Given a finitely generated field extension \(K\) of the rational numbers and an abelian variety \(C\) over \(K\), the authors consider the class of all abelian varieties over \(K\) which are isogenous (over \(K\)) to an abelian subvariety of a power of \(C\).

The authors show that there is a single, naturally constructed abelian variety \(C^{\flat}\) in the class whose ring of endomorphisms controls all isogenies in the class. Precisely, this means that if \(d\) is the discriminant of this ring then for any pair of isogenous abelian varieties in the class there exists an isogeny between them whose kernel has exponent at most \(d\).

Furthermore, the authors prove for any element \(A\) in the class, the same number \(d\) governs several invariants attached to \(A\) such as the smallest degree of a polarisation on \(A\), the discriminant of its ring of endomorphisms or the size of the invariant part of its geometric Brauer group. All these are bounded only in terms of \(d\) and the dimension of \(A\).

In the case where \(K\) is a number field the authors go further and show that the period theorem applies to \(C^{\flat}\) in a natural way and gives an explicit bound for \(d\) in terms of the degree of \(K\), the dimension of \(C^{\flat}\) and the stable Faltings height of \(C\). This, in turn, yields explicit upper bounds for all the previous quantities related to isogenies, polarisations, endomorphisms, and Brauer groups which significantly improve known results.