Decomposable Jacobians

TL;DR

This paper presents examples of smooth projective curves with Jacobians that are isogenous to a product of many Jacobians, highlighting complex decompositions in algebraic geometry.

Contribution

It provides explicit examples of curves with highly decomposable Jacobians, expanding understanding of Jacobian structures in algebraic geometry.

Findings

Constructed curves with Jacobians isogenous to large products

Demonstrated the existence of complex Jacobian decompositions

Extended known examples of Jacobian decompositions

Abstract

In this paper we give examples of smooth projective curves whose Jacobians are isogenus to a product of an arbitrarily high number of Jacobians

Equations

$$G:=N⋊B:=\left(⟨a_1,a_2⟩\times ⟨a_3,a_4⟩\times \cdots \times ⟨a_{2n-1},a_{2n}⟩\right)⋊⟨b⟩$$

$$|G|=2^{2n}3.$$

$$Q:=N\cap R⊴G$$

$$(n_1{)}^b:=b{n}_1{b}^{-1}=n_2\notin Q\text{mbox}and(n_1{)}^{b^2}=n_1{n}_2\notin Q.$$

$$2=|N:Q|=\frac{|⟨n_1,n_2⟩|}{|⟨n_1,n_2⟩\cap Q|}=4,$$

$$m:=\frac{2^{2n}-1}{3}$$

$$U=\ker (\psi )\cap \ker ({\psi }^b)=\ker (\psi )\cap \ker ({\psi }^b)\cap \ker (\psi {\psi }^b)={\cap }_{i=0}^2\ker ({\psi }^i).$$

$$\ker (\theta )=\bigcap _{g\in G}\ker ({\psi }^g)=\bigcap _{n\in N,i=0,1,2}\ker ({\psi }^{n{b}^i})=\bigcap _{i=0}^2\ker ({\psi }^{b^i})=U.$$

$$|\{\theta \in \mathrm{Irr}(G)/\mathrm{deg}(\theta )=3\}|=m=\frac{2^{2n}-1}{3}$$

$$U=M\cap N⟷M=UB⟷U=\ker (\theta ).$$

$${\rho }_M={\chi }_0\oplus \theta .$$

$$p_H:=\frac{1}{|H|}\sum _{h\in H}h$$

$${\mathcal{H}}_H:=p_H\mathbb{Q}[G]p_H=\mathbb{Q}[H\backslash G/H]$$

$$\mathbb{Q}[G]\to {\mathrm{End}}_{\mathbb{Q}}(A)$$

$$\mathrm{Im}(\alpha ):=\mathrm{Im}(n\alpha )\subset A$$

$$A_H:=\mathrm{Im}(p_H).$$

$$e_{W_i}=\frac{{\dim }_{\mathbb{C}}(V_i)}{|G|}\sum _{g\in G}{\mathrm{tr}}_{K_i/\mathbb{Q}}({\chi }_{V_i}(g^{-1}))g.$$

$${\rho }_H\simeq \sum _{i=1}^r{a}_i{W}_i,$$

$${\dim }_{\mathbb{Q}}({\tilde{W}}_i)={\dim }_{\mathbb{Q}}(W_i^H)=[L_i:\mathbb{Q}]{\dim }_{\mathbb{C}}(V_i^H).$$

$$f_{H,{\tilde{W}}_i}:=p_H{e}_{W_i}=e_{W_i}{p}_H.$$

$$p_H=\sum _{i=1}^u{f}_{H,{\tilde{W}}_i}.$$

$$A_{H,{\tilde{W}}_i}:=\mathrm{Im}(f_{H,{\tilde{W}}_i}),$$

$$+:A_{H,{\tilde{W}}_1}\times A_{H,{\tilde{W}}_2}\times \cdots \times A_{H,{\tilde{W}}_u}\to A_H.$$

$$G=H_1\cup H_2\cup \cdots \cup H_s$$

$$q_j:=\frac{1}{|H|}\sum _{h_j\in H_j}{h}_j$$

$$A_{H,{\tilde{W}}_i}=\{z\in A|q_j(z)={\chi }_{V_i}(q_j)z\text{mbox}forj=1,\dots ,s{\}}_0$$

$$\frac{2(g-1)}{|G|}=2(\gamma -1)+\sum _{j=1}^r\left(1-\frac{1}{m_j}\right).$$

$$G=⟨a_1,\cdots ,a_{\gamma },b_1,\cdots ,b_{\gamma },c_1,\cdots ,c_r/\prod _{i=1}^{\gamma }[a_i,b_i]\prod _{j=1}^r{c}_j=1,|c_j|=m_j\text{mbox}forj=1,...,r⟩$$

$$G:=N⋊B:=\left(⟨a_1,a_2⟩\times ⟨a_3,a_4⟩\times \cdots \times ⟨a_{2n-1},a_{2n}⟩\right)⋊⟨b⟩$$

$$X_H:=X/H.$$

Full text

The authors were partially supported by Grants Fondecyt 1190991 and CONICYT PAI Atracción de Capital Humano Avanzado del Extranjero PAI80160004

1. Introduction

In [5], Ekedahl and Serre gave examples of curves $X$ whose Jacobian is completely decomposable, i.e. isogenous to a product of elliptic curves. The highest genus of their examples is $g(X)=1297$. They asked among other things whether the genus of a curve with a completely decomposable Jacobian is bounded above. Although in the meantime many other examples of such Jacobians have been given (see e.g. [6]), no example of genus bigger than 1297 seems to be known.

