A local-global principle for isogenies of abelian surfaces

In this talk, we will study the problem of understanding if the property of admitting an isogeny of fixed degree satisfies the local-global principle. Specifically, we seek to address the following question. Let $A$ be an abelian variety and $K$ be a number field. Assume that, for all but finitely many primes $p$ in $K$, the abelian variety $A$ reduced modulo $p$ is isogenous to an abelian variety via an isogeny of fixed degree $N$. Is it true that $A$ is isogenous to an abelian variety via an isogeny of degree $N$? We will present the known results on the topic and some new results. This is joint work with Prof. Davide Lombardo (Università di Pisa).