On Isogenies of CM elliptic curves

In this talk I will report on joint work with Y. Li, T. Wang and J. Xia. Our main result gives lower bounds for the number of primes rational p such that the reductions of two fixed CM elliptic curves modulo a prime above p are m-isogenous.
These bounds are polynomial and understood asymptotically as m tends to infinity. After giving a complete statement of the result I plan to briefly summarise the proof. If time permits, I will highlight some useful results on classical modular forms that were crucial for our argument.