In this paper we consider an easier question, namely: can a Jacobian be isogenous to the product of arbitrary many Jacobians of the same genus (not necessarily equal to one)? In [4] we gave examples of Jacobians which are isogenous to an arbitrary number of Prym varieties of the same dimension. The main result of this paper is the following theorem (see Corollary 5.3).

Theorem 1.1**.**

Given any positive integer $N$, there exists a smooth projective curve $X$ whose Jacobian is isogenous to the product of $m\geq N$ Jacobian varieties of the same dimension.

To be more precise, in Corollary 5.2, for any positive integer $n\geq 2$ and any integer $t\geq 2n$ we give examples of curves of genus $m(t-3)$ whose Jacobians are isogenous to the product of $m=\frac{1}{3}(2^{2n}-1)$ Jacobians of curves of genus $\;t-3.$ The idea is to consider curves $X$ of genus $2^{2n}(t-3)+1$ with an action of the group $G=N\rtimes B\simeq\mathbb{Z}_{2}^{2n}\rtimes\mathbb{Z}_{3}$ of signature $(0;\underbrace{3,\dots,3}_{t})$. Then the quotient curves $X_{B}=X/B$ will have the asserted properties.

The $m$ Jacobians are Jacobians of subcovers of the curve $X,$ of genus $t-3,$ and they are in fact isomorphic to Prym varieties of étale double covers of other subcovers. Using these isomorphisms gives a second proof of the theorem, which actually gives a bit more, namely an estimate of the degree of the isogeny.

In Section 2 we investigate the representations of the group $G$. In Section 3 we recall some results on curves with $G$-action. In Section 4 we study the diagram of subcovers of the curve $X$. Section 5 contains the first proof of the above mentioned theorem. In Section 6 we give the proof using the Prym varieties and the trigonal construction. Finally in the last section we compute the group algebra decomposition of the Jacobian $JX$ of the curve $X$.

2. The group $G$

Consider the group $V_{4}\rtimes B\simeq A_{4}$ the alternating group of degree 4 with $V_{4}=\langle a_{1},a_{2}\rangle$ the Klein group and $B=\langle b\rangle\simeq\mathbb{Z}_{3}$, and for any positive integer $n$ let

[TABLE]

where the subgroups $\langle a_{j},a_{j+1}\rangle\simeq V_{4}$ for each $j$ and $\langle a_{j},a_{j+1}\rangle\rtimes\langle b\rangle\simeq A_{4}$ for $j=1,3,\dots 2n-1$. Obviously $G$ is of order

[TABLE]

Also $\;B\;$ is a $3$-Sylow subgroup of $\;G\;$ and $\;B=N_{G}(B)=C_{G}(B).$

In this section we will show that all subgroups $M$ of $G$ of index $4$ are maximal in $G$ (there is no subgroup strictly between it and $G$); if we consider only such $M$ containing $B$ (there is exactly one in each conjugacy class of subgroups of index four), they correspond bijectively to the subgroups $U$ of index four in $N$ that are normal in $G$, and also to the set of irreducible representations of $G$ of degree three. An example of such a subgroup is $\;M_{1}=\left(\langle a_{3},a_{4}\rangle\times\cdots\times\langle a_{2n-1},a_{2n}\rangle\right)\rtimes\langle b\rangle$ with $U_{1}=\langle a_{3},a_{4}\rangle\times\cdots\times\langle a_{2n-1},a_{2n}\rangle$. We need some preliminaries.

Lemma 2.1**.**

$\;G\;$* has no subgroups of index $\;2.\;$*

Proof.

Suppose $\;R\leq G\;$ such that $\;|G:R|=2.\;$ Then $\;R\unlhd G\;$ and by the Sylow Theorem $\;B\leq R.\;$ In this way, we have $\;G=NB=NR.\;$ This implies

[TABLE]

and $\;\;|N:Q|=2.\;$

Hence $\;N=\langle n_{1}\rangle Q\;$ for some $n_{1}\in N$. Since $\;Q\unlhd G,\;$ we have

[TABLE]

The last equation follows from the fact that $\langle n_{1},n_{2}\rangle\simeq V_{4}$ and $(n_{1})^{b^{2}}\neq n_{1},n_{2}$. Then $\langle n_{1},n_{2}\rangle\cap Q=\{1\}$ and hence

[TABLE]

a contradiction. ∎

Now let $\;M\leq G\;$ be any subgroup such that $\;|G:M|=4.\;$ As we saw above, there are such subgroups. According to Lemma 2.1, $M$ is maximal in $G$. Moreover, we have,

Lemma 2.2**.**

$\;G=NM\;\;$* ; $\;U=N\cap M\unlhd G\;$ and $\;|N:U|=4.$*

Proof.

Since $\;M\;$ is a maximal subgroup of $\;G\;$ and $\;N\unlhd G\;$ we have $\;G=NM.\;$

Since $\;N\unlhd G\;$ we have that $\;U=N\cap M\unlhd M.\;$ Also, $\;U\unlhd N\;$ since $\;N\;$ is abelian. Therefore, $\;U\unlhd NM=G.\;\;$ Also $\;4=|G:M|=|N:U|.$ ∎

According to a result of elementary number theory, the number

[TABLE]

is an integer for all positive integers $n$.

Lemma 2.3**.**

There are complex irreducible representations $\theta_{1},\dots,\theta_{m}$ of degree $3$ and complex irreducible representations $\chi_{0},\chi_{1},\chi_{2}$ of degree $1$ of $G$ ($\chi_{0}$ the trivial character). These are all the irreducible complex representations of $G$.

Proof.

