Etale double covers of cyclic p-gonal covers

TL;DR

This paper investigates the Galois groups of certain covers in algebraic geometry and explores relationships between Prym varieties and Jacobians, extending known constructions in the field.

Contribution

It computes Galois groups of specific covers and establishes new relations between Prym varieties and Jacobians, generalizing the trigonal construction.

Findings

Galois groups of the covers are explicitly computed.

Relations between Prym varieties and Jacobians are established.

The work generalizes the trigonal construction.

Abstract

This paper computes the Galois group of the Galois cover of the composition of an \'etale double cover of a cyclic $p$-gonal cover for any prime $p$. Moreover a relation between some of its Prym varieties and the Jacobian of a subcover is given. In a sense this generalizes the trigonal construction.

Equations

$$G=N⋊P$$

$$\alpha :\prod _{i=1}^{m}P(Y_i/X)\to JT.$$

$$Z\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\Delta }_i:=\{H{n}_1{g}_i,H{n}_2{g}_i\}$$

$$\Omega :={\Delta }_1\cup {\Delta }_2\cup \cdots \cup {\Delta }_p.$$

$$H_i:=g_i^{-1}H{g}_i.$$

$${\Delta }_{i}n=\{H{n}_1{g}_i,H{n}_2{g}_i\}{g}_i^{-1}{n}_i^{\prime }{g}_i=\{H{n}_1{n}_i^{\prime }{g}_i,H{n}_2{n}_i^{\prime }{g}_i\}={\Delta }_i.$$

$$\sigma :=(1,2,\dots ,p)(p+1,p+2,\dots ,2p).$$

$$N\cong ({\mathbb{Z}}_2{)}^{p-1}.$$

$$s_1:=t_1{t}_2\in N,$$

$${\sigma }^{-1}{t}_1{t}_2\sigma$$

$${\sigma }^{-2}{t}_1{t}_2{\sigma }^2$$

$$\cdots$$

$${\sigma }^{-(p-1)}{t}_1{t}_2{\sigma }^{p-1}$$

$$\prod _{i=1}^p{s}_i=(t_1{t}_2)({\sigma }^{-1}{t}_1{t}_2\sigma )({\sigma }^{-2}{t}_1{t}_2{\sigma }^2)\cdots ({\sigma }^{-(p-1)}{t}_1{t}_2{\sigma }^{p-1})=1.$$

$$2^{p-1}=|⟨s_1,s_2,\dots ,s_p⟩|\le |N|.$$

$${\Phi }^{\prime }:N\to N/H_1\times N/H_2\times \dots \times N/H_{p-1}$$

$$G=N⋊P\simeq {\mathbb{Z}}_2^{p-1}⋊{\mathbb{Z}}_p.$$

$$G=⟨s_1,\dots ,s_p,\sigma |\prod _{i=1}^p{s}_i=1,{\sigma }^p=1,s_1^2=1,{\sigma }^{-1}{s}_j\sigma =s_{j+1}\text{mbox}forj=1,\dots ,p-1⟩.$$

$$m:=\frac{1}{p}(2^{p-1}-1)$$

$$Y_j:=Z/R_j\text{mbox}forj=1,\dots ,m$$

$$Z\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\sum _{i=1}^m\dim P(Y_i/X)=\dim JT.$$

$$\{{\chi }_0\otimes {\rho }_j,{\mathrm{Ind}}_N^G({\chi }_i\otimes \eta )/0\le j\le p-1,1\le i\le m\}$$

$$J_{{\rho }_0}\times J_{\psi }\times J_{{\theta }_1}\times \cdots \times J_{{\theta }_m}\to JZ.$$

$$n_{\rho }=\frac{\dim V_{\rho }}{m_{\rho }},$$

$$B_{\psi }=J_{\psi },J_{{\theta }_j}\sim B_{{\theta }_j}^p,J_{{\rho }_0}=J{\mathbb{P}}^1=0$$

$$B_{{\theta }_j}\sim P(Y_j/X)\text{and }{J}_{\psi }\sim J(X).$$

$$JX\times \prod _{j=1}^{m}P(Y_j/X{)}^p\to JZ.$$

$${\nu }_i:Z\to Y_i\text{mbox}and\mu :Z\to T,$$

$$\alpha :=\sum _{i=1}^m\mathrm{Nm}\mu \circ {\nu }_i^{\ast }:\prod _{i=1}^{m}P(Y_i/X)\to JT.$$

Full text

Partially supported by Anillo ACT 1415 PIA-CONICYT

1. Introduction

In this paper we investigate the Galois group of the Galois cover of the composition of étale double coverings $Y\rightarrow X$ of cyclic covers $X\rightarrow\mathbb{P}^{1}$ of prime degree $p$. For $p=2$, Mumford shows in [6] that $Y\rightarrow\mathbb{P}^{1}$ is Galois with Galois group the Klein group of order 4 and the Prym variety $P(Y/X)$ is isomorphic as a principally polarized abelian variety to either a Jacobian or the product of 2 Jacobians.

