arXiv:1407.5191v1 [math.AG] 19 Jul 2014

STRONGLY CYCLIC COVERINGS OF CYCLIC CURVESCHARLES SIEGEL
Abstract. In this note, we introduce the notion of an unramified strongly
cyclic covering for a cyclic curve, a class that has similar properties to, andcontains, unramified double covers of hyperelliptic curves. We determine several of their basic properties, extending the theorems in [FGD06] to this larger
class. In particular, we will write down equations for smooth affine models,determine when they are isomorphic, and discuss the curves that they areramified cyclic covers of.

The algebra and geometry of double covers of hyperelliptic curves is well understood, and has been studied by, among many others, Farkas [Far76, Far87a, Far87b]
and Bujalance [Buj86], from the topological viewpoint and Fuertes and GonzálezDiez [FGD06] from a more algebraic standpoint. This can be done because thesquare-trivial line bundles on the curve can all be described in terms of the fixed
points of the hyperelliptic involution. This yields concrete equations, which canthen be used to get very strong geometric information about the covers that cannot be so readily obtained for a general curve.
One would like to generalize these results to understand concretely examples ofhigher degree cyclic coverings, as studied by Faber in [Fab88] and recently by Langeand Ortega in [LO11]. The natural generalization is to cyclic curves of degree d,because for any fixed point of the cyclic automorphism, p, the class dp does notdepend on which fixed point was chosen. Unfortunately, there are not enough ofthese points to describe every degree d cyclic covering. Below, we will study the setof cyclic coverings that can be written in terms of the fixed points, and will proveanalogues of the results for double covers of hyperelliptic curves in this case.
Below, we will not distinguish between an affine algebraic curve, its smooth andcomplete model, and its model as a compact Riemann surface.
Definition 1 (Cyclic Curve). A cyclic curve is a curve C with a morphism f :C → P1 of degree d and Galois group Z/dZ and Z/dZ ⊳ Aut(C).
Our goal will be to study a class of divisors on a cyclic curve. For the remainderof this note, let f : C → P1 be the cyclic gd1 on a curve of genus g, andP
x1 , . . . , xr therramification points of f . We want to look at the divisors of theform
i=1 ni xi andP
Prni 1of degree 0. Equivalently, we can write them as i=1 ni xi − d gd with 0 ≤ ni < d,and we will equivalent mean the divisor of degree 0 and the positive part by itself.
From here on, we fix the degree d to be a prime number, and we will assume thatthe map f is unramified over ∞, and we will identify P1 \ {∞} with the complexnumbers.
r−2
Proposition
points of order d in J (C)[d] that can be written
P2. There are din the form
ni xi as above.
1

2

CHARLES SIEGELProof. Following [Dol12], we let R = {x1 , . . . , xr } be the set of ramification points.
rThen we can look
Pr at Fd = {the set of functions R → Fd }. The hyperplane given by{µ : R → Fd | i=1 µ(xi ) = 0} gives the set of degree 0 divisors. The curve can beQrwritten
y d = i=1 (x − bi )di , where bi = f (xi ), for some set of 1 ≤ di < d such thatPdi ≡ 0 modulo d. We can see that this is trivial by comparing the lines y = 0
and z = 0, and so, finally, we quotient by this relation, and get Fr−2
which embeds
dinto J (C)[d], the points of order d on the Jacobian of C.
Elements α ∈ Fr−1
in the proof of P
Proposition 2 will also be identified withdr
their images in J (C)[d], and as divisors i=1 α(xi )xi , with α(xi ) interpreted as aninteger 0 ≤ α(xi ) < d.
Definition 3 (Strongly Cyclic Cover). The unramified cyclic cover corresponding
to a point above will be called a strongly cyclic cover. We will denote this coverby π : C̃ → C, and when context requires that we be more specific, we will use
subscripts.
This implies that only a small fraction of all cyclic covers are strongly cyclic.
For instance, if g(C) = 2 and d = 3, then there will be four ramification points andso there are at most 32 points of order 3 that are strongly cyclic, but there are 34
points of order 3 on the Jacobian of C.
This is in contrast with the d = 2 case. For hyperelliptic curves, there are 2g + 2
ramification points, so there are 22g points of this form, and so every point of order
two on a hyperelliptic Jacobian is one of these.
fπ
The Riemann-Hurwitz theorem applied to C̃ → C → P1 immediately gives us
Lemma 4. If g is the genus of C and g̃ is the genus of C̃, theng=g̃=r=(r − 2)(d − 1)
2(d − 1)(rd − 2d − 2)
22g+2d−1