For any irreducible character $\psi$ of $N$ and any element $g\in G$ we denote by $\psi^{g}$ the conjugate character of $N$ defined by $\psi^{g}(n):=\psi(gng^{-1})$, The stabilizer of the trivial character $\psi_{0}$ of $N$ is the group $G$, whereas the stabilizer of any non-trivial character of $N$ is trivial. Hence there are exactly $1+m$ orbits for the action of $G$ on the set of all irreducible characters of $N$. Let $\psi_{0},\psi_{1},\dots,\psi_{m}$ be a system of representatives of these orbits and $\;\nu_{0},\nu_{1},\nu_{2}\;$ the irreducible characters of $B.\;$ Then, according to [9, Proposition 25] the irreducible representations of $\;G\;$ are $\;\chi_{0}=\psi_{0}\otimes\nu_{0},\;\chi_{1}=\psi_{0}\otimes\nu_{1},\;\chi_{2}=\psi_{0}\otimes\nu_{2}\;$ and the induced representation $\;\theta_{i}=(\psi_{i}\otimes\nu_{0})^{G},\;$ for each $\;i=1,\dots m.$ ∎

This implies immediately,

Corollary 2.4**.**

The irreducible rational representations of $G$ are exactly the trivial representation $\chi_{0}$, the representations $\theta_{j}$ of degree $3$ for $j=1,\dots,m$, and the representation $\psi:=\chi_{1}\oplus\chi_{2}$ of degree $2$.

Lemma 2.5**.**

Let $\psi$ be a non-trivial irreducible character of $N$ and $\theta:=(\psi\otimes\nu_{0})^{G}$ the corresponding irreducible $3$-dimensional representation of $G$. Then

1. (i)

$|N:\ker(\psi)|=2;\;$** 2. (ii)

$\;U:=\ker(\psi)\cap\ker(\psi^{b})=\ker(\psi)\cap\ker(\psi\psi^{b})\;\;$* ; $\;|N:U|=4\;\;$ and $\;\;\ker(\theta)=U\unlhd G.$*

Proof.

(i): Since $\;N/\ker(\psi)\;$ is isomorphic to a finite cyclic subgroup of $\;{\mathbb{C}}^{\ast}\;$ and $\;N\cong({\mathbb{Z}}_{2})^{2n}\;$ we have $\;|N:\ker(\psi)|=2.\;$

(ii): Since $\;|N:\ker(\psi)|=2\;$ and $\;\psi\neq\psi^{b}\;$ we have $\;N=\ker(\psi)\ker(\psi^{b}).$

Hence $\;2=|N:\ker(\psi)|=|\ker(\psi^{b}):\ker(\psi)\cap\ker(\psi^{b})|.\;$ Therefore $\;|N:U|=4.$

We have $\;\ker(\psi)\cap\ker(\psi^{b})\leq\ker(\psi\psi^{b}).\;$ So

[TABLE]

It is well known that $\ker(\theta)=\bigcap_{g\in G}\ker(\psi^{g})$. Hence

[TABLE]

∎

Conversely, we have

Lemma 2.6**.**

Let $\;U\unlhd G\;$ such that $\;|N:U|=4.\;$ Then

1. (i)

There is a non-trivial character$\;\psi\;$* of $\;N\;$ such that $\;\theta=(\psi\otimes\nu_{0})^{G}\;$ is an irreducible representation of $G$ with $\;U=\ker(\theta);$* 2. (ii)

$\;M=UB\leq G\;$* and $\;|G:M|=4.$*

Proof.

(i): We have that $\;N/U\;$ is isomorphic to the Klein group of order $\;4.\;$ Consider $\;L\leq N\;$ such that $\;U\leq L\;\;$ and $\;|N:L|=2.\;$ Then $\;U\leq L^{b^{i}}\;$ since $\;U\unlhd G.$

There is a non-trivial character $\;\psi\;$ of $\;N\;$ such that $\;L=\ker(\psi).\;$ In this way, for the induced representation $\;\theta=(\psi\otimes\nu_{0})^{G},\;$ we have by the previous lemma that $\;U=\ker(\theta).$

(ii): Since $\;U\unlhd G\;$ with $[N:U]=4$, we have $\;M=UB\leq G$ and $|G:M|=4$. ∎

Combining these lemmas we get

Proposition 2.7**.**

1. (i)

There are canonical bijections between the following sets

• •

$\{U\unlhd G\;\;/\;\;|N:U|=4\};$**

• •

$\{M\leq G\;\;/\;\;|G:M|=4\ \textup{ and }\ B\leq M\};$**

• •

$\{\theta\in\operatorname{{Irr}}(G)\;\;/\;\;\deg(\theta)=3\}.$** 2. (ii)

[TABLE]

Proof.

(i): The bijections are given by

[TABLE]

If $\;\rho_{M}\;$ denotes the representation of $G$ induced by the trivial representation of $\;M,\;$ then for each such $M$,

[TABLE]

So $\theta$ is of degree 3.

(ii): There are $2^{2n}-1$ non-trivial irreducible characters $\psi$ of $N$. The group $B$ acts non-trivially on them. So Lemma 2.6 and part (i) of the proposition imply the assertion. ∎

Remark 2.8**.**

Let $\;U\unlhd G\;$ such that $\;|N:U|=4,\;$ $\;M=UB\;$ (the corresponding maximal subgroup), and $U=\ker(\theta)$ as in the Proposition 2.7.