For $p=3$, the trigonal construction tells us the principally polarized $P(Y/X)$ is isomorphic to a Jacobian of a tetragonal curve. In Section 2 we study the Galois group of the Galois closure $Z\rightarrow\mathbb{P}^{1}$ of $Y\rightarrow\mathbb{P}^{1}$ for an odd prime $p$. The main result of this section is

Theorem 2.6. Let $p$ be an odd prime, $Y\rightarrow X$ an étale double cover and $X\rightarrow\mathbb{P}^{1}$ a cyclic cover of degree $p$. Then $Y\rightarrow\mathbb{P}^{1}$ is not Galois. Denoting by $Z\rightarrow\mathbb{P}^{1}$ its Galois closure, its Galois group $G$ is

[TABLE]

*where $\;N\cong\mathbb{Z}_{2}^{p-1}\;$ and $\;P\cong\mathbb{Z}_{p},\;$ and $X=Z/N$, $Y=Z/H$, with $H$ a maximal subgroup of $N$.

There are $2^{p-1}-1$ maximal subgroups of $N$. The group $P$ acts on them by conjugation and there are $m:=\frac{1}{p}(2^{p-1}-1)$ conjugacy clases of such subgroups.

Let $\{Y_{i}\rightarrow X\;|\;i=1,\dots,m\}$ be the corresponding double covers. It is easy to see that they are all étale. If $T:=Z/P$, there is a natural homomorphism

[TABLE]

Our main result is:

Theorem 3.1. $\alpha:\displaystyle{\prod_{i=1}^{m}P(Y_{i}/X)}\rightarrow JT$* is an isogeny with kernel in the $2^{p-2}$-division points*.

As an immediate consequence we get examples of Jacobians with arbitrarily many isogeny factors of the same dimension. For $p=3$ this is not yet the trigonal construction, which, however, is an easy consequence, as we show in Section 4.

For the sake of completeness, we also consider the case $p=2$, i.e., we give a proof of Mumford’s theorem mentioned above. Note that Mumford gives only a short sketch of proof leaving the details to the reader. It seems to us that our proof is different from the one Mumford had in mind.

2. Étale covers of cyclic $p$-gonal covers

2.1. The structure of the Galois group

Let $p$ be a prime and $\varphi:X\rightarrow\mathbb{P}^{1}$ be a cyclic covering of degree $p$ ramified over $\beta$ points of $\mathbb{P}^{1}$, with $\beta\geq 3$. Observe that if $p=2$ then $\beta$ must be even.

Let $\psi:Y\rightarrow X$ be an étale double cover and $\widetilde{\varphi}:Z\rightarrow\mathbb{P}^{1}$ the Galois closure of the composed map $\varphi\circ\psi$. Let $G$ denote the Galois group of $\widetilde{\varphi}$ and $H$ and $N$ the subgroups of $G$ corresponding to $Y$ and $X$. So we have the following commutative diagram

[TABLE]

In this section we determine the structure of $G$.

Lemma 2.1**.**

The permutation representation $\rho$ of of $G$ on the right cosets of $H$ in $G$ has its image in the alternating group $A_{2p}$ of degree $2p$, and the non-trivial elements of $G$ fixing points in $Z$ have order $p$. Moreover, the representation $\rho:G\rightarrow A_{2p}$ is injective.

Proof.

Recall that $Y\rightarrow X$ is the double covering corresponding to the embedding $H\subset N$. Since $\varphi$ is cyclic of prime degree, the local monodromy of each of its branch points is a cycle of length $p$. Since $\psi$ is an étale double cover, every local monodromy of $\varphi\circ\psi$ above a branch point is the product of two disjoint cycles of length $p$ and hence in $A_{2p}$. Since $G$ is generated by these products, this gives the first assertion. The second assertion is clear. ∎

Corollary 2.2**.**

If $p=2$, the covering $\varphi\circ\psi$ is Galois with Galois group $G$ the Klein group of order $4$. In particular $Z=Y$.

Proof.

