An automorphism of an abelian variety induces a decomposition of the variety up to isogeny. There are two such results, namely the isotypical decomposition and Roan’s decomposition theorem. We show that they are essentially the same. Moreover, we generalize in a sense this result to abelian varieties with action of an arbitrary finite abelian group. An early version of this article was inadvertently published before all the revisions had been completed and then retracted [https://doi.org/10.1007/s00013-018-1244-3]. This article is the final peer reviewed version.
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Birkenhake, Ch., Lange, H.: Complex Abelian Varieties. Grundlehren der Mathematischen Wissenschaften, vol. 302, 2nd edn. Springer, New York (2004)
We would like to thank Gabriele Nebe for pointing out a mistake in the first version of the paper.
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The authors were partially supported by Grants Fondecyt 1190991, CONICYT PAI Atraccion de Capital Humano Avanzado del Extranjero PAI80160004 and Anillo ACT 1415 PIA-CONICYT.
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Carocca, A., Lange, H. & Rodríguez, R.E. Abelian varieties with finite abelian group action. Arch. Math. 112, 615–622 (2019). https://doi.org/10.1007/s00013-018-1291-9
- Abelian variety
- Finite groups
Mathematics Subject Classification