Consider $\;L\leq N\;$ such that $\;U\leq L$ and $\;|N:L|=2.$ Observe that for each $U$ there are three such $L$, forming a conjugacy class of subgroups of $G$. Also, consider $\;\psi$, the irreducible character of $\;N\;$ with $\;\ker(\psi)=L.\;$ Then

• •

$\;U\leq\ker(\psi^{b^{i}})=:L_{i}\;$ for $\;i=0,1,2.\;$ So $L_{0}=L$;

• •

$\;\theta=(\psi^{b^{i}}\otimes\nu_{0})^{G}\;$ for $\;i=0,1,2$;

• •

$\;\rho_{N}=\chi_{0}\oplus\chi_{1}\oplus\chi_{2}$;

• •

$\;\rho_{L_{i}}=\ \rho_{N}\oplus\theta\;$ for $\;i=0,1,2.$;

• •

$\;\rho_{M}=\chi_{0}\oplus\theta.$

3. Action of Hecke algebras of an abelian variety

Let $G$ be any finite group acting on an abelian variety $A$ over the field of complex numbers and let $H\leq G$ be a subgroup. In this section we recall a result of [4] together with the notation needed for it.

The element

[TABLE]

is an idempotent of the group algebra $\mathbb{Q}[G]$. The Hecke algebra for $H$ in $G$ is defined to be the subalgebra

[TABLE]

of $\mathbb{Q}[G]$. The action of $G$ on $A$ induces an algebra homomorphism

[TABLE]

in a natural way. We denote the elements of $\mathbb{Q}[G]$ and their images by the same letter. For any element $\alpha\in\mathbb{Q}[G]$ we define its image in $A$ by

[TABLE]

where $n$ is any positive integer such that $n\alpha$ is in $\operatorname{{End}}(A)$. It is an abelian subvariety which does not depend on the chosen integer $n$. Consider the abelian subvariety of $A$ given by

[TABLE]

Restricting (3.1) to $\mathcal{H}_{H}$ gives an algebra homomorphism $\mathcal{H}_{H}\rightarrow\operatorname{{End}}_{\mathbb{Q}}(A_{H}).$

Let $\{W_{1},\dots,W_{r}\}$ denote the irreducible rational representations of $G$. To any $W_{i}$ there corresponds an irreducible complex representation $V_{i}$, uniquely determined up to an element of the Galois group of $K_{i}$ over $\mathbb{Q}$, where $K_{i}$ is the field obtained by adjoining to $\mathbb{Q}$ the values of the character $\chi_{V_{i}}$ of $V_{i}$.

To each $W_{i}$ we can associate a central idempotent $e_{W_{i}}$ of $\mathbb{Q}[G]$ by

[TABLE]

Let $\rho_{H}$ denote the representation of $G$ induced by the trivial representation of $H$. It decomposes as

[TABLE]

with $a_{i}=\frac{1}{s_{i}}\dim_{\mathbb{C}}(V_{i}^{H})$ and $s_{i}$ the Schur index of $V_{i}$. Renumbering if necessary, let $\{W_{1},\dots,W_{u}\}$ denote the set of all irreducible rational representations of $G$ such that $a_{i}\neq 0$. Then there is a bijection from this set to the set $\{\widetilde{W}_{1},\dots,\widetilde{W}_{u}\}$ of all irreducible rational representations of the algebra $\mathcal{H}_{H}$. An analogous statement holds for the complex irreducible representations of $G$ and of $\mathcal{H}_{H}$. Let $\widetilde{V}_{i}$ denote the representation of $\mathcal{H}_{H}$ associated to the complex irreducible representation $V_{i}$ of $G$ and Galois associated to $\widetilde{W}_{i}.$ The dimension of $\widetilde{W}_{i}$ is given by

[TABLE]

where $L_{i}$ denotes the field of definition of the representation $V_{i}$. Recall that $s_{i}=[L_{i}:K_{i}]$ is the Schur index of $V_{i}$.

For $i=1,\dots,u$ consider the central idempotents of $\mathcal{H}_{H}$ given by

[TABLE]

Then $p_{H}$ decomposes as

[TABLE]

Defining for $i=1,\dots,u$ the abelian subvarieties

[TABLE]

one obtains the following isogeny decomposition of $A_{H}$, given by the addition map

[TABLE]

It is uniquely determined by $H$ and the action of $G$ and called the isotypical decomposition of $A_{H}$. So in order to describe the abelian subvariety $A_{H}$, it suffices to describe the $A_{H,\widetilde{W}_{i}}$. This is done by [4, Theorem 4.3]. For it we need one more notation. Consider the decomposition of $G$ into double cosets of $H$ in $G$,

[TABLE]

with $H_{1}=H$. Then a basis for the Hecke algebra $\mathcal{H}_{H}$ is given by the elements

[TABLE]

for $j=1,\dots,s$. If $\chi_{V_{i}}$ denotes the character of the representation $V_{i}$, then [4, Theorem 4.3] says,

Theorem 3.1**.**

Suppose $\dim V_{i}^{H}=1$ and $K_{i}=\mathbb{Q}$. Then

[TABLE]

where the index [math] means the connected component containing [math].