According to Lemma 2.1, $G$ is a subgroup of $A_{4}$, generated by elements which are products of two disjoint cycles of length $2$. Hence $G$ is the Klein group of order four. ∎

Proposition 2.3**.**

Suppose $p$ is an odd prime. Then the covering $\varphi\circ\psi:Y\rightarrow\mathbb{P}^{1}$ cannot be Galois.

Proof.

Since groups of order $2p$ cannot be generated by elements of order $p$, the covering $\varphi\circ\psi:Y\rightarrow\mathbb{P}^{1}$ cannot be Galois. ∎

For the rest of this section we assume that $p$ is an odd prime; hence the covering $Y\rightarrow\mathbb{P}^{1}$ is not Galois, so $Z\neq Y$ and $H$ and $N$ are the proper subgroups of $G$ corresponding to $Y$ and $X$ respectively, as in Diagram (2.1).

Let $\{1=g_{1},g_{2},\dots,g_{p}\}$ denote a complete set of representatives of right cosets of $N$ in $G$ and $\{1=n_{1},n_{2}\}$ denote a complete set of representatives of right cosets of $H$ in $N$. Then the set $\{n_{i}g_{j}:i=1,\dots,p,j=1,2\}$ is a complete set of representatives of right cosets of $H$ in $G$.

For $i=1,\dots,p$ consider

[TABLE]

as a set of two elements. Then the right action of $G$ on the right cosets of $H$ in $G$ induces a transitive action of $G$ on the set

[TABLE]

This is just the right action of $G$ on the right cosets of $N$ in $G$. Now denote for $i=1,\dots,p$,

[TABLE]

Clearly each $H_{i}$ is a normal subgroup of index 2 in $N$.

Lemma 2.4**.**

1. (i)

Any element of $H_{i}$ stabilizes each of the two points of $\Delta_{i}$; 2. (ii)

$N$* is the stabilizer of each set $\Delta_{i}$.*

Proof.

For (i) use that that $H$ is normal in $N$. By definition, $N$ is the normal subgroup of $G$ corresponding to the covering $X\rightarrow\mathbb{P}^{1}$. Since the $\Delta_{i}$ represent a right cosets of $N$ in $G$, this implies (ii). One can also see this directly: suppose $n\in N$. For any $i,1\leq i\leq p$ there is an $n^{\prime}_{i}\in N$ such that $n=g_{i}^{-1}n^{\prime}_{i}g_{i}$. Then we have

[TABLE]

Since $\;G\;$ does not stabilize $\;\Delta_{i}\;$ and $\;N\;$ has prime index in $\;G,\;$ we conclude that $\;N\;$ is the stabilizer of $\;\Delta_{i}.$ ∎

Recall the representation $\rho:G\rightarrow A_{2p}$. Since $N$ is a normal subgroup of index $p$ in $G$, we may enumerate the right cosets $\Delta_{i}$ of $N$ in $G$ in such a way that we can identify the set $\Delta_{i}$ with the set $\{i,p+i\}$ and the action of $G$ on the $\Delta_{i}$ corresponds to the permutation (right-)action of the group $A_{2p}$ on the set $\{1,\dots,2p\}$. Moreover, fixing a branch point, we may enumerate its branches in such a way that the local monodromy around this point is given by the cycle

[TABLE]

Lemma 2.5**.**

[TABLE]

Proof.

Consider, for $i=1,\dots,p,\;$ the transposition $t_{i}:=(i,p+i)$. Certainly $t_{i}$ is not contained in $G$, since $G\subset A_{2p}$. However, we have

[TABLE]

since it stabilizes each set $\{(i,p+i)\}$ and so is in $N$ by Lemma 2.4 and the identifications. Moreover,

[TABLE]

which gives

[TABLE]

Since the cycles $s_{i}$ pairwise commute, and clearly there is no non-trivial relation between the cycles $s_{1},s_{2},\dots s_{p-1}$, this implies

[TABLE]

Since a non-trivial element of $\;H_{1}\cap H_{2}\cap...\cap H_{p-1}\;$ would be the transposition exchanging the two points of $\;\Delta_{p}\;$, which is not in $\;A_{2p}\;$, we have $\;H_{1}\cap H_{2}\cap...\cap H_{p-1}=\{1\}.\;$ Consider the group homomorphism

