A local-global principle for isogenies of elliptic curves

Let E be an elliptic curve defined over a number field.  In this talk, we will address the following question: if the reduction of E modulo almost all primes has a rational cyclic isogeny of degree N, is E forced to as well?  Building on the work of Sutherland, Anni, and Banwait--Cremona when N is prime, we will focus on the case that N is composite.