4. The diagram of subcovers of $X$

We start this section recalling some basic properties on groups action on smooth projective curves. Let $\;X\;$ be a smooth projective curve of genus $\;g,\;G$ a finite group acting on $\;X\;$ and $\;X\rightarrow X_{G}\;$ be the quotient projection. This cover may be partially characterized by a vector of numbers $\;(\gamma;m_{1},\cdots,m_{r})\;$ where $\;\gamma\;$ is the genus of $\;X_{G},$ the integer $\;0\leq r\leq 2g+2\;$ is the number of branch points of the cover and the integers $\;m_{j}\;$ are the orders of the cyclic subgroups $\;G_{j}\;$ of $\;G\;$ which fix points on $\;X$. We call $(\gamma;m_{1},\cdots,m_{r})$ the branching data of $G$ on $\;X$. These numbers satisfy the Riemann-Hurwitz formula

[TABLE]

A $(2\gamma+r)-$tuple $\left(a_{1},\cdots,a_{\gamma},b_{1},\cdots,b_{\gamma},c_{1},\cdots,c_{r}\right)$ of elements of $\;G\;$ is called a *generating vector of type * $(\gamma;m_{1},\cdots,m_{r})$ if

[TABLE]

where $[a_{i},b_{i}]=a_{i}b_{i}a_{i}^{-1}b_{i}^{-1}.$

Riemann’s Existence Theorem gives us the following theorem (see [2, Proposition 2.1])

Theorem 4.1**.**

The group $G$ acts on a smooth projective curve $X\;$ of genus $\;g$ with branching data $(\gamma;m_{1},\cdots,m_{r})$ if and only if $G$ has a generating vector of type $(\gamma;m_{1},\cdots,m_{r})$ satisfying the Riemann-Hurwitz formula (4.1).

From now on let $G=N\rtimes B\simeq\mathbb{Z}_{2}^{2n}\rtimes\mathbb{Z}_{3}$ be the group of Section 2. Recall that

[TABLE]

where for each $j=1,3,\dots 2n-1$ the subgroups $\langle a_{j},a_{j+1}\rangle$ are isomorphic to Klein group and the subgroups $\langle a_{j},a_{j+1}\rangle\rtimes\langle b\rangle$ are isomorphic to the alternating group of degree four.

Remark 4.2**.**

For $\;n=1,\;$ we have that $\;G$ is isomorphic to the alternating group of degree four. It is easy to check that $G$ acts on a smooth projective curve $X$ of genus $\;g=4(t-3)+1\;$ with branching data $(0;\underbrace{3,\cdots,3}_{t})\;$ and $\;t\geq 4.$

In this case, if $\;P(X_{\langle a_{1}\rangle}\rightarrow X_{N})$ denotes the Prym variety of $X_{\langle a_{1}\rangle}\rightarrow X_{N}$ then, by the trigonal construction (see [7]), we have $P(X_{\langle a_{1}\rangle}\rightarrow X_{N})$ is isomorphic to $JX_{B}$ as principally polarized abelian variety.

Lemma 4.3**.**

For $\;n>1\;$ the group $G$ acts on a smooth projective curve $X$ of genus $\;g=2^{2n}(t-3)+1\;$ with branching data $(0;\underbrace{3,\cdots,3}_{t})\;$ for $\;t\geq 2n.$

Proof.

We illustrate by giving a generating vector for $\;t=2n.$

For $\;n=2,\;$ consider the elements $\;y_{1}=a_{1}b,\;y_{2}=b,\;y_{3}=a_{3}b^{2}\;$ and $\;y_{4}=b^{2}a_{4}a_{1}.\;$

Since $\;b^{2}a_{3}b=a_{4}\;$ we have $\;y_{1}y_{2}y_{3}y_{4}=a_{1}bba_{3}b^{2}b^{2}a_{4}a_{1}=a_{1}b^{2}a_{3}ba_{4}a_{1}=a_{1}a_{4}a_{4}a_{1}=1.\;$ Also, it is clear that $\;G=(\langle a_{1},a_{2}\rangle\times\langle a_{3},a_{4}\rangle)\rtimes\mathbb{Z}_{3}=\langle y_{1},y_{2},y_{3},y_{4}\rangle.\;$ In this way $\;(y_{1},y_{2},y_{3},y_{4})\;$ is a generating vector of type $\;(0;3,3,3,3)\;$ for $\;G.$

For $\;n=3,\;$ is it clear that $\;(y_{1}=a_{1}b,\;y_{2}=b,\;y_{3}=a_{3}b^{2},\;y_{4}=b^{2}a_{4}a_{1},\;y_{5}=a_{5}b,y_{6}=b^{2}a_{5})\;$ is a generating vector of type $\;(0;3,3,3,3,3,3)\;$ for $\;G=(\langle a_{1},a_{2}\rangle\times\langle a_{3},a_{4}\rangle\times\langle a_{5},a_{6}\rangle)\rtimes\mathbb{Z}_{3}.$

Now, for $\;n>3\;$ we can apply a similar procedure as the one described above in both cases $n$ even or odd.

∎

Let $X$ denote a smooth projective curve with an action of $G$ with signature $(0;\underbrace{3,\dots,3}_{t}).$ The action induces an action of $G$ on the Jacobian $JX$ of $X$. In this section we will study the subcovers of $X$ which will be important for us.

For any subgroup $H$ of $G$ let $X_{H}$ denote the quotient curve

[TABLE]

According to Proposition 2.7 the subgroup $N$ has exactly $m=\frac{2^{2n}-1}{3}$ subgroups of index $4\>$ which are normal in $G$. Let $U_{1},\dots,U_{m}$ be these subgroups and $\;M_{1},\dots,M_{m}\;$ denote the corresponding maximal subgroups of $\;G.$ For any $U_{j}$ we choose a subgroup $L_{j}\leq N$ containing $U_{j}$ with $[N:L_{j}]=2$. According to Remark 2.8, the conjugacy class of $L_{j}$ is uniquely determined. With these notations we have the following diagram of covers of curves.

