Fields of definition of abelian subvarieties (Q2086421)

Let \(A\) be an abelian variety over a field \(K\) of characteristic zero. In this paper, the author shows that, if no isotypic component of \(A_{\overline{K}}\) is simple, then there are infinitely many abelian subvarieties of \(A_{\overline{K}}\) whose field of definition is the one of the geometric endomorphisms of \(A\). The idea of the proof is to relate the set of abelian subvarieties to the rational points of a Grassmannian \(Gr(D)\) over a skew field \(D\). The Galois action of \(\mathrm{Gal}(\overline{K}/K)\) on subvarieties factors through the finite group \(G=\mathrm{Gal}(K_A/K)\), where \(K_A\) denotes the field of definition of the endomorphisms of \(A_{\overline{K}}\). The fixed points of a nontrivial element of \(G\) are given by the rational points of a proper closed subvariety of \(Gr(D)\). Since the finite union of these subvarieties cannot cover the whole \(Gr(D)\), one finds infinitely many subvarieties by density of the rational points of \(Gr(D)\).