Jacobian varieties with group algebra decomposition not affordable by Prym varieties

For the whole of this section, let $G$ denote a finite group. Let $\varsigma\colon G\to\operatorname{GL}(V)$ be a complex irreducible representation of $G$, $L$ its field of definition and $K$ its character field, so we have $K\subseteq L$. The integer $\operatorname{s}(\varsigma)\coloneq[L:K]$ is called the Schur index of the representation (see [book:serre77]*section 12.2). According to [book:lange22]*equation (2.19), there is a unique (up to isomorphism) rational irreducible representation $\rho\colon G\to\operatorname{GL}(W)$ such that

where $\chi_{\varphi}$ denotes the character of a representation $\varphi$. For simplicity we will not distinguish between $\rho$ and $\rho\otimes\mathbb{C}$.

Set $\operatorname{Irr}_{\mathbb{Q}}(G)\coloneq\{\rho_{1},\ldots,\rho_{r}\}$, the set of irreducible rational representations of $G$ up to isomorphism. To each representation $\rho_{i}$ corresponds a unique central idempotent of the group-algebra $\mathbb{Q}[G]$ denoted by $e_{i}$ such that $\mathbb{Q}[G]e_{i}$ affords $\rho_{i}$ and $e_{1}+\cdots+e_{r}=1$. These idempotents induces a unique decomposition of $\mathbb{Q}[G]$ into simple sub-algebras

called isotypical decomposition of $\mathbb{Q}[G]$ (see [book:lange22, equation 2.27]). Moreover, each simple sub-algebra $\mathbb{Q}[G]e_{i}$ can be (non-uniquely) decomposed into $n_{i}\coloneq\deg(\varsigma_{i})/\operatorname{s}(\varsigma_{i})$ indecomposable left $\mathbb{Q}[G]$-modules, where $\varsigma_{i}$ is a complex irreducible representation Galois associated to $\rho_{i}$. More precisely, there are primitive orthogonal idempotents $q_{i,1},\ldots,q_{i,n_{i}}$ such that $e_{i}=q_{i,1}+\cdots+q_{i,n_{i}}$ (see [book:lange22, equation 2.31]). This idempotents induce the group algebra decomposition:

We now introduce a particular kind of representations that will be important in section 2.4. For a subgroup $H\subset G$ we set $\rho_{H}\coloneq\operatorname{Ind}_{H}^{G}(1_{H})$, where $1_{H}$ denotes the trivial representation of $H$. Since $1_{H}$ is a rational representation, $\rho_{H}$ is also rational.

Consider an abelian variety $A$ with a $G$-action. The action of $G$ induces an algebra homomorphism $\mathbb{Q}[G]\to\operatorname{End}_{\mathbb{Q}}(A)\coloneqq\operatorname{End}(A)\otimes_{\mathbb{Z}}\mathbb{Q}$; even if it is not a monomorphism, we do not distinguish between elements of $\mathbb{Q}[G]$ and their images in $\operatorname{End}_{\mathbb{Q}}(A)$. Each element $\gamma\in\mathbb{Q}[G]$ defines an abelian subvariety $\operatorname{Im}(\gamma)\coloneqq\operatorname{Im}(a\gamma)$, where $a$ is any integer such that $a\gamma\in\operatorname{End}(X)$. This definition does not depend on the choice of $a$. The isotypical decomposition of $\mathbb{Q}[G]$ of equation (5) induces a decomposition of $A$ as stated in the following result.

This isogeny is the isotypical decomposition of $A$. Moreover, the decomposition in equation (6) yields the following result.

Let $X$ be a compact Riemann surface with a $G$-action; that is, there is a monomorphism from $G$ to the automorphism group $\operatorname{Aut}(X)$ of $X$. This action defines a Galois covering $\pi_{G}\colon X\to Y$, where $Y\coloneq X/G$ (see [book:lange22]*section 3.1). For a pair of subgroups $H\subset N$ of $G$, the induced maps $\pi^{H}_{N}\colon X/H\to X/N$ are called intermediate coverings of $\pi_{G}$. The genus and ramification data of the intermediate coverings are determined by the geometric signature

of $\pi_{G}$; that is, the genus $g$ of $Y$ and the stabilizer subgroups $G_{j}$ of $G$ with respect to branch points in each different orbit of branch points of $\pi_{G}$; the subgroups $G_{j}$ are determined up to conjugacy and reenumeration. The signature of the $G$-action on $X$ is $(g;n_{1},\ldots,n_{s})$, where $n_{i}\coloneq\lvert G_{i}\rvert$ for $i=1,\ldots,s$.

With respect to the existence of a Galois covering with a given geometric signature, we have the following result, which is a direct consequence of [book:lange22]*Theorem 3.1.4

For a compact Riemann surface $X$, we denote its Jacobian variety by $\operatorname{J}(X)$; it is a principally polarized abelian variety. Given a (ramified) covering map $f\colon X\to Y$ between compact Riemann surfaces, the pullback $f^{*}\colon\operatorname{J}(Y)\to\operatorname{J}(X)$ is an isogeny onto its image $f^{*}\operatorname{J}(Y)$.

