Abelian subvarieties
It is shown that if \(X\) is an abelian variety defined over a field \(K\), the number of orbits of the set of abelian subvarieties of \(X\) under the natural action of the group of automorphisms of \(X\) is finite. It follows that there are, up to isomorphism, only finitely many abelian subvarieties of \(X\). The argument is brief, elegant, and elementary: One proves that given a finite-dimensional semi-simple \(\mathbb{Q}\)-algebra \(F\), a finitely generated right \(F\)-module \(M\) and a lattice \(L\) in \(M\), the number of orbits of the set of \(F\)-submodules of \(M\) under the action of the group \(G\) of automorphisms \(\sigma\) of \(M\) such that \(\sigma L=L\), is finite. This is applied to \(M=F= \text{End} X \otimes\mathbb{Q}\), \(L= \text{End} X\) and \(G= \Aut X\).
finiteness of number of abelian subvarieties of abelian variety
scientific article; zbMATH DE number 869790