section we give a brief introduction to Hurwitz spaces. For further details the reader can check H () among many other authors. Let be a curve of genus and be a covering of degree with branch points. We denote the branch points by and let . Choose loops around such that
acts on the fiber by path lifting, inducing a transitive subgroup of the symmetric group (determined by up to conjugacy in ). It is called the monodromy group of . The images of in form a tuple of permutations called a tuple of branch cycles of . We call such a tuple the signature of . The covering is of type if it has as tuple of branch cycles relative to some homotopy basis of .
Two coverings and are weakly equivalent (resp. equivalent) if there is a homeomorphism and an analytic automorphism of such that (resp., ). Such classes are denoted by (resp., ). The Hurwitz space (resp., symmetrized Hurwitz space ) is the set of weak equivalence classes (resp., equivalence) of covers of type , it carries a natural structure of an quasiprojective variety.
Let denote the conjugacy class of in and . The set of Nielsen classes is
The braid group acts on as
where . We have if and only if the tuples , are in the same braid orbit .
Let be the moduli space of genus curves. We have morphisms where . Each component of has the same image in . We denote by This causes no confusion since for a fixed we are in one of the cases of Table 1.
Next, we see how this applies to our particular situation. The family of covers as in Table 1, have monodromy group or . We denote the set of branch points of by . The branch cycle description of is as in Table 1. Since we have at least branch points which have the same ramification then there is an action of permuting these branch points (i.e., which correspond to the ramification type ). Notice that in case 1 there is an action of on the set of branch points. The symmetrized Hurwitz space is birationally isomorphic to the locus of hyperelliptic curves in hyperelliptic moduli with reduced automorphism group . It will be our goal to determine this locus for any . We summarize the results of this section in the next lemma.
Let be a genus hyperelliptic curve with reduced automorphism group isomorphic to and denote the locus of such curves in the hyperelliptic moduli . Then, and the signature of the covering are given in Table 1. Further, each locus is -dimensional irreducible subvariety of the hyperelliptic moduli .
The moduli dimension of these families of covers is , where is the number of branch points of the cover . The ramification of each branch point is of the type . The Hurwitz-Riemann formula determines the number of branch points in each case. ∎
3.2 Parametrization of families
In this section we state the equations of curves in each case of Table 1. Continuing with the notation of section 4.1 we have Thus the places of are roots of the polynomial
Then, the equation of the curve for all cases 1-8 is where is respectively
Since we know in each case, then it is an elementary exercise to compute the equation of the curve for all cases of Table 1. In our case we can apply the above when or . In the first case we have
Then, By replacing with we determine the equation of the curve in each case. In the second case we determine using and .
4 Isomorphism classes of hyperelliptic curves with reduced automorphism group
In this section we discuss the invariants of hyperelliptic curves with reduced automorphism group . Such invariants are needed to describe the loci and discuss the field of definition of such curves. We will consider the coefficients of our curves as variables in order to study the relations among the different function fields that will be introduced.
To get a description of for each case of Table 1, we need invariants which would classify the isomorphism classes of hyperelliptic genus curves. These invariants are generators of the fixed field of acting on the -dimensional space of binary forms of degree .
We use the symbolic method of classical invariant theory to construct invariants of binary forms. Let and be binary forms of degree and respectively. We denote by their -transvection; see Sh1 () for details. For the rest of this paper denotes a binary form of degree . Invariants (resp., covariants) of binary forms are denoted by (resp., ) where the subscript denotes the degree (resp., the order). We define the following covariants and invariants:
The -invariants are called absolute invariants. We define the following absolute invariants:
We will only perform computations on subvarieties of dimension , hence don’t need other absolute invariants. Next we will give necessary conditions on these invariants for the corresponding curve to have reduced automorphism group and full automorphism group or .
Let be a hyperelliptic curve with genus such that . Then the invariants are zero for .
In all cases, it can be directly computed that the corresponding ’s are zero. ∎
Let be a genus hyperelliptic curve such that . Then, is isomorphic to a curve given by the equation , with
where or . Such equation is called the normal equation of the curve . The following
are called dihedral invariants. Assume (the other case is similar). From the definition of the invariants we have
for each . Notice that solving this linear system we have , for each . For we have the equation
which is a quadratic polynomial in . It is shown in GS () that is a rational variety and . The next theorem determines a relation between dihedral invariants.
Let be a genus hyperelliptic curve with and its corresponding dihedral invariants. Then
i) If is odd then and
ii) If is even then and
i) This follows from Theorem 3, i) in GS ().
ii) The equation of is given by where is a polynomial of degree in . Computing invariants is the same as in part i). In this case the involutions of lift to elements of order 4 in . From Theorem 3, ii) in GS () we have the equation of part ii). This completes the proof. ∎
Since the discriminant of the quadratic in Eq. (2) is zero (see Thm. 4.1) we have . Hence, . Let be a hyperelliptic curve with reduced automorphism group and equation Since are invariants under any coordinate change then is an extension of . This curve can be normalized by means of the transformation , which gives
Notice that . Since all the other ’s can be expressed in terms of then we have that .
The cover has branch points. Let . Then, the isomorphism class of the corresponding curve is determined up to permutation of