Theorem 5. Let C be a cyclic curve and α ∈ Fr−1a nonzero element of the kerneldQrof the map to J (C)[d]. Then C is given by the equation y d = i=1 (x − bi )α(xi ) .
QLet β ∈ Fr−1
give a curve Cβ with equation z d = ri=1 (x − bi )β(xi ) . Then:d(1) the normalization of C̃β = C ×P1 Cβ is the unramified strongly cyclic covercorresponding to β, and every strongly cyclic cover occurs in this way,(2) two strongly cyclic covers C̃β1 → C and C̃β2 → C are isomorphic if andonly if β1 + β2 is a multiple of α, in which case the change of coordinates
is:

yj,
(x, y, z) 7→ x, y, ζdz
where ζd is a primitive dth root of unity,(3) Let Aβ = {xi |β(xi ) 6= 0}. Then C̃β is a ramified d-cyclic cover of a curve of(d−1)(|Aβ |−2)(d−1)(|Aiα−β |−2)genus g0 =and curves of genera gi =where22i = 1, . . . , d − 1.

STRONGLY CYCLIC COVERINGS OF CYCLIC CURVES

3(4) The number of strongly
cyclic covers of Cβ that cover a curve of genus
kk(k−2)(d−1)
ris k
1 − d1 dk−1 − (−1).
g=2dWe note that in part 2, although not isomorphic as covers of C, C̃β1 and C̃β2 are
isomorphic as curves if β2 is a multiple of β1 . Specifically, if µ is the line bundle
corresponding to β1 , then they are given by the relative spectra OC ⊕ µ⊕ . . .⊕ µd−1 ,and OC ⊕ µk ⊕ . . . ⊕ µ(d−1)k , which are isomorphic because d and k are relatively
prime, and d is the order of µ.
For part 4, the result of Bujalance [Buj86] states the d = 2 case of this slightly
more strongly, as a partition over half of the genera that appear. For higher degree,
the situation is much less clear, just as it is in part 3, because the relation between
β and the iα − β’s is more complex.
Proof. To prove the first point, it suffices to show that if fi : Ci → P1 for i = 1, 2 bemaps of degree d with the same Galois group. Assume further that Branch(f2 ) ⊂Branch(f1 ) and the ramification type over each common branch point is the same.
Then the normalization of C1 ×P1 C2 → C2 is unramified.
This is a local issue near a branch point. So, without loss of generality, we cantake fi : C → C to be z k and wk for some k. Then, the fiber product is the curvez k = wk in C2 and the structure map to C is z k . The fiber over 0 is singular, but
it has k distinct branches, so that the normalization is unramified over 0.
Part two follows by a direct computation, where f (x) is defined by z1yz2 = f (x)and z1 , z2 are the z-coordinates for the covers given by β1 and β2 .
Now, we show that this is the only case. Let φ : C̃β1 → C̃β2 be an isomorphism
over C. The field C(C̃β1 ) is generated by x, y, z, so we’ll denote it by C(x, y, z)
and ignore the relations in the notation. So there is a rational function
R(x, y, z) ∈QC(x, y, z) so that φ(x, y, z) = (x, y, R(x, y, z)), and R(x, y, z)d = ri=1 (x − bi )β2 (xi ) .
Under the action of multiplying y and z by dth roots of unity, the fixed fields ofindex d are C(x, y) = C(Cβ1 ), C(x, z) = C(Cβ2 ), and C(x, y i /z) = C(Ciα−β ). Let
w be y, z or y i /z for some i.
Then, we can write R(x, y, z) = a(x) + b(x)w. Then, as R(x, y, z)d ∈ C(x), wePd−2
have a(x)d + b(x)d wd + a(x)b(x)w( i=0 fi (x)wi ) ∈ C(x). So then either w ∈ C(x)or a(x)b(x) = 0, and if b(x) = 0, then R(x, y, z) Q
∈ C(x), thus, a(x) = 0. So
R = bw. Raising to the dth power, we get that ri=1 (x − bi )β2 (xi ) = bd wd =Q
bd ri=1 (x − bi )χ(xi ) where χ is β1 ,β2 or iα − β1 , depending on w. We divide and getQbd = ri=1 (x − bi )β2 (xi )−χ(xi ) , and as |β2 (xi ) − χ(xi )| < d, we must have b(x)d = 1,so b(x) = ζd for some root of unity. Thus, β2 = χ, and we have β2 = iα − β1 forsome i.
Part three is just the map C̃β → Cβ combined with part two.
Part four is a fairly straightforward counting argument. We need to have exactly
k ramification
points, we need to count the number of β’s such that |Aβ | = k. Thisis kr times the number of linear functions {x1 , . . . , xk } → Fd whose values add
up to zero and none of them are zero. This can be counted by inclusion-exclusion,
kkPgiving us ki=0 (−1)i ki dk−i−1 , and this simplifies to 1 − d1 dk−1 − (−1)
d , giving
us the count.
Corollary 6. Any degree d cyclic curve given by y d = f (x) such that f (x) ∈ Q[x]and which splits over Q into the product of two polynomials of degrees divisible byd, then the curve admits a smooth strongly cyclic cover defined over Q.

