Uniform boundedness of \(p\)-primary torsion of abelian schemes (Q421017)

The main object of the paper under review is an abelian variety \(A\) defined over a field \(k\) which is finitely generated over \(\mathbb Q\). Given a prime \(p\) and a character \(\chi: \Gamma_k\to \mathbb Z^*_p\) of the absolute Galois group of \(k\), which is assumed to be non-Tate (i.e., does not appear as a subrepresentation of the \(p\)-adic representation associated with \(A\)), the authors prove that the order of the group \(A[p^{\infty }](\chi )\) of \(p\)-primary torsion of \(A\) is uniformly bounded provided \(A\) varies within a one-dimensional family. This result is deduced from a geometric result of a similar spirit for abelian varieties over function fields of curves, combined with the Mordell conjecture (Faltings' theorem). As an application, the authors establish the one-dimensional case of the modular tower conjecture.