[TABLE]

defined by $\;\Phi^{\prime}(n)=(H_{1}n,H_{2}n,\ldots,H_{p-1}n).\;$ Since $\;\ker(\Phi^{\prime})=\displaystyle\bigcap_{i=1}^{p-1}H_{i}=\{1\},\;$ we have $\;N\cong N/\ker{\Phi^{\prime}}\lesssim({\mathbb{Z}}_{2})^{p-1}.\;$ Hence $\;N\cong({\mathbb{Z}}_{2})^{p-1}.\;$ ∎

Theorem 2.6**.**

Let $X\rightarrow\mathbb{P}^{1}$ be a cyclic covering of degree an odd prime $p$, and $Y\rightarrow X$ be an étale double covering. Let $Z\rightarrow\mathbb{P}^{1}$ be the Galois closure of the composition $Y\rightarrow\mathbb{P}^{1}$ with Galois group $G$. With the notation of above, if $P$ denotes the subgroup of $G$ generated by the cycle $\sigma$, then $G$ is the semi-direct product

[TABLE]

Proof.

Since $\;N\;$ is a normal subgroup of index $\;p\;$ in $\;G\;$ and $\;|N|=2^{p-1}\;$ we have $\;G=N\rtimes P.$ ∎

A presentation of $G$ is given as

[TABLE]

2.2. The subcovers of $Z$

Let $p$ denote an odd prime. According to a well-known result of elementary number theory, the number

[TABLE]

is an integer. The abelian group $\;N\;$ has exactly $\;m\;$ conjugacy classes of maximal subgroups with respect to the action of $\;P.\;$ For each $\;1\leq j\leq m\;$ consider $\;R_{j}\;$ a representative of the corresponding conjugacy class of maximal subgroups of $\;N.\;$ Here $\;R_{1}=H.\;$

To each subgroup $\;R_{j}\;$ corresponds a double covering of $X$. Let

[TABLE]

denote the corresponding curves. In particular, $Y_{1}=Y$.

Denoting moreover $T:=Z/P$, we have the following diagram

[TABLE]

Lemma 2.7**.**

The map $Z\rightarrow X$ of degree $2^{p-1}$ is étale. In particular, all covers $Y_{i}\rightarrow X$ of the above diagram are étale.

Proof.

This follows immediately from Lemma 2.1 ∎

Proposition 2.8**.**

Let $\beta$ denote the number of branch values of $X\rightarrow\mathbb{P}^{1}$, with $\beta\geq 3$. Then the genera of the curves in the above diagram are:

• •

$g(X)=\frac{p-1}{2}(\beta-2)$;

• •

$g(Y_{i})=(p-1)(\beta-2)-1$;

• •

$g(Z)=2^{p-2}(p-1)\beta-(p2^{p-1}-1)$;

• •

$g(T)=\frac{2^{p-1}-1}{p}(\frac{p-1}{2}\beta-p)$.

Proof.

The first 3 assertions are obvious, since $X\rightarrow\mathbb{P}^{1}$ is totally ramified and $Z\rightarrow X$ is étale. For the last assertion note that over each branch value of $T\rightarrow\mathbb{P}^{1}$ there are $m=\frac{2^{p-1}-1}{p}$ branch points of index $p-1$ and one point étale over $\mathbb{P}^{1}$. So the Hurwitz formula gives the assertion. ∎

As an immediate consequence we obtain the following result.

Corollary 2.9**.**

If $P(Y_{i}/X)$ denotes the Prym variety of $Y_{i}/X$, we have

[TABLE]

This suggests that there is a relation between the Prym varieties $P(Y_{i}/X)$ and the Jacobian $JT$. The aim of this paper is to study the relation.

2.3. The rational representations of $G$

We follow [9, Section 8.2] to determine the irreducible representations of a semidirect product $G=N\rtimes P.\;$ Let $\widehat{N}$ be the character group of $N$. The group $P$ acts on $\widehat{N}$ in the usual way. The stabilizer in $\;P\;$ of the trivial character $\chi_{0}$ of $\;N\;$ is $P$ itself, whereas the stabilizer of any non-trivial complex irreducible character of $\;N\;$ consists of $\;\{1\}$ only. Hence there are exactly $1+m$ orbits of the action of $P$ on $\widehat{N}$, with $m$ as in (2.5). Let $\chi_{0},\chi_{1},\dots,\chi_{m}$ be a system of representatives of these orbits, $\;\rho_{0},\dots,\rho_{p-1}$ ($\rho_{0}$ the trivial representation) the irreducible representations of the cyclic group $P\;$ and $\;\eta\;$ the trivial character of $\;\{1\}.$

According to [9, Proposition 25]

[TABLE]

is the set of all complex irreducible representations of $G.$ The next result follows immediately

Corollary 2.10**.**

The rational irreducible representations of $G$ are exactly the trivial representation $\rho_{0}=\chi_{0}\otimes\rho_{0}$, the representations $\theta_{i}=\operatorname{{Ind}}_{N}^{G}(\chi_{i}\otimes\eta)\;$ of degree $\;p$ for $i=1,\dots,m$, and the representation $\psi:=(\chi_{0}\otimes\rho_{1})\oplus\dots\oplus(\chi_{0}\otimes\rho_{p-1})$ of degree $p-1$.

