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Abelian varieties with group action. (English) Zbl 1072.14053

The aim of the present paper is to prove some general results on abelian varieties with action by a finite group, and to deduce from those general facts a method to decompose the Jacobian $J\left(X\right)$ of a curve $X$ in terms of a finite group action on that curve.

Let $G$ be a finite group acting on an abelian variety $A$. Let ${W}_{1},\cdots ,{W}_{r}$ denote the irreducible $ℚ$-representations of $G$ and ${n}_{i}:={dim}_{{D}_{i}}\left({W}_{i}\right)$, where ${D}_{i}:={\text{End}}_{G}\left({W}_{i}\right)$ for $1\le i\le r$. Then the authors’ first main theorem states that there are abelian subvarieties ${B}_{1},\cdots ,{B}_{r}$ of $A$ and an isogeny $A\sim {B}_{1}^{{n}_{1}}×\cdots ×{B}_{r}^{{n}_{r}}$. This refined isotypical decomposition is then applied to Jacobians of smooth projective curves equipped with the action of a finite automorphism group. If $G$ is a finite group acting on such a curve of genus $g\ge 2$ , then the quotient map $\pi :X\to Y:=X/G$ induces, via the general main theorem, an isogeny decomposition

$J\left(X\right)\sim {\pi }^{*}J\left(Y\right)×{B}_{2}^{{n}_{2}}×\cdots ×{B}_{r}^{{n}_{r}}$

of $J\left(X\right)$ into certain abelian varieties, the components of which can be linked to certain suitable Prym varieties arising from the $G$-action. Finally, the authors work out several concrete examples of particular group actions on curves, including the symmetric group of degree 4, the alternating group of degree 5, the dihedral groups of order $2p$ and $4p$ and the quaternion group, which illustrate the general results by explicit computations.

##### MSC:
 14K02 Isogeny 14H40 Jacobians, Prym varieties 14L30 Group actions on varieties or schemes (quotients)