4CHARLES SIEGEL

Finally, we look at the branch points of those strongly cyclic covers which arealso d-cyclic curves. These are given by pairs of branch points, bi , bj and we requirethat the corresponding values of β are β(xi ) = 1 and β(xj ) = d − 1. Then, thecover is given by the equationsrY

yd =(x − bk )β(xk )z d = (x − bi )(x − bj )d−1 .

k=1

We make the substitution z = t(x − aj ), which transforms the equation in z toa −a td

x−aitd = x−a, which we can then solve for x to get x = i1−tjd .
jFrom here, it becomes a straightforward manipulation:yd===r Ya i − a j td

1 − td− ak

k=1r  dYβ(xk )β(xk )t (ak − aj ) − (ak − ai )1 − tdk=1β(xk )td (ai − aj )d Y td (ak − aj ) − (ak − ai )(1 − td )d1 − tdk6=i,j==Yβ(xk )
td (ai − aj )dP
td (ak − aj ) − (ak − ai )β(x)dk
(1 − t ) k
k6=i,jβ(xk )dd
YY t (ai − aj )ak − aiβ(x)dkP(ak − aj )t −ak − aj(1 − td ) k β(xk )k6=i,jNow, we set

k6=i,jPy(1 − td )( β(xk ))/dw=
1/dQβ(xk )(a−a)
t(ai − aj )kj
k6=i,jand obtain the equation

β(xk )Y 
ak − ai2w =t −ak − ajdk6=i,jQr
Corollary 7. Let C be a d-cyclic curve with equation y d = k=1 (x − ai )β(xi ) .
Then for each ordered pair (except when d = 2, then unordered) of point (ai , 0)and (aj , 0), there is a smooth strongly cyclic cover, with additional equation z d =(x − ai )(x − aj )d−1 , Fi,j : Ci,j → C where Ci,j is itself d-cyclic, given by y d =β(xk )
Q
ak −ai2x−
withk6=i,j
ak −aj1/d 
Q(a−a)d wt(ai − aj )kjk6=i,j ai − aj x
PFi,j (x, y) = ,
.
1 − xd(1 − td ) β(xj )/dThis can be characterized, up to equivalence, by the fact that the two chosen points
are the only ones not covered by ramification points.

STRONGLY CYCLIC COVERINGS OF CYCLIC CURVES

5References[Buj86]E. Bujalance. A classification of unramified double coverings of hyperelliptic Riemannsurfaces. Arch. Math. (Basel), 47(1):93–96, 1986.
[Dol12] Igor V. Dolgachev. Classical algebraic geometry. Cambridge University Press, Cambridge, 2012. A modern view.
[Fab88] Carel Faber. Prym varieties of triple cyclic covers. Math. Z., 199(1):61–79, 1988.
[Far76] H. M. Farkas. Unramified double coverings of hyperelliptic surfaces. J. Analyse Math.,
30:150–155, 1976.
[Far87a] H. M. Farkas. Unramified coverings of hyperelliptic Riemann surfaces. In Complex analysis, I (College Park, Md., 1985–86), volume 1275 of Lecture Notes in Math., pages
113–130. Springer, Berlin, 1987.
[Far87b] H. M. Farkas. Unramified double coverings of hyperelliptic surfaces. II. Proc. Amer.
Math. Soc., 101(3):470–474, 1987.
[FGD06] Yolanda Fuertes and Gabino González-Diez. Smooth double coverings of hyperellipticcurves. In The geometry of Riemann surfaces and abelian varieties, volume 397 of Contemp. Math., pages 73–77. Amer. Math. Soc., Providence, RI, 2006.
[LO11] Herbert Lange and Angela Ortega. Prym varieties of cyclic coverings. Geom. Dedicata,150:391–403, 2011.
(Charles Siegel) Kavli Institute for the Physics and Mathematics of the Universe(WPI), Todai Institutes for Advanced Study, the University of Tokyo
E-mail address, Charles Siegel: charles.siegel@ipmu.jp
URL: http://db.ipmu.jp/member/personal/2754en.html