According to [1, Proposition 13.6.1] the rational irreducible representations of $G$ correspond canonically and bijectively to a set of $G$-stable abelian subvarieties of the Jacobian $JZ$ of $Z$ such that the addition map is an isogeny. If the abelian subvariety of $JZ$ corresponding to the rational irreducible representation $\rho$ of $G$ is denoted by $J_{\rho}$, the isotypical decomposition of $JZ$ is the isogeny given by the addition map

[TABLE]

Furthermore, according to [1, Proposition 13.6.2] and [2], for each rational irreducible representation $\rho$ of $G$ there exist abelian subvarities $B_{\rho}$ of $J_{\rho}$ such that $B_{\rho}^{n_{\rho}}$ is isogenous to $J_{\rho}$, with

[TABLE]

where $V_{\rho}$ is a complex irreducible representation of $G$ Galois associated to $\rho$ and $m_{\rho}$ is the Schur index of $V_{\rho}$.

The subvarieties $B_{\rho}$ are, in general, determined only up to isogeny, with $B_{\rho_{0}}=J_{\rho_{0}}=J(Z/G)$. In our case, we have $\;m_{\rho_{0}}=m_{\psi}=m_{\theta_{j}}=1,\;\;\dim(V_{0})=\dim(V_{\psi})=1\;$ and $\;\dim(V_{\theta_{j}})=p,\;$ hence

[TABLE]

where $\sim$ denotes isogeny.

Furthermore, it follows from [2, Corollary 5.6] that

[TABLE]

Therefore the group algebra decomposition of $JZ$ is given by

[TABLE]

3. The isogeny $\alpha$

Let the notation be as in Section 1 and for $i=1,\dots,m$ denote

[TABLE]

the maps of diagram (2.6), so that $\nu_{i}^{*}:JY_{i}\rightarrow JZ$ and $\operatorname{{Nm}}\mu:JZ\rightarrow JT$ are the induced homomorphisms of the corresponding Jacobians. Then the addition map gives a homomorphism

[TABLE]

According to Corollary 2.9, $\displaystyle\prod_{i=1}^{m}P(Y_{i}/X)$ and $JT$ are of the same dimension. The aim of this section is the proof of the following theorem.

Theorem 3.1**.**

$\alpha:\displaystyle\prod_{i=1}^{m}P(Y_{i}/X)\rightarrow JT$* is an isogeny with kernel contained in the $2^{p-2}$-division points.*

For this we use the following result (for the proof see [8, Corollary 2.7]).

Proposition 3.2**.**

Let $f:Z\rightarrow X:=Z/N$ be a Galois cover of smooth projective curves with Galois group $N$ and $H\subset G$ a subgroup. Denote by $\nu:Z\rightarrow Y:=Z/H$ and $\varphi:Y\rightarrow X$ the corresponding covers. If $\{g_{1},\dots,g_{r}\}$ is a complete set of representatives of $G/H$, then we have

[TABLE]

Now denote for $i=1,\dots,m$,

[TABLE]

and let

[TABLE]

Recall from (2.4) that $G=N\rtimes P$ with

[TABLE]

with $s_{i}$ and $\sigma$ as in Section 1. The group $P$ acts by conjugation on the elements of $N$ by

[TABLE]

Recall furthermore that $R_{i}$ is the subgroup of $N$ giving the cover $Y_{i}\rightarrow X$. Then it is easy to see that we have the following commutative diagram

[TABLE]

with $\beta=(\operatorname{{Nm}}\nu_{1},\operatorname{{Nm}}\nu_{2},\dots,\operatorname{{Nm}}\nu_{m})$.

For $i=1,\dots,m$ consider the following subdiagram

[TABLE]

with $\alpha_{i}:=\operatorname{{Nm}}\mu\circ\nu_{i}^{*},\;C_{i}:=\operatorname{{Nm}}\mu(A_{i})$ and $B_{i}:=\mu^{*}(C_{i})$.