[TABLE]

Lemma 4.4**.**

Let $X\rightarrow\mathbb{P}^{1}$ be Galois with group $G=N\rtimes B\simeq\mathbb{Z}_{2}^{2n}\rtimes\mathbb{Z}_{3}$ and action with signature $(0;\underbrace{3,\dots,3}_{t})$. Then

1. (i)

$X_{N}\rightarrow\mathbb{P}^{1}$* is totally ramified and $X\rightarrow X_{N}$ is étale;* 2. (ii)

Over each of the $t$ branch points $p_{i}$ of $X\rightarrow\mathbb{P}^{1}$ the map $X_{B}\rightarrow\mathbb{P}_{1}$ admits $m$ ramification points of ramification index $2$ and $1$ point which is étale over $\mathbb{P}^{1}$.

Proof.

Let $p\in\mathbb{P}^{1}$ be a branch point of $X\rightarrow\mathbb{P}^{1}$.

(i): Since $X_{N}\rightarrow\mathbb{P}^{1}$ is cyclic of degree 3, it is either étale over $p$ or totally ramified. Suppose it is étale. Since $p$ is a branch point of $X\rightarrow\mathbb{P}^{1}$, it is also a branch point of the Galois cover $X\rightarrow X_{N}$ of type $3$ (i.e. all points of $X$ over $p$ are of ramification degree 2 with respect to $X\rightarrow X_{N}$). This gives a contradiction, since $2^{2n}$ is not divisible by 3. This completes the proof of (i).

(ii): Let $p_{1},\dots,p_{t}$ denote the branch points of $X\rightarrow\mathbb{P}^{1}$ and $b_{i}\in X_{B}$ be a point in the fibre over $p_{i}$. Then either $b_{i}$ is of ramification degree 2 over $p_{i}$ or étale over $p_{i}$. In the first case the map $X\rightarrow X_{B}$ is étale over $b_{i}$ and in the second case totally ramified over $b_{i}$.

Let $r_{i}$ be the number of points in the fibre in $X_{B}$ over $p_{i}$ at which the map $X_{B}\rightarrow\mathbb{P}^{1}$ is ramified. According to the Riemann-Hurwitz (4.1) formula we get

[TABLE]

where the inequality follows from the inequalities $r_{i}\leq m$. So we have

[TABLE]

On the other hand, [8, Corollary 3.4] gives a method to compute the genus of $X_{B}.$ Let $\;G_{j}\cong{\mathbb{Z}}_{3}\;$ be the stabilizer of the branch point $p_{j}\;$ with $\;j=1,\dots t.\;$ Then

[TABLE]

Hence the equivalence (4.3) gives the assertion. ∎

From the Riemann-Hurwitz (4.1) formula we immediately get from Lemma 4.4,

Lemma 4.5**.**

Under these assumptions we have

• •

$g(X)=2^{2n}(t-3)+1$;

• •

$g(X_{N})=t-2$;

• •

$g(X_{L_{i}})=2t-5$;

• •

$g(X_{U_{i}})=4t-11$;

• •

$g(X_{M_{i}})=t-3$;

• •

$g(X_{B})=m(t-3)$.

Corollary 4.6**.**

If $P(X_{L_{i}}\rightarrow X_{N})$ denotes the Prym variety of $X_{L_{i}}\rightarrow X_{N}$, we have

[TABLE]

Proof.

$\dim P(X_{L_{i}}\rightarrow X_{N})=g(X_{L_{i}})-g(X_{N})$. So Lemma 4.5 implies the assertion. ∎

Corollary 4.6 suggests that there is a relation between these Pryms and $J(X_{B})$ which we investigate in the next section. The second equality is explained by the following proposition.

Proposition 4.7**.**

For $i$ in $1,\ldots,m$ there is a canonical isomorphism of principally polarized abelian varieties

[TABLE]

Proof.

Observe that for each $i$ in $1,\ldots,m$, $G/U_{i}\cong A_{4}$ acts on the corresponding curve $X_{U_{i}}$ (of genus $4t-11)$) and the involutions in $A_{4}$ act without fixed points (since $X\to X_{N}$ is étale). It then follows from the trigonal construction (see [7]) that $P(X_{L_{i}}\rightarrow X_{N})$ is isomorphic to $JM_{i}$ as principally polarized abelian varieties. ∎

5. The isogeny decomposition of the Jacobian of $X_{B}$

Let the notations be as above. For $i=1,\dots,m$ denote by $\mu_{i}:X_{B}\rightarrow X_{M_{i}}$ the covering given by $B\leq M_{i}$. The pull back homomorphisms

[TABLE]

are isogenies onto their images. Considering the composition of these isogenies with the addition map we get a canonical homomorphism

[TABLE]

The main result of this section is the following theorem.

Theorem 5.1**.**

The homomorphism $a:\prod_{i=1}^{m}JX_{M_{i}}\rightarrow JX_{B}$ is an isogeny.

Proof.

