Poincaré's reducibility theorem with \(G\)-action
Let \(G\) be a finite group acting on an abelian variety \(A\) defined over an algebraically closed field \(k\). The authors prove the following analogue of Poincaré's reducibility theorem [\textit{H. Poincaré}, Am. J. Math. 8, 289--342 (1886; JFM 18.0421.02); Œuvres IV, 318--378 (1950; Zbl 0041.37402)]: there exist \(G\)-simple abelian subvarieties \(B_1,\dots,B_r\) of \(A\) such that the addition map induces a \(G\)-equivariant isogeny \(\mu:B_1\times\dots\times B_r\to A\). Moreover, this decomposition is unique, in the sense that if there is another set \(B_1',\dots,B_s'\) of \(G\)-simple abelian subvarieties of \(A\) such that the addition map induces a \(G\)-equivariant isogeny \(\mu':B_1'\times\dots\times B_s'\to A\), then \(r=s\) and there exists a permutation \(\sigma\) of degree \(r\) such that for \(i=1,\dots,r\) there is a \(G\)-isogeny \(B_i\to B_{\sigma(i)}'\).
scientific article; zbMATH DE number 2114296