Proposition 3.3**.**

For $i=1,\dots,m$ the map $\operatorname{{Nm}}\nu_{i}\circ\mu^{*}\circ\alpha_{i}:P(Y_{i}/X)\rightarrow P(Y_{i}/X)$ is multiplication by $2^{p-2}$.

Proof.

Since $\nu_{i}^{*}:P(Y_{i}/X)\rightarrow A_{i}$ is an isogeny, it suffices to show that the composition

[TABLE]

is multiplication by $2^{p-2}$.

Now from Proposition 3.2 we deduce

[TABLE]

since any $n\in N\setminus R_{i}$ induces the non-trivial involution of $Y_{i}/X$ and $A_{i}$ is the image of $P(Y_{i}/X)$.

Now for any $z\in A_{i}$,

[TABLE]

By equation (3.5) we have

[TABLE]

and for $k=1,\dots,p-1$,

[TABLE]

since $R_{i}\neq\sigma^{-k}R_{i}\sigma^{k}$ and considering that half of the elements of the subgroup $\sigma^{k}R_{i}\sigma^{k}$ belong to $R_{i}$, hence fix $z$, and the other half belongs to $N\setminus R_{i}$ and hence sends $z$ to $-z$. Together this completes the proof of the proposition. ∎

Proof of Theorem 3.1.

Since

[TABLE]

Proposition 3.2 implies that $\beta\circ\mu^{*}\circ\alpha$ is multiplication by $2^{p-2}$. In particular $\alpha$ has finite kernel. But according to Corollary 2.9, $\displaystyle\prod_{i=1}^{m}P(Y_{i}/X)$ and $JT$ have the same dimension. So $\alpha$ is an isogeny. ∎

Corollary 3.4**.**

Given any positive integer $N$, there exist smooth projective curves $Y$ whose Jacobian is isogenous to the product of $m\geq N$ principally polarized abelian varieties of the same dimension.

Proof.

Choose a prime $p$ such that $\frac{1}{p}(2^{p-1}-1)\geq N$. This is equivalent to $2^{p-1}>pN$. Hence there are infinitely many primes with this property. According to Theorem 3.1, the Jacobian $JT$ has the property of the corollary. ∎

We thank the referee for suggesting the following remark and Elham Izadi for the contents of it.

Remark 3.5**.**

There is a slight relation of Corollary 3.4 and a question of Ekedahl and Serre, whether for any positive integer $g$ there is a smooth curve of genus $g$ whose Jacobian is isogenous to a product of elliptic curves. Izadi showed in [4] that, if there is a complete subvariety of codimension $g$ in the moduli space, then there exist smooth curves of genus $g$ whose Jacobian is isogenous to the product of elliptic curves. As was later proved by Keel and Sadun however in [5], there are no such subvarieties in characteristic 0.

4. The case $p=3$

In this case we have $m=1$, so let $Y_{1}=:Y$, $\nu_{1}=:\nu$ and $A_{1}=:A$. Moreover, the subgroup $N$ is the Klein group of order $4$. Diagram (2.6) simplifies to

[TABLE]

Theorem 4.1**.**

The map $\alpha=\nu^{*}\circ\operatorname{{Nm}}\mu:P(Y/X)\rightarrow JT$ is an isogeny with kernel the group $P(Y/X)[2]$ of all two-division points.

Proof.

From Theorem 3.1 we know that $\ker\alpha\subseteq P(Y/X)[2]$. On the other hand, $\mu^{*}$ is injective, since $\mu:Z\rightarrow T$ is ramified. Hence from diagram (3.4) we have $\ker(\operatorname{{Nm}}\mu|_{A})=\ker(1+\sigma+\sigma^{2})|_{A})$. So we get

[TABLE]

Let $\gamma:Y\rightarrow X$ denote the double covering and $\epsilon:Z\rightarrow X$ the composition

[TABLE]

Since $N$ is a normal subgroup of $G$, the automorphism $\sigma$ descends to an automorphism $\overline{\sigma}:X\rightarrow X$, also of order 3. This is the automorphism giving the cyclic covering $X\rightarrow\mathbb{P}^{1}$.