Recall that for any subgroup $H$ of $G$, $\rho_{H}$ denotes the representation of $G$ induced by the trivial representation of $H$. Moreover, $\chi_{0}$ denotes the trivial representation and $\theta_{i},\;i=1,\dots,m$ the irreducible rational represetations of degree 3 of $G$. It is easy to see that

[TABLE]

Let $f_{B,\widetilde{\chi}_{0}}$ and $f_{B,\widetilde{\theta}_{i}}$ denote the corresponding central idempotents of the Hecke algebra $\mathcal{H}_{B}$. Then according to (3.4), the idempotent $p_{B}$ decomposes as

[TABLE]

Considering the idempotents as elements of $\operatorname{{End}}_{\mathbb{Q}}(JX_{B})$,

[TABLE]

since $g(X_{G})=0$ and

[TABLE]

since $\theta_{i}=\rho_{M_{i}}-\chi_{0}$ according to Remark 2.8. Hence according to (3.5), the addition map gives an isogeny

[TABLE]

Combining this with the isogenies $\mu_{i}^{*}$ we get the isogeny $a:\prod_{i=1}^{m}JX_{M_{i}}\rightarrow JX_{B}$ as claimed. ∎

Combing several results, we can say,

Corollary 5.2**.**

For any positive integer $n\geq 2$ consider the integer $m=\frac{1}{3}(2^{2n}-1)$. Then there exist curves of genus $m(t-3)$ for any integer $t\geq 2n$ whose Jacobian is isogenous to the product of $m$ Jacobians.

Proof.

According to Lemma 4.3 there exist curves $X$ of genus $2^{2n}(t-3)+1$ with an action of the group $G=N\rtimes B$ of Section 2 and with $g(X_{B})=m(t-3)$. So Theorem 5.1 gives the assertion. ∎

Corollary 5.3**.**

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

Proof.

Choose a positive integer $n$ such that $m=\frac{1}{3}(2^{2n}-1)\geq N$. This is equivalent to $2^{2n}>3N$. According to the previous corollary there exist curves whose Jacobian is isogenous to the product of $m\geq N$ Jacobians. ∎

Remark 5.4**.**

The isogeny $+:\prod_{i=1}^{m}JX_{B,\widetilde{\theta_{i}}}\longrightarrow JX_{B}$ is the isotypical decomposition of $JX_{B}$ with respect to the Hecke algebra action of $\mathcal{H}_{B}$ on $JX_{B}$.

In order to describe the isotypical components $JX_{B,\widetilde{\theta_{k}}}=\mu_{k}^{*}(JX_{M_{k}})$ of $JX_{B}$ by equations we will apply Theorem 3.1. There are $m$ conjugacy classes of involutions in $G$, and $m+1$ double cosets in $B\backslash G/B$,

[TABLE]

with $H_{0}=B$. All other $H_{i}$ have representatives $j_{i}$ in exactly one of the conjugacy classes of involutions. So

[TABLE]

where $j_{i}$ is an involution of $G$. According to Section 3, a basis of $\mathbb{Q}[B\backslash G/B]$ given by

[TABLE]

Lemma 5.5**.**

1. (i)

$|H_{i}|=9$* for $i=1,\dots,m$;* 2. (ii)

for all $k=1,\dots,m$ we have for all $h_{i}\in H_{i}$ and $i=1,\dots,m$,

[TABLE]

Proof.

(i): Clearly $|H_{i}|\leq 9$. Hence $3+9m=2^{2n}3=|G|=\sum_{i=0}^{m}|H_{i}|\leq 3+9m$ which implies the assertion.

(ii): The first equation is obvious. For the second observe that $\chi_{\theta_{k}}(g)=0$ for all elements $g$ of order three in $G$, and that each $H_{i}$ is composed by the three elements in the conjugacy class of $j_{i}$ and six elements of order three. This implies the assertion. ∎

The following theorem gives the description of the isotypical components of $JX_{B}$.

Theorem 5.6**.**

[TABLE]

Proof.

According to (5.1) we have $\dim\theta_{k}^{B}=1$. Since moreover $\theta_{k}$ is defined over $\mathbb{Q}$, we can apply Theorem 3.1 to give

[TABLE]

where the last equation follows from Lemma 5.5(ii). ∎

6. The degree of the isogeny $a$

In Section 5 we used the right hand side of the diagram (4.2) to decompose the Jacobian of the curve $X_{B}$. By Proposition 4.7 we know that for all $i$ there is a canonical isomorphism of principally polarized abelian varieties $P(X_{L_{i}}\rightarrow X_{N})\rightarrow JX_{M_{i}}$. In this section we will see that one can also use the left hand side of the diagram to decompose the Jacobian $JX_{B}$, here as a product of Prym varieties. In view of Proposition 4.7, this is the same decomposition as above, however has the advantage that it gives in addition something about the degree of the isogeny.

Let the notation be as in the last section. For $n=1,\dots,m$ we denote

[TABLE]

the maps of diagram (4.2). The maps $\nu_{i}^{*}:JX_{L_{i}}\rightarrow JX$ and $\operatorname{{Nm}}\mu:JX\rightarrow JX_{B}$ are the induced homomorphisms of the associated Jacobians. Then the addition map gives a homomorphism

[TABLE]

According to Corollary 4.6, $\prod_{i=1}^{m}P(X_{L_{i}}\rightarrow X_{N})$ and $JX_{B}$ are of the same dimension. To be more precise, we have

Theorem 6.1**.**

$\alpha:\prod_{i=1}^{m}P(X_{L_{i}}\rightarrow X_{N})\longrightarrow JX_{B}$* is an isogeny with kernel contained the the $2^{2n-1}$-division points.*

Applying the isomorphism of Proposition 4.7, the homomorphism $\alpha$ coincides with the isogeny $a$ of Theorem 5.1. The new result is the estimate of its degree, namely

[TABLE]

which follows from the fact that the group of 2-division points of $JX_{B}$ is of order $2^{2g(X_{B})}$. The proof is analogous to the proof of [4, Theorem 3.1] (also see [3, Proposition 3.2]). For the convenience of the reader we give full details. We need the following proposition.

Proposition 6.2**.**

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

[TABLE]

Proof.

