Wojciech J. Gajda

Abstract:

Let E be an elliptic curve (or an abelian variety) defined over a number field K. In this talk, we will study the problem of understanding if the property of admitting an isogeny of a fixed degree satisfies the local-global principle. This means that we will study the following. Assume that, for all but finitely many primes p in K, the elliptic curve E reduced modulo p is isogenous to an elliptic curve via an isogeny of fixed degree N. Is it true that E is isogenous to an elliptic curve via an isogeny of degree N ? We will present the known results on the topic and we present new results in higher dimension, i.e. for abelian surfaces. This is joint work with Prof. Davide Lombardo (Università di Pisa).