Suppose $\eta$ is the two-division point of $JX$ giving the double cover $\gamma$ and let $\eta^{\bot}$ be the subgroup of $JX[2]$ orthogonal with respect to the Weil form $e_{2\lambda}$ associated to twice the canonical polarization $\lambda$ of $JX$. Then from [6] we know that

[TABLE]

This gives

[TABLE]

But $JX=\ker(1+\overline{\sigma}+\overline{\sigma}^{2})$. In particular for all $x\in\eta^{\bot}$ we have $\epsilon^{*}(1+\overline{\sigma}+\overline{\sigma}^{2})(x)=0$. Together this implies $\ker\alpha=P(Y/X)[2]$. ∎

As an immediate consequence we get a version of the trigonal construction in the special case of an étale cover of a cyclic trigonal cover $X\rightarrow\mathbb{P}^{1}$.

Corollary 4.2**.**

Let the notation be as in Theorem 4.1. The isogeny $\alpha:P(Y/X)\rightarrow JT$ induces an isomorphism of principally polarized abelian varieties

[TABLE]

where $\;\widehat{}$ denotes the dual abelian variety.

Proof.

Let $\lambda_{P}$ denote the polarization on $P(Y/X)$ induced by the canonical polarization of $JY$. It is twice a principal polarization. According to Theorem 4.1, $\alpha$ has kernel $P(Y/X)[2]$ which coincides with the kernel of the polarization $\lambda_{P}$. Hence $\alpha$ factors as follows, with $\overline{\alpha}\;$ an isomorphism,

[TABLE]

It remains to show that $\overline{\alpha}$ respects the principal polarizations. If we denote by $\lambda_{1}$ the polarization of $\widehat{P(Y/X)}$ induced via $\overline{\alpha}$ from the canonical polarization $\lambda_{JT}$ of $JT$, we may complete diagram 4.2 to the following one.

[TABLE]

It now follows from the commutativity of this diagram that $\lambda_{1}$ is principal and that $\ker(\lambda_{1}\circ\lambda_{P})=P(Y/X)[2]$. Hence $\lambda_{1}$ is the canonical principal polarization on $\widehat{P(Y/X)}$ as claimed. ∎

5. Estimate of the kernel of $\alpha$ for odd $p$

We show that the same proof as in the last section gives for any odd prime $p$ a lower bound for the order of the kernel of the isogeny $\alpha:\displaystyle\prod_{i=1}^{m}P(Y_{i}/X)\rightarrow JT$. We have the following result.

Proposition 5.1**.**

With the notation of above we have for any odd prime $p$,

[TABLE]

Furthermore, for $p>3$, $\operatorname{Ker}\alpha$ cannot be equal to $\displaystyle\prod_{i=1}^{m}P(Y_{i}/X)[2]$.

Proof.

For the first assertion it suffices to see that $\operatorname{Ker}\alpha_{i}$ contains $P(Y_{i}/X)[2]$. But since $\mu^{*}$ is injective, $\mu$ being ramified of prime degree, it follows from diagram (3.4) and Theorem 3.1 that

[TABLE]

Hence for the proof of the first assertion is suffices to show that for any $z\in P(Y_{i}/X)[2]$ we have

[TABLE]

This follows with the same proof as in the proof of Theorem 4.1 for $p=3$.

Finally, if we had $\operatorname{Ker}\alpha=\displaystyle\prod_{i=1}^{m}P(Y_{i}/X)[2]$, the same proof as for Corollary 4.2 would provide an isomorphism of principally polarized abelian varieties $\displaystyle\prod_{i=1}^{m}\widehat{P(Y_{i}/X)}\simeq JT$. For $p>3$, i.e. $m>1,\;$ this contradicts the fact that the canonical polarization of $JT$ is irreducible. ∎

6. The case $p=2$

Let $Y\rightarrow X$ be an étale double covering of a double covering $X\rightarrow\mathbb{P}^{1}$. According to Corollary 2.2, the composition $Y\rightarrow\mathbb{P}^{1}$ is Galois, with Galois group the Klein group

[TABLE]

Denoting $Y_{r}:=Y/\langle r\rangle$ and similarly $Y_{s}$ and $Y_{rs}$, we have the following diagram of double coverings,

[TABLE]

We assume that $\nu_{s}$ is étale and $Y_{s}\rightarrow\mathbb{P}^{1}$ is ramified over $2\beta$ points of $\mathbb{P}_{1}$, with $\beta\geq 3$ (so that $\dim P(Y/Y_{s})>0$). Each branch point of $Y_{s}\rightarrow\mathbb{P}^{1}$ is a branch point of exactly one of the maps $Y_{r}\rightarrow\mathbb{P}^{1}$ and $Y_{rs}\rightarrow\mathbb{P}^{1}$. So if $2\beta_{r}$ respectively $2\beta_{rs}$ denote the number of branch points of $Y_{r}\rightarrow\mathbb{P}^{1}$ respectively $Y_{rs}\rightarrow\mathbb{P}^{1}$, we have

[TABLE]

The genera of the curves are:

[TABLE]

In particular, $\dim P(Y/Y_{s})=g(Y_{r})+g(Y_{rs})$.

