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Zbl 1072.14053
Lange, Herbert; Recillas Pishmish, Sevin
Abelian varieties with group action.
(English)
[J] J. Reine Angew. Math. 575, 135-155 (2004). ISSN 0075-4102; ISSN 1435-5345/e

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 $J(X)$ 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 $W_1,\dots, W_r$ denote the irreducible $\bbfQ$-representations of $G$ and $n_i:= \dim_{D_i}(W_i)$, where $D_i:= \text{End}_G(W_i)$ for $1\le i\le r$. Then the authors' first main theorem states that there are abelian subvarieties $B_1,\dots, B_r$ of $A$ and an isogeny $A\sim B^{n_1}_1\times\cdots\times B^{n_r}_r$. 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 $$J(X)\sim \pi^*J(Y)\times B^{n_2}_2\times\cdots\times B^{n_r}_r$$ of $J(X)$ 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.
[Werner Kleinert (Berlin)]
MSC 2000:
*14K02 Isogeny
14H40 Jacobians
14L30 Group actions on varieties or schemes

Keywords: algebraic curves; Jacobians; Prym varieties; isogeny of abelian varieties; decomposition theorems

Cited in: Zbl 1109.14028 Zbl 1070.14032

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