This is a special case of [4, Corollary 3.3] ∎

Denote for $i=1,\dots,m$,

[TABLE]

and let

[TABLE]

Recall that $\sigma$ generates the group $B$. Then 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 $\widetilde{A}_{i}:=\mu^{*}(C_{i})$.

Proposition 6.3**.**

For $i=1,\dots,m$ the map

[TABLE]

is multiplication by $2^{2n-1}$.

For the proof of the proposition we need the following lemma.

Lemma 6.4**.**

For any two subgroups $H_{1}\neq H_{2}$ of $N$ of index $2$ we have

[TABLE]

Proof.

Since $N=H_{1}H_{2}$, we have $N/H_{1}\simeq H_{2}/(H_{1}\cap H_{2}),$ which implies the assertion. ∎

Proof of Proposition 6.3.

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

[TABLE]

is multiplication by $2^{2n-1}$.

Now from Proposition 6.2 we deduce

[TABLE]

For the proof of (6.3) we apply Proposition 6.2 with $H=L_{i}$. Let $g_{1}$ be any element of $L_{i}$ and $g_{2}$ be any element of $N\setminus L_{i}$; then $\{g_{1},g_{2}\}$ is a complete set of representatives of $N/L_{i}$. So Proposition 6.2 gives

[TABLE]

But $z\in JX^{L_{i}}$ implies that $g_{1}(z)=z$. This gives

[TABLE]

But $X_{L_{i}}/X_{N}$ is a double cover, which implies $g_{2}(z)=-z$. Since $g_{1}$ and $g_{2}$ are arbitrary elements of $L_{i}$ and $N\setminus L_{i}$ respectively, this gives the assertion.

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

[TABLE]

By equation (6.3) and Lemma 6.4 we have

[TABLE]

and for $k=1$ and 2,

[TABLE]

since $L_{i}\neq\sigma^{-k}L_{i}\sigma^{k}$ and by Lemma 6.4 half of the elements of the subgroup $\sigma^{-k}L_{i}\sigma^{k}$ belong to $L_{i}$, hence fix $z$, and the other half belongs to $N\setminus L_{i}$ and hence sends $z$ to $-z$. Together they complete the proof of the proposition. ∎

Proof of Theorem 6.1.

Since

[TABLE]

Proposition 6.3 implies that $\beta\circ\mu^{*}\circ\alpha$ is multiplication by $2^{2n-1}$. In particular $\alpha$ has finite kernel. But according to Corollary 4.6, $\prod_{i=1}^{m}P(X_{L_{i}}\rightarrow X_{N})$ and $JX_{B}$ have the same dimension. So $\alpha$ is an isogeny. ∎

Remark 6.5**.**

As we noticed already above, the isogenies $a$ and $\alpha$ are compatible with respect to the isomorphism of Proposition 4.7. Hence Theorem 6.1 is also the isotypical decomposition of $JX_{B}$.

7. The group algebra decomposition of $JX$

Let $G=N\rtimes B$ as act on the curve $X$ as above. For the trivial subgroup $H=\{1\}$ the Hecke algebra $\mathcal{H}_{\{1\}}$ coincides with the whole group algebra $\mathbb{Q}[G]$ and we have $JX_{\{1\}}=JX$ and $\widetilde{W}_{i}=W_{i}$ for all $i$. According to (3.5) and [4, equation (2.3)] the isotypical decomposition of $JX$ is

[TABLE]

where we write for simplicity $JX_{\{1\},W}=JX_{W}$ for each irreducible rational representation $W$ of $G$, for which we refer to Corollary 2.4. Note that $JX_{\chi_{0}}=0$, since $X_{G}=\mathbb{P}^{1}$.

This decomposition can be decomposed further. In fact, as outlined in [4, Section 2], for each $\theta_{i}$ there is a not uniquely determined abelian subvariety $B_{i}$ such that $JX_{\theta_{i}}\sim B_{i}^{3}$. On the other hand, at least by these methods $JX_{\psi}$ cannot be decomposed further. Hence we get a decomposition

[TABLE]

which is called the group algebra decomposition of $JX$ (see [1, Section 13.6]).

Lemma 7.1**.**

1. (i)

$JX_{\psi}=JX_{N}$; 2. (ii)

$B_{i}\sim P(X_{L_{i}}\rightarrow X_{N})$* for $i=1,\dots,m$.*

Corollary 7.2**.**

A group algebra decomposition of $JX$ is

[TABLE]

Proof of the Lemma 7.1.

(i): The covering $X_{N}\rightarrow\mathbb{P}^{1}$ is Galois with group of order 3. There are 2 irreducible rational representation of this group, namely $\psi$ and the trivial representation $\chi_{0}$. To $\psi$ corresponds the Prym variety $P(X_{N}\rightarrow\mathbb{P}^{1})$ and to the trivial representation $J\mathbb{P}^{1}=0$. This implies $JX_{\psi}\sim JX_{N}$ and thus $JX_{\psi}=JX_{N}$, since the isotypical decomposition is uniquely determined.

(ii): According to Theorem 6.1, $\alpha(P(X_{L_{i}}\rightarrow X_{N}))$ contained in $JX_{B}$ and thus in $JX$, since $X\rightarrow X_{B}$ is ramified. So clearly it is contained in $JX_{\theta_{i}}$. Since one can conclude from Lemma 4.5 that $\dim JX_{\theta_{i}}=\dim B_{i}$, this implies that there is an isogeny $JX_{\theta_{i}}\sim B_{i}$.

The last isogeny is a consequence of Proposition 4.7. ∎