Proposition 6.1**.**

The following map is an isogeny,

[TABLE]

with kernel consisting at most of two-division points.

Proof.

First we claim that $\operatorname{{Im}}(\alpha)\subset P(Y/Y_{s})$. Note first that the automorphism $s$ descends to an automorphism $\overline{s}$ of $Y_{r}$ and we have for any $x\in JY_{r}$

[TABLE]

where the last equation follows from Proposition 3.2. An analogous equation is valid for $\nu_{rs}^{*}$. So we have

[TABLE]

which implies the assertion.

It remains to show that $\ker\alpha$ consists of $2$-division points, since $g(Y_{r})+g(Y_{rs})=\dim P(Y/Y_{s})$. For this it suffices to show that the composed map

[TABLE]

is multiplication by 2. But $\operatorname{{Nm}}\nu_{r}\circ\nu_{r}^{*}=\deg\nu_{r}=2$ and the same is valid for $\nu_{rs}$. This completes the proof of the proposition. ∎

Proposition 6.1 implies

[TABLE]

Since $\nu_{r}$ and $\nu_{rs}$ are ramified, the homomorphisms $\nu_{r}^{*}$ and $\nu_{rs}^{*}$ are injective. Hence we get

[TABLE]

The following theorem is due to Mumford (see [6, page 356]).

Theorem 6.2**.**

Let the notation be as above. Then we have:

1. (i)

the map

[TABLE]

is an isomorphism; 2. (ii)

the isomorphism $\alpha$ respects the canonical principal polarizations.

Proof.

(i): According to (6.2) it suffices to show that the images of $JY_{r}[2]$ via $\nu_{r}^{*}$ and $JY_{rs}[2]$ via $\nu_{rs}^{*}$ in $JY[2]$ intersect only in $0\in JY$. Now, fixing a theta characteristic of JY, the 2-division points of $JY$ correspond in a natural way bijectively to the theta characteristics of $Y$. An analogous statement is valid for $JY_{r}$ and $JY_{rs}$. Using this, the assertion follows from the fact that the theta characteristics of $Y$ which are pullbacks from theta characteristics of $Y_{r}$ are disjoint from those which are are pullbacks from theta characteristics of $Y_{rs}$.

But this follows from the fact that, according to what we have said right after the diagram 6.1, the branch points $b_{1},\dots,b_{2\beta}$ of $Y_{s}\rightarrow\mathbb{P}^{1}$ can be enumerated in such a way that $b_{1},\dots,b_{2\beta_{r}}$ are the branch points of $Y_{r}\rightarrow\mathbb{P}^{1}$ and that $b_{2\beta_{r}+1},\dots b_{2\beta}$ are the branch points of $Y_{rs}\rightarrow\mathbb{P}^{1}$. For this, note only that all theta characteristics of a hyperelliptic curve are sums of ramification points of the hyperelliptic covering (see for example [7, Section III, 5]).

(ii): From the proof of Proposition 6.1 we know that the composition

[TABLE]

with $\gamma:=(\operatorname{{Nm}}\nu_{r},\operatorname{{Nm}}\nu_{rs})$, is multiplication by 2. If $\theta:=\theta_{JY\times JY_{rs}}$ denotes the canonical polarization of $JY_{r}\times JY_{rs}$, this implies that $(\gamma\circ\alpha)^{-1}(\theta)=4\theta$ (see [1, Corollary 2.3.6]). Since $\alpha$ is an isomorphism, it follows that $\gamma^{-1}(\theta)$ is the fourth power of a principal polarization, say $\gamma^{-1}(\theta)=4\Xi$.

Now $\alpha:JY_{r}\times JY_{rs}\rightarrow P(Y/Y_{s})$ is an isomorphism. The canonical principal polarization of $JY$ restricts to $\nu_{r}^{*}(JY_{r})$ as twice a principal one, and to $\nu_{rs}^{*}(JY_{rs})$ as twice a principal one, the restriction to $P(Y/Y_{s})$ is $2\Xi$. Then (ii) follows from the fact that the map $\alpha$ is $G$-equivariant, since both varieties are the eigen-subvarieties of $-1$ for the same element of $G$, namely $\sigma$. ∎