Now consider a Galois covering $\pi_{G}\colon X\to Y$ as in the previous subsection and consider two subgroups $H$ and $N$ of $G$ such that $H\subset N$. The following result gives a decomposition of $\operatorname{J}(X/H)$ and $\operatorname{P}(\pi^{H}_{N})$ into a product of group algebra components of $\operatorname{J}(X)$ up to isogeny, we call it the group algebra decomposition of $\operatorname{J}(X/H)$ and $\operatorname{P}(\pi^{H}_{N})$, respectively.

Since $B_{1}\sim\operatorname{J}(X/G)$ , we are concerned into describing the group algebra components $B_{i}$ for $i=2,\ldots,r$ as Prym varieties. The following corollary gives a characterization of components that are isogenous to Prym varieties of intermediate coverings (see [book:lange22]*Corollary 3.5.10).

For some values of $i$ we may have $\dim B_{i}=0$, and hence $B_{i}\sim\operatorname{P}(\pi_{N}^{H})$ for any intermediate covering with trivial Prym variety (in particular for $H=N$). In contrast, item 1 of Proposition 1 implies that there is no equivariant isogeny between non-trivial components $B_{i}$ and $B_{j}$ if $i\neq j$. Thereby, if in the group algebra decomposition of $\operatorname{P}(\pi^{H}_{N})$ there are at least two different non-trivial group algebra components $B_{i}$ and $B_{j}$ with $t_{i},t_{j}\geq 1$ or one non-trivial $B_{i}$ with $t_{i}\geq 2$, then $\operatorname{P}(\pi^{H}_{N})$ is not isogenous to any $B_{i}$. This shows the importance of knowing the dimension of each $B_{i}$; it depends only on the geometric signature of $\pi_{G}$, in equation (8), and may be computed as follows.

Since $T$ is abelian and $G=T\rtimes\Lambda$, we will use the Wigner–Mackey method of little groups (see [book:serre77]*section 8.2) in order to determine the complex irreducible representations of $G$, which, according to the following proposition, should be $q$ up to isomorphism.

Let $\zeta_{k}\in\mathbb{C}^{\times}$ denote the primitive $k$-root of unity $\mathrm{e}^{2\piup\mathrm{i}/k}$ for each positive integer $k$. Since $T$ is naturally isomorphic to the additive group $\mathbb{F}_{q}$, according to [book:lidl97]*Theorem 5.7, the irreducible characters of $T$ are

Now we describe the rational irreducible representations of $G$ based on the results of sections 2.1 and 3.1, we keep the notation of the section 3.1. Let $\operatorname{\sigma_{0}}(n)$ denote the number of positive divisors of an integer $n$.

For convenience, let $d_{1},\ldots,d_{r}$ denote the positive divisors of $q-1$ in decreasing order. With this notation $\xi_{d_{1}}=\xi_{q-1}$ is equivalent to $1_{G}$.

In order to decompose the Prym varieties of the intermediate coverings of a Galois covering $\pi_{G}\colon X\to Y$ through Proposition 4, we give a characterization of the subgroups of $G$ up to conjugacy. Recall that, for each divisor $m$ of $n$, the field $\mathbb{F}_{q}$ is an extension of $\mathbb{F}_{p^{m}}$ of degree $n/m$ (see [book:lidl97]*Theorem 2.6). In particular, since $T\cong\mathbb{F}_{q}$, it is a $(n/m)$-dimensional vector space over any field $\mathbb{F}_{p^{m}}$ with $m\mid n$, where scalar multiplication is given by

For cleaner notation, let $\Lambda_{d}$ denote the unique subgroup of $\Lambda$ of order $d$, namely $\langle\lambda_{(q-1)/d}\rangle$.

It is possible for two subgroups of the form $H=V\rtimes\Lambda_{d}$ and $N=W\rtimes\Lambda_{d}$ with $V\neq W$ to be conjugates, a necessary condition is that $\dim_{\mathbb{F}_{p^{m}}}(V)=\dim_{\mathbb{F}_{p^{m}}}(W)$, but it is not a sufficient condition in general (a standard counting argument shows that there are non-conjugate linear subspaces of the same dimension $k$ if $2\leq k\leq n/m-2$). Nevertheless, according to Remark 1, the decomposition of the representations $\rho_{H}$ and $\rho_{N}$ only depends on the cardinality of the $G$-conjugacy classes of $H$ and $N$, respectively; therefore, Proposition 7 implies that $\rho_{N}\cong\rho_{H}$ if and only if $\dim_{\mathbb{F}_{p^{m}}}(V)=\dim_{\mathbb{F}_{p^{m}}}(W)$. This motivates the following definition.

Note that for a fixed divisor $d\mid q-1$, there are subgroups of $G$ of type $(d,k)$ if and only if $k$ is a multiple of $m$, as stated in Theorem 12. This assumption will be implicit each time we discuss about a subgroup of a given type; we could have defined $k$ as $\dim_{\mathbb{F}_{p^{m}}}(V)$ instead of $\dim_{\mathbb{F}_{p}}(V)$ and then we would have $k\in\{1,\ldots,n/m\}$, but setting $k\coloneq\dim_{\mathbb{F}_{p}}(V)$ keeps a cleaner and more intuitive notation.